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Sec 2.1 Geometry โ Parallel Lines and Angles Name: PARALLEL LINES
1. Give an alternate name for angle โก๐ using 3 points:
2. Angles โก๐จ๐ฉ๐ฌ and โก๐ช๐ฉ๐ฎ can best be described as:
3. Angles โก๐ and โก๐ can best be described as:
4. The line ๐ฎ๐ฏ โก can best be described as a:
5. Which angle corresponds to โก๐ซ๐ฌ๐ฉ :
6. Angles โก๐ญ๐ฌ๐ฉ and โก๐ช๐ฉ๐ฌ can best be described as:
7. Angles โก๐ and โก๐ can best be described as:
8. Which angle is an alternate interior angle with โก๐ช๐ฉ๐ฌ :
9. Angles โก๐ฎ๐ฉ๐ช and โก๐ฉ๐ฌ๐ญ can best be described as:
10. Angles โก๐ and โก๐ can best be described as:
11. Which angle is an alternate exterior angle with โก๐จ๐ฉ๐ฎ :
12. Which angle is a vertical angle to โก๐จ๐ฉ๐ฎ :
13. Which angle can be described as consecutive exterior angle with โก๐ :
14. Any two angles that sum to 180 ฬcan be described as angles.
M. Winking Unit 2-1 page 19
โ C
โ A
โ F
โ D
These symbols imply the two
lines are parallel.
โ E
โ B
โ H
โ G
INTERIOR
EXTERIOR
EXTERIOR
1 2
3 4
5 6
7 8
WORD BANK:
Alternate Interior Angles
Alternate Exterior Angles Corresponding Angles
Vertical Angles
Consecutive Interior Angles โก๐ฎ๐ฉ๐ช
Consecutive Exterior Angles
โก๐ซ๐ฌ๐ฏ โก๐ Transversal
โก๐
โก๐จ๐ฉ๐ฎ
โก๐ฏ๐ฌ๐ญ Congruent
Supplementary
TRIANGLEโs INTERIOR ANGLE SUM
1. a. First, Create a random triangle on a piece of patty papers.
b. Using your pencil, write a number inside each interior angle a
label.
c. Next, cut out the triangle.
d. Finally, tear off or cut each of the angles from the triangle
e. Using tape, carefully put all 3 angles next to one another so that
they all have the same vertex and the edges are touching but they arenโt overlapping
2. What is the measure of a straight angle or the angle that creates a line by using two
opposite rays from a common vertex?
3. Collectively does the sum of your 3 interior angles of a triangle form a straight angle?
What about others in your class?
4. Make a conjecture about the sum of the interior angles of a triangle. Do you think your conjecture will always be true? (please explain using complete sentences)
1
2
3
1
2
3
โข Paste or Tape your 3 vertices here:
โข Common Vertex
M. Winking Unit 2-1 page 20
5. More formally, why do the 3 interior angles of any triangle sum to 180 ฬ ?
Consider โABC. The segment ๐จ๐ฉฬ ฬ ฬ ฬ is extended into a line and a parallel line is constructed
through the opposite vertex. So, ๐จ๐ฉ โก โฅ ๐ช๐ซ โก .
a. Why is โก๐ โ โก๐ ?
b. Why is โก๐ โ โก๐ ?
c. Why is ๐โก๐ + ๐โก๐ + ๐โก๐ = ๐๐๐ยฐ ?
d. Using substitution we can replace ๐โก๐ with ๐โก๐ and ๐โก๐ with ๐โก๐ to show that the
interior angles of a triangle must always sum to 180 ฬ.
( ) + ๐โก๐ + ( ) = ๐๐๐ยฐ
Write the angle number in the and then write the letter that corresponds with the number based on the code at the bottom in the box.
7. Angle 2 and Angle are alternate exterior angles.
8. Angle 7 and Angle are alternate exterior angles.
9. Angle 4 and Angle are corresponding angles.
10. Angle 5 and Angle are consecutive interior angles.
11. Angle 3 and Angle are alternate interior angles.
12. Angle 7 and Angle are consecutive exterior angles.
13. Angle 6 and Angle are vertical angles.
14. Angle 2 and Angle are a linear pair and on the same side of the transversal.
15. Angle 1 and Angle are corresponding angles.
1=D 2=U 3= L 4 = A 5 = N 6 = I 7 = E 8 = C
What type of Geometry is this?
1 2
3 4
5 6
7 8
M. Winking Unit 2-1 page 21
16. Given lines p and q are parallel, find the value of x that makes each diagram true.
a. b.
17. Given lines p and q are parallel, find the value of x that makes each diagram true.
a. b.
18. Given lines m and n are parallel, find the value y of that makes each diagram true.
a. b.
40ยบ
m
n
130ยบ
yยบ
(6x+5)ยบ
(2x+15)ยบ
p
q
x = x =
40ยบ
m
n 130ยบ yยบ
y = y =
p
q (6x+5)ยบ
(2x+45)ยบ
M. Winking Unit 2-1 page 22
x = x =
p
q
(x)ยบ
140ยบ
(x)ยบ
155ยบ p
q
19. ANGLE PUZZLE. Find ๐โก๐จ๐ฌ๐ญ
๐โก๐ด๐ธ๐น =
20. Converse of AIA, AEA, CIA, CEA. Which sets of lines are parallel and explain why?
a. b.
c. d.
โ
โ
โ
โ
A
B C
D
E
F
โ G
Given:
๐โก๐ซ๐ฌ๐ญ = ๐๐ยฐ
๐โก๐จ๐ฉ๐ฎ = ๐๐ยฐ โก๐ฉ๐จ๐ฌ is a right angle โก๐ช๐ฎ๐ฌ and โก๐ซ๐ฌ๐ฎ are supplementary
t
r 54ยบ
28ยบ
h
g 120ยบ
p
q 70ยบ
120ยบ
r
n
m 75ยบ
105ยบ
p
q
j
M. Winking Unit 2-1 page 23
Sec 2.2 Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.
**Using a ruler measure the two lengths to make sure they have the same measure.
2. [COPY ANGLE] Construct an angle with ray ๐ฏ๐ฐโโโโ โ and congruent to the angle โก๐ซ๐ฌ๐ญ
3.
4.
**Using a protractor measure the two angles to make sure they have the same measure.
A B
โ C
โ F E
โ D
โ H
โ I
Step I: Open the compass to any length and create an arc on the original angle with the needle at the vertex
Step 2: Leave the compass set to the exact same length and create a new arc on the new ray with the needle at the end point of the ray.
Step 3: Open the compass to the length of the intercepted arc in the original angle and draw a small arc to be sure the created arcs intersect on the ray of the angle.
Step 4: Leave the compass open to the exact same length as it was from step 3. Then with the needle at the intersection of the second arc and the copied ray, create an arc that intersects the first arc.
