geometry 1 unit 4 congruent triangles casa grande union high school fall 2008
TRANSCRIPT
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Geometry 1 Unit 4Congruent Triangles
Casa Grande Union High SchoolFall 2008
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Geometry 1Unit 4
4.1 Triangles and Angles
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Scalene- No sides congruent
Equilateral- all 3 sides are congruent
Isosceles- at least 2 sides are congruent
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Acute Triangle- All anglesare less than 90°
Obtuse Triangle- 1 angle greater than 90 ° Right Triangle- 1 angle
measuring 90°
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Classifying Triangles Example 1
◦ Name each triangle by its sides and angles
A. B.
C.
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Vertex (plural vertices)◦ The points joining the sides of a triangle
Adjacent sides◦ Sides sharing a common vertex
Side AB is adjacent to side BC
A
BC
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Interior angle◦ Angle on the inside of a triangle
Exterior angle◦ Angle outside the triangle that is formed by extending
one side
A
B
C
Interior angleExterior angle
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Triangle Sum Theorem◦ The sum of the three interior angles of a triangle
is 180º
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Exterior Angle Theorem◦ The measure of the exterior angle of a triangle is
equal to the sum of the two nonadjacent interior angles. Example: m∠1=m∠A+ m∠B
B
AC
1
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Example 2
Find the measure of each angle.
2x + 10
x x + 2
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Example 3
Given that ∠ A is 50º and ∠B is 34º, what is the measure of ∠BCD?
What is the measure of ∠ACB? DA
B
C
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Right triangle vocabulary
Legs◦ Sides that form the right angle
Hypotenuse◦ Side opposite the right angle
Legs
Hypotenuse
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Corollary to the Triangle Sum Theorem◦ The acute angles of a right triangle are
complementary. m∠A+ m∠B = 90°
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Example 4
A. Given the following triangle, what is the length of the hypotenuse?
B. What are the length of the legs?
C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement?
13
12
5
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Legs◦ The two congruent sides of an isosceles triangle.
Base◦ The noncongruent side of an isosceles triangle.
Base Angles◦ The two angles that contain the base of an isosceles
triangle.
Vertex Angle◦ The noncongruent angle in an isosceles triangle.
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Legs
Base Angles
Vertex Angle
Base
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Example 5
A. Given the following isosceles triangle, what is the measurement of segment AC?
B. What is the measurement of angle A?
A
B C75º
15
7
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Example 6
Find the missing measures
80°
53°
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Example 7
Given: ∆ABC with mC = 90°Prove: mA + mB = 90°
Statement Reason
1. mC = 90°
2. mA + mB + mC = 180°
3. mA + mB + 90° = 180°
4. mA + mB = 90°
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Geometry 1Unit 4
4.2 Congruence and Triangles
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Congruent Figures Congruent Figures
◦ Figures are congruent if corresponding sides and angles have equal measures.
Corresponding Angles of Congruent Figures◦ When two figures are congruent, the angles that
are in corresponding positions are congruent.
Corresponding Sides of Congruent Figures◦ When two figures are congruent, the sides that
are in corresponding positions are congruent.
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Congruent Figures For the triangles below, ∆ABC ≅ ∆PQR
◦ The notation shows congruence and correspondence.
◦ When writing congruence statements, be sure to list corresponding angles in the same order.
A
B
C
P
Q
R
Corresponding Angles
Corresponding Sides
A ≅ P AB ≅ PQ
B ≅ Q BC ≅ QR
C ≅ R CA ≅ RP
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Complete the congruence statement for the two given triangles:
DEFWhat side corresponds with DE?
What angle corresponds with E?
D
E
F
S
V
T
Example 1
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Example 2
In the diagram, ABCD ≅ KJHLa. Find the value of x.b. Find the value of y.
A B
D C
9 cm
6 cm86°
91°
113°
J
H
KL
(5y – 12)°
(4x – 3) cm
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Third Angles Theorem◦ If 2 angles of 1 triangle are congruent to 2 angles
of another triangle, then the third angles are also congruent.
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Example 3
Given ABC PQR, find the values of x and y.
A
B CP
QR
85°
50°
(6y – 4)°
(10x + 5)°
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Example 4 Decide whether the triangles are congruent.
Justify your answer.
F
HE
G
J
58°
58°
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Example 5
Given: MN ≅ QP, MN || PQ, O is the midpoint of MQ and PN.
Prove: ∆MNO ≅ ∆QPO
P
M
O
Q
N Statements Reasons
1.
2. Alt. Interior Angles Theorem
3. Vertical Angles Theorem
4.O is the midpoint of MQ and PN
5. Def of Midpoint
6. ∆MNO ≅ ∆QPO
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Geometry 1Unit 4
4.3 Proving Triangles are Congruent: SSS and SAS
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Warm-Up
Complete the following statementBIG
B
I
G
R
A
T
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Definitions Included Angle
◦ An angle that is between two given sides.
Included Side ◦ A side that is between two given angles.
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Example 1 Use the diagram.
Name the included angle between the pair of given sides.
KP
J L
PLandKP
LKandPK
KLandJK
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Triangle Congruence Shortcut SSS
◦ If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.
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Triangle Congruence Shortcuts 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.
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Example 2 Complete the congruence statement. Name the congruence shortcut used.
ST
U
V
WSTW
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Example 3 Determine if the following are congruent. Name the congruence shortcut used.
J
H
I
L
MN
HIJ LMN
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Example 4 Complete the congruence statement. Name the congruence shortcut used.
B
OX
C
A
R
XBO
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Example 5 Complete the congruence statement. Name the congruence shortcut used.
SPQ
S
P
Q
T
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Constructing Congruent Triangles Construct segment DE as a segment
congruent to AB Open your compass to the length of AC.
Place the point of your compass on point D and strike an arc.
Open the compass to the width of BC. Place the point of your compass on E and strike an arc. Label the point where the arcs intersect as F.
A
C
B
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Example 6Given: AB ≅ PB, MB ⊥ APProve: ∆MBA ≅ ∆MBP
AB P
M
Statements Reasons
1. MB ⊥ AP
2. Perpendicular lines form right angles
3. Right angles are congruent
4. AB ≅ PB
5. MB ≅ MB
6.
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Example 7
Use SSS to show that ∆NPM ≅∆DFEN(-5, 1)P(-1, 6)M(-1, 1)D(6, 1)F(2, 6)E(2, 1)
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Geometry 1Unit 4
4.4 Proving Triangles are Congruent: ASA and AAS
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Triangle Congruence Shortcuts 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.
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Triangle Congruence Shortcuts AAS
◦ If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.
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Example 1