Step 5: Finally, use a straight edge to draw a ray from the endpoint of the original bottom ray through the intersection of the two created arcs.
M. Winking Unit 2-2 page 24
3. [Perpendicular Bisector] Construct a perpendicular bisector to the segment ๐จ๐ฉฬ ฬ ฬ ฬ .
**Using a ruler measure the two halves of the segment to make sure they have the same measure.
Step I: Open the compass to a length that is certainly larger than half of the segment length. Draw an arc with the needle at the left endpoint of the segment.
Step 2: Leave the compass open to the same length from step 1 and create another arc only this time starting with the needle at the right endpoint.
Step 3: Using a straight edge and a pencil, draw a line that passes through the intersection point of the two arcs.
A B
M. Winking Unit 2-2 page 25
4. [Angle Bisector] Construct an angle bisector of the angle โก๐ซ๐ฌ๐ญ
**Using a ruler measure the two halves of the segment to make sure they have the same measure.
Step I: Open the compass to any length and create an arc with the needle of the compass at the vertex of the angle.
Step 2: Open the compass so that the needle is on one intersection of the arc and the angle and the pencil is at the other intersection of the arc and the angle.
Step 3: Create an arc with the opened length determined in step 2 towards the interior of the angle.
Step 4: Move the compass so that the needle and pencil are on the opposite intersection points that they were on in step 2 and create another arc to the exterior of the angle.
Step 5: Using a straight edge, draw ray from the vertex of the angle through were the two outer arcs intersect.
E
โ D
โ F
M. Winking Unit 2-2 page 26
5. [Perpendicular to a Line Through a Point] Construct a perpendicular line to ๐จ๐ฉโกโ โโ โ through point C.
Step I: Start by placing the needle on the point not on the line. Then, open your compass to a length such that when you draw an arc it will pass through the line twice and draw the arc.
Step 2: Open the compass so that the needle is on one intersection of the arc and the pencil is on the other intersection of the arc. Then, create a arc above and below the line.
Step 3: Opening the compass to the same length as the previous step and switching the needle and pencil to the opposite intersections of arcs that they were on in step 2, create another arc above and below the line.
Step 3: Using a straight edge, draw a line passing through the intersection of the two most recent arc intersections. The line should incidentally also pass through the point not on the line.
โ C
A B โ โ
M. Winking Unit 2-2 page 27
6. [Hexagon inscribed in a Circle] Construct a circle with radius ๐จ๐ฉฬ ฬ ฬ ฬ and an inscribed regular hexagon.
Step I: Start by placing the needle on the point that is to be the center of the circle and the pencil on the other endpoint of the radius. Create the entire circle with the compass.
Step 2: With the compass still open to the exact length of the radius place the needle on the endpoint of the radius that is on the circle and with the pencil mark an intersection on the circle with a new arc.
Step 3: Leaving the compass still open to the same length move the needle to the new intersection point and mark another intersection point on the circle again. Continue repeating this process until you have made it all the way around the circle.
A B
Step 4: The last mark should end right were you started with the needle. Then, connect each consecutive arc intersection with a segment.
M. Winking Unit 2-2 page 28
7. [Triangle inscribed in a Circle] Construct a circle with radius ๐จ๐ฉฬ ฬ ฬ ฬ and an inscribed regular triangle.
Step I: Start by placing the needle on the point that is to be the center of the circle and the pencil on the other endpoint of the radius. Create the entire circle with the compass.
Step 2: With the compass still open to the exact length of the radius place the needle on the endpoint of the radius that is on the circle and with the pencil mark an intersection on the circle with a new arc.
Step 3: Leaving the compass still open to the same length move the needle to the new intersection point and mark another intersection point on the circle again. Continue repeating this process until you have made it all the way around the circle.
A B
Step 4: The last mark should end right were you started with the needle. Then, connect every other consecutive arc intersection with a segment.
M. Winking Unit 2-2 page 29
8. [Square inscribed in a Circle] Construct a circle with radius ๐จ๐ฉฬ ฬ ฬ ฬ and an inscribed square.
Step I: Start by placing the needle on the point that is to be the center of the circle and the pencil on the other endpoint of the radius. Create the entire circle with the compass.
Step 2: Line your straight edge up with the radius and extend the radius segment to create a diameter.
Step 3: Create a perpendicular bisector of the newly created diameter (see previous construction #3 if needed)
A B
Step 4: Connect the each endpoint of the diameter with each endpoint of where the perpendicular bisector intersects the circle.
M. Winking Unit 2-2 page 30
9. [Construct a Parallel Line given a point and a line] Construct a parallel line to ๐จ๐ฉโกโ โโ โ through point C.
Step I: Start drawing a general line that passes through point O and
intersects line ๐๐โกโ โโ โโ .
Step 2: Open the compass so that the needle is on the intersection of the new line just
created and line ๐๐โกโ โโ โโ . Then, open the compass a little (it needs to be less than the distance to reach point O ). Create an arc as shown below.
Step 3: Leave the compass set to the same opening and recreate another arc of the same radius but with the needle at point O.
Step 4: Put the compass needle on the intersection of the transversal line and first arc that you created as shown below and open the pen to the intersection of the first
arc and line ๐๐โกโ โโ โโ . Create a small arc to verify the intersection of the arcs mark the correct intersection.
Step 5: Leave the compass open to the same length as the previous step and put the compass needle on the intersection of the transversal line and the second arc that you created. Then, create an arc of the same radius to intersect the second arc created as shown below.
Step 6: D
A B
โ C
M. Winking Unit 2-2 page 31
Sec 2.3 Geometry โ Dilations Name: [Example Dilation]: Dilate the ABCD by a factor of ๐.๐ from point E.
1. Dilate the โ ABC by a factor of ๐
๐ from point D.
a. Measure the length of
๐จ๐ฉฬ ฬ ฬ ฬ in centimeters to
the nearest tenth.
AB =
b. Measure the length of
๐จโฒ๐ฉโฒฬ ฬ ฬ ฬ ฬ ฬ in centimeters to
the nearest tenth.
AโBโ =
c. Determine the value of
AโBโ divided by AB.
๐จโ๐ฉโ
๐จ๐ฉ =
d. What might you conclude about the scale factor and the ratio of dilated segment
measure to its pre-image?
e. Measure angle โก๐ฉ๐จ๐ช and the angle โก๐ฉโฒ๐จโฒ๐ชโฒ using a protractor.
๐โก๐ฉ๐จ๐ช = ๐โก๐ฉโฒ๐จโฒ๐ชโฒ =
f. What might you conclude about each pair of corresponding angles?
Step I: Measure the distance from the point of dilation to a point to be dilated (preferably using centimeters).
Step 2: Multiply the measured distance by the scale factor.
๐. ๐๐๐ ร ๐. ๐ = ๐ ๐๐
Step I: With the ruler in the same place as it was in step #1, mark a point at the measured distance determined in step #2 as the image of the original point. Repeat the process for all points that are to be dilated. Usually, denoted with the same letter but a single quote after it (referred to as a prime).
โ D
โ A
โ B
โ C
M. Winking Unit 2-3 page 32
2. Consider the following picture in which BCDE has been dilated from point A.
a. What is the scale factor
of the dilation based on
the sides?
b. What is the area of
BCDE?
c. What is the area of
BโCโDโEโ?
d. What is the value of the area of BโCโDโEโ divided by area of BCDE?
e. What might you conclude about the ratio of two dilated shapes sides compared to the
ratio of their areas?
3. Dilate the line ๐จ๐ฉ โก by a factor of ๐. ๐ from point C.
How could you characterize the lines ๐จ๐ฉ โก and ๐จโฒ๐ฉโฒ โก ?
โ C
โ A
โ B
M. Winking Unit 2-3 page 33
4. Dilate the line ๐จ๐ฉ โก by a factor of ๐. ๐ from point C.
How could you characterize the lines ๐จ๐ฉ โก and ๐จโฒ๐ฉโฒ โก ?
5. Consider the following picture in which rectangular prism A has been dilated from point G.
a. What is the scale factor
of the dilation based on
the sides?
b. What is the volume of
rectangular prism A?
c. What is the volume of
rectangular prism Aโ?
d. What is the value of the volume of prism Aโ divided by volume of prism A?
e. What might you conclude about the ratio of two dilated solids sides compared to the ratio
of their volumes?
โ A
โ B
โ C
M. Winking Unit 2-3 page 34
Coordinate Dilations Name: 1. Plot the following points and connect the
consecutive points.
A(5,1)
B(8,2)
C(9,3)
D(9,7)
E(7,8)
F(8,7)
G(7,5)
H(6,4)
I(5,6)
J(6,5)
K(5,7)
L(4,6)
M(4,4)
N(3,5)
O(2,7)
P(3,8)
Q(1,7)
R(1,3)
S(2,2)
A(5,1)
2. First use the dilation rule D:(x,y) โ (0.5x, 0.5y)
using the points from the previous problem, plot the newly created points, and connect the consecutive points.
Aโ( )
Bโ( )
Cโ( )
Dโ( )
Eโ( )
Fโ( )
Gโ( )
Hโ( )
Iโ( )
Jโ( )
Kโ( )
Lโ( )
Mโ( )
Nโ( )
Oโ( )
Pโ( )
Qโ( )
Rโ( )
Sโ( )
Aโ( )
3. What would happen with the rule: D:(x,y) โ (3x, 3y)?
4. What would happen with the rule: D:(x,y) โ (0.5x, 1y)?
5. What would happen with the rule: R:(x,y) โ (โ 1x, 1y)?
6. What would happen with the rule: R:(x,y) โ (1x, โ 1y)?
M. Winking Unit 2-3 page 35
7. Figure FCDE has been dilated to create to create FโCโDโEโ. a. What is the dilation scale
factor?
b. What is the location of the center of dilation?
c. What is the ratio of the areas?
8. Which point would be the center of dilation?
9. The different sizes of soft drink cups at a movie theater are created by using dilations. If the large is 8 inches tall and the medium is 6 inches tall, answer the following.
a. What is the scale factor of the dilation from a large to a medium drink?
b. What is the ratio of the volumes of the two drinks?
c. If the large holds 30 ounces, how much does the medium hold?
โข โข
โข โข A
B
C
D
8 i
n.
6 i
n.
M. Winking Unit 2-3 page 36
โABC ~ โDEF
๐ฌ๐ญ
๐ฉ๐ช=
๐ซ๐ฌ
๐จ๐ฉ=
๐ญ๐ซ
๐ช๐ฉ=
๐
๐= ๐. ๐
๐ญ๐ฌ
๐จ๐ฉ=
๐ฌ๐ซ
๐ฉ๐ช= ๐
โABC ~ โFED
Sec 2.4 Geometry โ Similar Figures Name: Two figures are considered to be SIMILAR if the two figures have the same shape but may differ in size. To be similar by definition, all corresponding sides have the same ratio OR all corresponding angles are congruent. Alternately, if one figure can be considered a transformation (rotating, reflection, translation, or dilation) of the other then they are also similar.
Two triangles are similar if one of the following is true:
1) (AA) Two corresponding pairs of angles are congruent. 2) (SSS) Each pair of corresponding sides has the same ratio. 3) (SAS) Two pairs of corresponding sides have the same ratio and the angle
between the two corresponding pairs the angle is congruent.
Determine whether the following figures are similar. If so, write the similarity ratio and a similarity statement. If not, explain why not. 1. 2. 3.
โABC ~ โFDE
Notice that in the similarity
statement above that
corresponding angles must
match up.
โก๐จ๐ฉ๐ช โ โก๐ญ๐ฌ๐ซ
M. Winking Unit 2-4 page 37
Assuming the following figures are similar use the properties of similar figures to find the unknown.
4. 5. 6.
7. 8.
9. Given the similarity statement โABC ~ โDEF and the following measures, find the
requested measures. It may help to draw a picture.
AB = 8
AC = 10
DE = 20
EF = 30
๐โก๐จ๐ฉ๐ช = ๐๐ยฐ
๐โก๐ฌ๐ญ๐ซ = ๐๐ยฐ
a. Find the measure of DF =
b. Find the measure of BC =
c. Find the measure of ๐โก๐ซ๐ฌ๐ญ =
d. Find the measure of ๐โก๐ฉ๐ช๐จ =
e. Find the measure of ๐โก๐ช๐จ๐ฉ =
f. Which angles are ACUTE?
g. Which angles are OBTUSE?
2x โ 4
3
x + 3 4
x = y = n =
9
t
18
8
t =
3
x
7
x =
18
M. Winking Unit 2-4 page 38
16. Explain why the reason the triangles are similar and find the measure of the requested side. A) B) Triangle mid-segment
17. Using (SSS, AA, SAS) which triangles can you determine must be similar? (explain why)
A) B) C)
D) E)
10
Similar? YES NO
SSS AA SAS
circle one
Similar? YES NO
SSS AA SAS
circle one
Similar? YES NO
SSS AA SAS
circle one
Similar? YES NO
SSS AA SAS
circle one
Similar? YES NO
SSS AA SAS
circle one
What is the measure of TU?
E
M. Winking Unit 2-4 page 39
What is the measure of AU?
c
5 cm 6 cm
x 3 cm
6 cm
18. Using some type of similar figure find the unknown lengths.
A. B. C.
C. D.
22a.
x = 22b.
x =
22c.
y =
22c.
x = 22d.
x =
HINT:
E
M
N
O
P
Q
X
Y
Z
T
V
M. Winking Unit 2-4 page 40
Prove the Pythagorean Theorem (a2 + b2 = c2) using similar right triangles:
M. Winking Unit 2-4 page 41
72 in. 21 in.
62 i
n.
mirror
10. Thales was one of the first to see the power of the property of ratios and similar figures. He realized that he could use this property to measure heights and distances over immeasurable surfaces. Once, he was asked by a great Egyptian Pharaoh if he knew of a way to measure the height of the Great Pyramids. He looked at the Sun, the shadow that the pyramid cast, and his 6 foot staff. By the drawing below can you figure out how he found the height of the pyramid?
11. Using similar devices he was
able to measure ships distances off shore. This proved to be a great advantage in war at the time. How far from the shore is the ship in the diagram?
12. Using a mirror you can also create similar triangles (Thanks to the
properties of reflection similar triangles are created). Can you find the height of the flag pole?
13. Use your knowledge of special right triangles to measure
something that would otherwise be immeasurable.
35
f t
5 ft
3 ft
Height of Pyramid:
Distance from Shore:
Height of the Flag Pole:
M. Winking Unit 2-4 page 42
14. Find the unknown area based on the pictures below.
15. If the small can holds 20 gallons how much will the big trashcan hold (assuming they
are similar shapes)
16. If the smaller spray bottle holds 37 fl. oz. , then how much does the larger one hold
assuming they are similar shapes?
17. The smaller of the two cars is a Matchbox car set at the usual 64
1th scale (the length)
and it takes 0.003 fluid ounces to paint the car. If the smaller is a perfect scale of the actual car and the ratios of the paint remains the same then how many gallons of paint will be needed for the real car? (128 fl. oz = 1 gallon)
2 in. 3 in.
Scale Similarity Length Area Volume
Factors Ratios
Scale Similarity Length Area Volume
Factors Ratios
Scale Similarity Length Area Volume
Factors Ratios
36 in2
?? in2
5 in. 15 in.
Scale Length Area
Factors
2 ft
4 ft
Area of the small square:
Volume of the Large Trashcan:
Volume of the Large Spray Bottle:
2 in. 128 in.
Amount of Paint Needed:
M. Winking Unit 2-4 page 43
18. If the following are similar determine the length of the unknown sides.
A.
B.
C.
D.
18a.
x =
Volume = 20 cm3
Volume = 4 cm3
Volume = 90 cm3 Volume = ??cm
3
18 b.
V =
18d.
V =
18c.
x =
M. Winking Unit 2-4 page 44
Sec 2.5 Geometry โ Congruence Name: Any two congruent figures can be mapped onto one another using a series of rigid or isometric transformation (reflections, rotations, and translations). โ See GSP Lab (Transformations) โ
Each of the following pairs of figures shown below are congruent. Write a congruence statement for each and tell whether or not a reflection would be needed to map the pre-image onto the image. 1. 2. 3. 4. Given the following congruencies find the requested unknown angle. 5. ONMABC โโ โ 6. TORGEM โโ โ
42ยฐ 57ยฐ
=โ MNOm
(3x+20)ยฐ
50ยฐ (3x)ยฐ
=โ GEMmM. Winking Unit 2-5 page 45
Given the following congruencies find the requested unknown side.
7. โฟTRI โ โฟANG 8. FUNMTH โโ โ
The following pairs of triangles are congruent. Provide a suggested transformation or series of transformations that can map one triangle onto the other congruent triangle. (In each diagram ABC DEFโ โ โ ) 9. 10. 11. 12.
8
17
=GN =UF
Circle which transformation(s) could be used to map โABC onto โDEF.
Translation Reflection Rotation Dilation
Circle which transformation(s) could be used to map โABC onto โDEF.
Translation Reflection Rotation Dilation
Circle which transformation(s) could be used to map โABC onto โDEF.
Translation Reflection Rotation Dilation
Circle which transformation(s) could be used to map โABC onto โDEF.
Translation Reflection Rotation Dilation
M. Winking Unit 2-5 page 46
Angle Puzzles. (angles are not drawn to scale)
9. Find ๐๐โก๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ 10. Find ๐๐โก๐ฎ๐ฎ๐ฎ๐ฎ๐ฎ๐ฎ
Given: โข ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ bisects โก๐ท๐ท๐ด๐ด๐ท๐ท โข ๐๐โก๐ท๐ท๐ด๐ด๐ท๐ท = 50ยฐ โข โก๐ด๐ด๐ท๐ท๐ด๐ด is a right angle
Given: โข ๐บ๐บ๐บ๐บ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects โก๐น๐น๐บ๐บ๐น๐น โข ๐๐โก๐น๐น๐บ๐บ๐น๐น = 60ยฐ โข ๐๐โก๐บ๐บ๐น๐น๐บ๐บ = 75ยฐ โข โก๐น๐น๐บ๐บ๐น๐น is a right angle
I
F
H
G
โ
โ
โ
โ โ
โ
โ
โ
E
A
B
โ
C
D
M. Winking Unit 2-5 page 47
๐๐โก๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ = ๐๐โก๐ฎ๐ฎ๐ฎ๐ฎ๐ฎ๐ฎ =
Sec 2.6 Geometry โ Triangle Proofs Name: COMMON POTENTIAL REASONS FOR PROOFS
Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point that divides a segment into two congruent segments. Definition of Angle Bisector: The ray that divides an angle into two congruent angles. Definition of Perpendicular Lines: Lines that intersect to form right angles or 90ยฐ Definition of Supplementary Angles: Any two angles that have a sum of 180ยฐ Definition of a Straight Line: An undefined term in geometry, a line is a straight path that has no thickness and
extends forever. It also forms a straight angle which measures 180ยฐ
Reflexive Property of Equality: any measure is equal to itself (a = a) Reflexive Property of Congruence: any figure is congruent to itself (๐น๐น๐น๐น๐น๐น๐น๐น๐น๐น๐น๐น ๐ด๐ด โ ๐น๐น๐น๐น๐น๐น๐น๐น๐น๐น๐น๐น ๐ด๐ด) Addition Property of Equality: if a = b, then a + c = b + c Subtraction Property of Equality: if a = b, then a โ c = b โ c Multiplication Property of Equality: if a = b, then ac = bc Division Property of Equality: if a = b, then a
c= b
c
Transitive Property: if a = b & b =c then a = c OR if a โ b & b โ c then a โ c.
Segment Addition Postulate: If point B is between Point A and C then AB + BC = AC Angle Addition Postulate: If point S is in the interior of โ PQR, then mโ PQS + mโ SQR = mโ PQR Side โ Side โ Side Postulate (SSS) : If three sides of one triangle are congruent to three sides of another triangle,
then the triangles are congruent. Side โ Angle โ Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to two sides
and the included angle of another triangle, then the triangles are congruent. Angle โ Side โ Angle Postulate (ASA): If two angles and the included side of one triangle are congruent to two
angles and the included side of another triangle, then the triangles are congruent. Angle โ Angle โ Side Postulate (AAS) : If two angles and the non-included side of one triangle are congruent to
two angles and the non-included side of another triangle, then the triangles are congruent Hypotenuse โ Leg Postulate (HL): If a hypotenuse and a leg of one right triangle are congruent to a hypotenuse
and a leg of another right triangle, then the triangles are congruent
Right Angle Theorem (R.A.T.): All right angles are congruent. Vertical Angle Theorem (V.A.T.): Vertical angles are congruent. Triangle Sum Theorem: The three angles of a triangle sum to 180ยฐ Linear Pair Theorem: If two angles form a linear pair then they are adjacent and are supplementary. Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third
pair of angles are congruent. Alternate Interior Angle Theorem (and converse): Alternate interior angles are congruent if and only if the
transversal that passes through two lines that are parallel. Alternate Exterior Angle Theorem (and converse): Alternate exterior angles are congruent if and only if the
transversal that passes through two lines that are parallel. Corresponding Angle Theorem (and converse) : Corresponding angles are congruent if and only if the
transversal that passes through two lines that are parallel. Same-Side Interior Angles Theorem (and converse) : Same Side Interior Angles are supplementary if and only if
the transversal that passes through two lines that are parallel. Pythagorean Theorem (and converse): A triangle is right triangle if and only if the given the length of the legs a
and b and hypotenuse c have the relationship a2+b2 = c2
Isosceles Triangle Theorem (and converse): A triangle is isosceles if and only if its base angles are congruent. Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to a side of the triangle, and its length is
half the length of that side. CPCTC: Corresponding Parts of Congruent Triangles are Congruent by definition of congruence.
1. Tell which of the following triangle provide enough information to show that they must be congruent. If they are congruent, state which theorem suggests they are congruent (SAS, ASA, SSS, AAS, HL) and write a congruence statement.
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
SSS SAS ASA AAS HL Not Enough Information
Circle one of the following:
Congruence Statement if necessary:
M. Winking Unit 2-6 page 49
2. Prove which of the following triangles congruent if possible by filling in the missing blanks:
a. Given ๐ช๐ช๐ช๐ช๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ช๐ช๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐จ๐จ๐จ๐จ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ
b. Given ๐ท๐ท๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and Point A is the
midpoint of ๐ท๐ท๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
c. Given ๐ฝ๐ฝ๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐น๐น๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ท๐ท๐น๐น๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐ฝ๐ฝ๐ฝ๐ฝ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ
Statements Reasons
1. ๐ถ๐ถ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
2. ๐ถ๐ถ๐ถ๐ถ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐ด๐ด๐ด๐ด๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ
3. โก๐ถ๐ถ๐ถ๐ถ๐ด๐ด โ โก๐ด๐ด๐ด๐ด๐ถ๐ถ
4. ๐ถ๐ถ๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ถ๐ถ๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
5. โ๐ถ๐ถ๐ถ๐ถ๐ด๐ด โ โ๐ด๐ด๐ด๐ด๐ถ๐ถ
Statements Reasons
1. Given
2. Given
3. Definition of Midpoint
4. Reflexive property of congruence
5. By steps 1,3,4 and SSS
Statements Reasons
1. Given
2. Given
3.
4.
5. โ๐๐๐๐๐๐ โ โ๐ธ๐ธ๐ธ๐ธ๐๐
M. Winking Unit 2-6 page 50
Prove the Isosceles Triangle Theorem and the rest of the suggested proofs. d. Given โ๐๐๐๐๐๐ is isosceles and point R is
the midpoint of ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
e. Given point I is the midpoint of ๐ฟ๐ฟ๐ฟ๐ฟ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and
point I is the midpoint of ๐จ๐จ๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
f. Given โก๐๐๐๐๐๐ and โก๐๐๐๐๐๐ are right angles
and ๐ฟ๐ฟ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ป๐ป๐ป๐ป๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
Statements Reasons 1. โ๐๐๐๐๐๐ is
isosceles
2. ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
3. R is the midpoint of ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
4. ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
5. ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
6. โ๐๐๐๐๐๐ โ โ๐๐๐๐๐๐
7. โก๐๐๐๐๐๐ โ โก๐๐๐๐๐๐
Statements Reasons 1. I is the midpoint
of ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
2. Definition of Midpoint
3. I is the midpoint of ๐ด๐ด๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
4.
5. โก๐ด๐ด๐ด๐ด๐๐ โ โก๐๐๐ด๐ด๐๐
6. โ๐ด๐ด๐๐๐ด๐ด โ โ๐๐๐๐๐ด๐ด
Statements Reasons 1. โก๐๐๐ด๐ด๐๐ & โก๐๐๐ป๐ป๐๐
are right angles
2. ๐๐๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ป๐ป๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
3. ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
4. โ๐๐๐ด๐ด๐๐ โ โ๐๐๐ป๐ป๐๐
M. Winking Unit 2-6 page 51
Prove the suggested proofs by filling in the missing blanks.
g. Given ๐ฎ๐ฎ๐ช๐ช๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐ท๐ท๐ท๐ท๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ and ๐ฎ๐ฎ๐ท๐ท๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐ช๐ช๐ท๐ท๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ
h. Given ๐น๐น๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ป๐ป๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ , โก๐๐๐๐๐๐ โ โก๐๐๐๐๐๐ and
๐ป๐ป๐ฝ๐ฝ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐ฟ๐ฟ๐น๐น๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ
Statements Reasons
1. ๐บ๐บ๐ถ๐ถ๏ฟฝโ๏ฟฝ๏ฟฝโ โฅ ๐๐๐๐๏ฟฝโ๏ฟฝ๏ฟฝโ
2.
3. ๐บ๐บ๐๐๏ฟฝโ๏ฟฝ๏ฟฝโ โฅ ๐ถ๐ถ๐๐๏ฟฝโ๏ฟฝ๏ฟฝโ
4.
5.
6. โ๐บ๐บ๐ถ๐ถ๐๐ โ โ๐๐๐๐๐ถ๐ถ
Statements Reasons
1. ๐๐๐ธ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
2. ๐น๐น๐ฝ๐ฝ = ๐ป๐ป๐ฝ๐ฝ
3. ๐น๐น๐ฝ๐ฝ + ๐๐๐ธ๐ธ = ๐ป๐ป๐ฝ๐ฝ + ๐ธ๐ธ๐๐
4. ๐น๐น๐ฝ๐ฝ + ๐๐๐ธ๐ธ = ๐๐๐ธ๐ธ and ๐ป๐ป๐ฝ๐ฝ + ๐ธ๐ธ๐๐ = ๐๐๐๐
5. ๐๐๐ธ๐ธ = ๐๐๐๐ Substitution Property
6. ๐๐๐ธ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
7. ๐ป๐ป๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โฅ ๐๐๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ
8. โก๐๐๐๐๐ธ๐ธ โ โก๐ป๐ป๐๐๐๐
9. โก๐ป๐ป๐๐๐๐ โ โก๐๐๐ธ๐ธ๐๐
10. โ๐ป๐ป๐๐๐๐ โ โ๐๐๐ธ๐ธ๐๐
M. Winking Unit 2-6 page 52
Prove the suggested proofs by filling in the missing blanks. i. Given โก๐๐๐๐๐๐ โ โก๐๐๐๐๐๐ , โก๐๐๐๐๐๐ โ โก๐๐๐๐๐๐,
and Point A is the midpoint of ๐ฝ๐ฝ๐น๐น๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ii.
j. Given that ๐จ๐จ๐ฎ๐ฎ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ bisects โก๐๐๐๐๐๐. Also, ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐จ๐จ๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are radii of the same circle with center A.
Statements Reasons
1. โก๐ป๐ป๐๐๐ด๐ด โ โก๐๐๐ป๐ป๐ด๐ด
2. โ๐๐๐ด๐ด๐ป๐ป is an isosceles triangle
3. ๐๐๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ป๐ป๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
4. A is the midpoint of ๐ธ๐ธ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
5. ๐ธ๐ธ๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ด๐ด๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
6. โก๐ป๐ป๐ด๐ด๐๐ โ โก๐๐๐ด๐ด๐ธ๐ธ
7. โ๐ป๐ป๐ด๐ด๐๐ โ โ๐๐๐ด๐ด๐ธ๐ธ
Statements Reasons
1. ๐ด๐ด๐บ๐บ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ bisects โก๐๐๐ด๐ด๐ธ๐ธ
2. Definition of Angle Bisector
3. Radii of the same circle are congruent.
4. Reflexive property of congruence
5. โ๐ด๐ด๐๐๐บ๐บ โ โ๐ด๐ด๐ธ๐ธ๐บ๐บ
M. Winking Unit 2-6 page 53
Prove the suggested proofs by filling in the missing blanks. i. Given:
โข โก๐๐ โ โก๐๐ โข ๐จ๐จ๐ช๐ช๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ฎ๐ฎ๐ฎ๐ฎ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โข ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐๐ forms an isosceles triangle
with base ๐ช๐ช๐ฝ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Prove: โ๐ด๐ด๐ถ๐ถ๐ถ๐ถ โ โ๐บ๐บ๐น๐น๐ธ๐ธ
Prove the suggested proofs by filling in the missing blanks. k. Given:
โข The circle has a center at point C โข ๐ซ๐ซ๐๐๐ซ๐ซ๐๐ forms an isosceles
triangle with base ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
Prove: โ๐ด๐ด๐ถ๐ถ๐ถ๐ถ โ โ๐ด๐ด๐ถ๐ถ๐ถ๐ถ
Statements Reasons
1. โก1 โ โก9
2. ๐ด๐ด๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐บ๐บ๐น๐น๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
3. โก2 โ โก4
4. โ๐ถ๐ถ๐ด๐ด๐ธ๐ธ is an isosceles triangle
5. Isosceles Triangle Theorem
6. โก6 โ โก7
7. โก2 โ โก7
8.
Statements Reasons 1. โ๐ด๐ด๐ถ๐ถ๐ด๐ด is an isosceles
triangle w/ base ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
2. ๐ด๐ด๐ธ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ด๐ด๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
3. The circle is centered at point C
4. ๐ด๐ด๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ด๐ด๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
5. Reflexive Property of Congruence
6. โ๐ด๐ด๐ถ๐ถ๐ถ๐ถ โ โ๐ด๐ด๐ถ๐ถ๐ถ๐ถ
M. Winking Unit 2-6 page 54
Sec 2.7 Geometry โ Locus of Points & Triangle Centers Name: 1. Anna is stuck out on a swimming dock in the lake. If the picture below is a scale drawing, how far
must she swim to get to shore (convert using the scale provided)?
2. Measure the distance (cm) from the point to the line in each of the following problems a. b.
3. Describe how you attempted to measure the distance from a point to a line??
Locus of Points. 1. Mrs. and Mr. Scitamehtam were looking to purchase a new house. The map below shows where
both of the two work. They both wish to be completely fair to each other in selecting a new house that is equal in distance from each of their respective places of work. Using a ruler or other construction utility can you show all such possible locations.
SCALE
10 ft
โ
โ
Mr. Scitamehtamโs place of employment
Mrs. Scitamehtamโs place of employment
M. Winking Unit 2-7 page 55
DEFINITIONS: 1. MEDIAN of a triangle: All triangleโs have 3 medians. Each median
is a segment with one endpoint on the midpoint of a side of a triangle and the other endpoint at the opposite vertex of the triangle.
2. ALTITUDE of a triangle: All triangleโs have 3 altitudes. Each
altitude is a line that passes through a vertex of the triangle and is perpendicular to the opposite side.
CONSTRUCTIONS:
1. Construct all 3 medians of the triangle below using either a straight edge & compass or a MIRA
2. Construct all 3 altitudes of the triangle
below using either a compass compass and straight edge or a mira
M. Winking Unit 2-7 page 56
3. Construct all 3 perpendicular bisectors of each side of the triangle below using either a compass compass and straight edge or a mira
4. Construct all 3 angle bisectors of the triangle below using either a
compass and straight edge or a mira
Each one of these constructions should create a common center of a triangle. 1. Circumcenter: Is the center of a circle that perfectly passes through each vertex. 2. Incenter: Is the center of largest possible circle that still completely fits inside the circle. 3. Centroid: Is the center that is the center of gravity of the triangle (balancing point) and there is a
constant ratio between the distance from the centroid to the midpoint and centroid to the vertex. 4. Orthocenter: Is the fourth common center but has no unique properties other than it is
on the EULER line. CAN YOU GUESS WHICH CENTER IS WHICH?
(Hint: LOOK at the BOLD letters in each prompt.
M. Winking Unit 2-7 page 57
Additional information about the Common Triangle Centersโฆ.
It is the INCENTER created by the ANGLE BISECTORS
It is the point that all sides are equidistant from and the center of a circle that is circumscribed by the triangle.
It is the CENTROID
created by the MEDIANS.
It is the ORTHOCENTER
and it is created by the
ALTITUDES of the
triangle.
It is the CIRCUMCENTER and it is created by the
PERPENDICULAR BISECTORS
It is the point that is equidistant from all of the triangleโs vertices
and the center of a circle that circumscribes the triangle.
p.34
If you wanted to balance a cut out of this triangle on one finger, you would hold the triangle at the centroid. It represents the center of gravity.
The only additional interesting fact about the orthocenter is that is a point on Eulerโs Line
(the line also includes the Centroid and Circumcenter.
M. Winking Unit 2-7 page 58
There are some interesting uses of some of the common triangle centers. Identify which center would be best for each situation.
1. A person wishes to make a triangular table with a single post holding up the table. Which triangle center should the post connect to that would help keep the table the most balanced and stable?
2. A person has a triangular piece of scrap ornate fabric. The person wants to create a circular table cloth. Which center might help her cut out the biggest possible circle from the fabric?
3. The space shuttle is monitoring 3 GPS satellites and has positioned itself so that it is
equidistant from each of the satellites. Which triangle center might have helped the pilot determine this particular location?
M. Winking Unit 2-7 page 59
Sec 2.8 Geometry โ Polygons & Quadrilaterals Name:
Polygon: A closed plane figure formed by three or more segments such that each segment intersects or connects end to end to form a closed shape.
Simple Polygon: A polygon in which sides only share each endpoint with one other side.
Regular Polygon: A polygon that is both equilateral and equiangular
Determine whether each figure below is a polygon or not a polygon.
1. 2. 3.
4. 5. 6.
Concave Polygon: A polygon in which a diagonal can be drawn such that part of one of the diagonals contains point in the exterior of the polygon. Convex Polygon: A polygon in which no diagonal contains points in the exterior of the polygon.
Determine whether each figure below is a Convex or Concave. 7. 8. 9.
Circle one of the following: Not a
Polygon It is a
Polygon Name if Polygon:
Circle one of the following: Not a
Polygon It is a
Polygon Name if Polygon:
Circle one of the following: Not a
Polygon It is a
Polygon Name if Polygon:
Circle one of the following: Not a
Polygon It is a
Polygon Name if Polygon:
Circle one of the following: Not a
Polygon It is a
Polygon Name if Polygon:
Circle one of the following: Not a
Polygon It is a
Polygon Name if Polygon:
Circle one of the following:
CONCAVE CONVEX
Circle one of the following:
CONCAVE CONVEX
Circle one of the following:
CONCAVE CONVEX
M. Winking Unit 2-8 page 60
Quadrilateral: A four-sided polygon. Parallelogram: A quadrilateral with two pairs of parallel sides. Trapezoid: A quadrilateral with exactly one pair of parallel sides. Rectangle: A quadrilateral with four right angles. Rhombus: A quadrilateral with four congruent sides. Square: A quadrilateral with four congruent sides and four right angles. Kite: A quadrilateral with exactly two pairs of congruent consecutive sides
but opposite sides are not congruent.
The Hierarchy of Quadrilaterals
Answer each of the following with ALWAYS, SOMETIMES, or NEVER
____________________10. A square is a rhombus.
____________________11. A rhombus is a square.
____________________12. A trapezoid is a parallelogram.
____________________13. A rectangle is a rhombus.
____________________14. A kite is a concave quadrilateral.
____________________15. A parallelogram is a rectangle.
____________________16. A rhombus is a trapezoid.
____________________17. A convex quadrilateral is a trapezoid.
____________________18. A rectangle is a parallelogram.
Convex Quadrilateral Concave Quadrilateral
Parallelograms
Rhombi. Squares . Rectangles.
Trapezoids .
Kites .
M. Winking Unit 2-8 page 61
Using your knowledge of congruent triangles and parallel triangles determine the highest level and most appropriate definition of a quadrilateral for each ABCD quadrilateral below. Each definition in the word wall should be used exactly once. Briefly explain why for each quadrilateral. 1. 2.
3. 4.
5. 6.
7. 8.
Both dashed figures can be assumed to be circles.
โก๐ด๐ด๐ด๐ด๐ด๐ด and โก๐ถ๐ถ๐ด๐ด๐ด๐ด are complimentary
Square
Parallelogram Kite
Rhombus Rectangle Trapezoid
Concave-Quadrilateral Convex-Quadrilateral
Please assume that different congruent
marks represent incongruent angles
M. Winking Unit 2-8 page 62
Sec 2.9 Geometry โ Polygon Angles Name:
The Asimov Museum has contracted with a company that provides Robotic Security Squads to guard the exhibits during the hours the museum is closed. The robots are designed to patrol the hallways around the exhibits and are equipped with cameras and sensors that detect motion. Each robot is assigned to patrol the area around a specific exhibit. They are designed to maintain a consistent distance from the wall of the exhibits. Since the shape of the exhibits change over time, the museum staff must program the robots to turn the corners of the exhibit. Below, you will find a map of the museum's current exhibits and the path to be followed by the robot. One robot is assigned to patrol each exhibit. When a robot reaches a corner, it will stop, turn through a programmed angle, and then continue its patrol. Your job is to determine the angles that R1, R2, R3, and R4 will need to turn as they patrol their area. Keep in mind the direction in which the robot is traveling and make sure it always faces forward as it moves around the exhibits.
Try measuring the angles on the following pages using a protractor or using the Geometerโs Sketchpad animation found at
http://gwinnett.k12.ga.us/PhoenixHS/math/grade09/unit03/Unit%203%20-%20Robot%20Task.gsp
Robot 1
Robot 2 Robot 3
Robot 4
Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com
p.63
1. What angles will Robot 1 need to turn? What is the total of these turns?
2. What angles will Robot 2 need to turn? What is the total of these turns?
Exterior Angle at Vertex โAโ:
Exterior Angle at Vertex โBโ:
Exterior Angle at Vertex โCโ:
Exterior Angle at Vertex โDโ:
+
TOTAL:
Exterior Angle at Vertex โEโ:
Exterior Angle at Vertex โFโ:
Exterior Angle at Vertex โGโ:
Exterior Angle at Vertex โHโ:
Exterior Angle at Vertex โIโ:
+
TOTAL:
Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com
p.64
3. What angles will Robot 3 need to turn? What is the total of these turns?
4. What angles will Robot 4 need to turn? What is the total of these turns?
Exterior Angle at Vertex โJโ:
Exterior Angle at Vertex โKโ:
Exterior Angle at Vertex โLโ:
Exterior Angle at Vertex โMโ:
Exterior Angle at Vertex โNโ:
Exterior Angle at Vertex โOโ:
+
TOTAL:
Exterior Angle at Vertex โPโ:
Exterior Angle at Vertex โQโ:
Exterior Angle at Vertex โRโ:
Exterior Angle at Vertex โSโ:
Exterior Angle at Vertex โTโ:
Exterior Angle at Vertex โUโ:
Exterior Angle at Vertex โVโ:
+
TOTAL:
Or Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ
Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com p.65
5. What do you notice about the sum of the angles? Do you think this will always be true? (please explain using complete sentences)
6. The museum also requested a captain robot, CR, to patrol the entire exhibit area. What angles will
CR need to turn? What is the total of these turns?
Exterior Angle at Vertex โAโ:
Exterior Angle at Vertex โBโ:
Exterior Angle at Vertex โCโ:
Exterior Angle at Vertex โDโ:
Exterior Angle at Vertex โEโ:
Exterior Angle at Vertex โFโ:
Exterior Angle at Vertex โGโ:
Exterior Angle at Vertex โHโ: +
TOTAL:
Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com
p.66
7. What makes Captain Robotโs path different from the other robotโs paths (other than the number of sides being different)? (please explain using complete sentences)
(RECALL FROM EARLIER IN THIS UNIT) 8. a. Create a random triangle on a piece of patty papers.
b. Write a number inside each interior angle c. Cut out the triangle d. Tear off or cut each of the angles from the triangle e. Paste all 3 angles next to one another so that they
all have the same vertex and the edges are touching but they arenโt overlapping
9. What is the measure of a straight angle or the angle that creates a line by using two opposite rays from a common vertex?
10. Collectively does the sum of your 3 interior angles of a triangle form a straight angle? What about
others in your class?
11. Make a conjecture about the sum of the interior angles of a triangle. Do you think your conjecture will always be true? (please explain using complete sentences)
1
2
3
1
2
3
โข Paste or Tape your 3 vertices here:
โข Common Vertex
Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com
p.67
โข
12. Determine the measure of the interior angles of the polygons of the polygons formed by Exhibits A through D.
Exhibit A (Quadrilateral) Exhibit B (Pentagon) Exhibit C (Hexagon) Exhibit D (Heptagon)
13. Look at the sums of the angles in each of the polygons, do you notice a pattern?
Can you briefly describe the pattern?
Interior Angle at Vertex โAโ:
Interior Angle at Vertex โBโ:
Interior Angle at Vertex โCโ:
Interior Angle at Vertex โDโ:
+
TOTAL:
Interior Angle at Vertex โEโ:
Interior Angle at Vertex โFโ:
Interior Angle at Vertex โGโ:
Interior Angle at Vertex โHโ:
Interior Angle at Vertex โIโ:
+
TOTAL:
Interior Angle at Vertex โJโ:
Interior Angle at Vertex โKโ:
Interior Angle at Vertex โLโ:
Interior Angle at Vertex โMโ:
Interior Angle at Vertex โNโ
Interior Angle at Vertex โOโ:
+
TOTAL:
Interior Angle at Vertex โPโ:
Interior Angle at Vertex โQโ:
Interior Angle at Vertex โRโ:
Interior Angle at Vertex โSโ:
Interior Angle at Vertex โTโ:
Interior Angle at Vertex โUโ
Interior Angle at Vertex โVโ:
+
TOTAL:
Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com
p.68
14. Using your hypothesis from the previous problem can you determine the sum of the interior angles of the following: A. Octagon (8-Sides) B. Dodecagon (12-sides) C. Icosagon (20 sides) D. A polygon with n-sides
15. Using the information from above what would be the measure of a single interior angle each were regular polygons? A. Regular Octagon B. Regular Dodecagon C. Regular Icosagon D. A regular polygon with n-sides
16. The museum intends to create regular polygons for its 6th exhibition. For regular polygons the robot will make the same โturnโ at each vertex. Can you determine the angle of each turn for the robot for a regular Pentagon? a. A regular Pentagon? b. A regular Hexagon? c. A regular Nonagon?
17. A sixth exhibit was added to the museum. The robot patrolling this exhibit will make 15ยบ turns at each vertex. How many sides must the exhibit have and what is the name of the polygon? What makes it possible for the robot to make the same turn each time (what type of polygon must the exhibit be)?
18. Find the value of x in each diagram below.
a. b. c.
Original Task developed by Janet Davis at Georgia Department of Education 2016 ยฉ Modified, Updated, and Edited by M. Winking e-mail: mmwinking@gmail.com
p.69
Sec 2.10 Geometry โ Quadrilateral Properties Name:
Name all of the properties of a parallelogram and its diagonals.
1. Opposite Sides are parallel 2. Opposite Sides are congruent 3. Opposite Angles are congruent
4. Consecutive Angles are supplementary 5. Diagonals bisect each other
The properties of a rectangle and its diagonals: 1. All angles are right 2. Opposite Sides are parallel 3. Opposite Sides are congruent
4. Diagonals bisect each other 5. Diagonals are congruent
The properties of a rhombus and its diagonals:
1. Opposite angles are congruent 2. Opposite Sides are parallel 3. All Sides are congruent
4. Consecutive angles are supplementary 5. Diagonals are perpendicular 6. Diagonals bisect interior angles.
The properties of a square and its diagonals: 1. All angles are right 2. Opposite Sides are parallel 3. All Sides are congruent 4. Diagonals bisect each other
5. Diagonals are perpendicular 6. Diagonals bisect angles 7. Diagonals are congruent
M. Winking Unit 2-10 page 70
Find the value of x in each diagram below using properties of quadrilaterals.
1. 2.
3. 4.
5. 6.
(3x + 10)ยฐ
(8x +5 )ยฐ
50ยฐ
xยฐ
AC = 9x + 4 BD = 6x + 16
1.
x =
2.
x =
3.
x =
4.
x =
5.
x =
6.
x =
8
x
x
2x
45
M. Winking Unit 2-10 page 71
Plot points A(-3, -1), B(-1, 2), C(4, 2), and D(2, -1).
1. What specialized geometric figure is quadrilateral
ABCD? Support your answer mathematically.
2. Draw the diagonals of ABCD. Find the coordinates of
the midpoint of each diagonal. What do you notice?
3. Find the slopes of the diagonals of ABCD. What do
you notice?
4. The diagonals of ABCD create four small triangles. Are any of these triangles congruent to any of the
others? Why or why not?
Plot points E(1, 2), F(2, 5), G(4, 3) and H(5, 6).
5. What specialized geometric figure is quadrilateral
EFHG? Support your answer mathematically using two
different methods.
6. Draw the diagonals of EFHG. Find the coordinates of
the midpoint of each diagonal. What do you notice?
7. Find the slopes of the diagonals of EFHG. What do you
notice?
8. The diagonals of EFHG create four small triangles. Are
any of these triangles congruent to any of the others?
Why or why not?
M. Winking Unit 2-10 page 72
Plot points P(4, 1), W(-2, 3), M(2,-5), and K(-6, -4).
9. What specialized geometric figure is quadrilateral
PWKM? Support your answer mathematically.
10. Draw the diagonals of PWKM. Find the coordinates of
the midpoint of each diagonal. What do you notice?
11. Find the lengths of the diagonals of PWKM. What do
you notice?
12. Find the slopes of the diagonals of PWKM. What do
you notice?
13. The diagonals of ABCD create four small triangles.
Are any of these triangles congruent to any of the
others? Why or why not?
Plot points A(1, 0), B(-1, 2), and C(2, 5).
14. Find the coordinates of a fourth point D that would
make ABCD a rectangle. Justify that ABCD is a
rectangle.
15. Find the coordinates of a fourth point D that would
make ABCD a parallelogram that is not also a
rectangle. Justify that ABCD is a parallelogram but is
not a rectangle.
M. Winking Unit 2-10 page 73
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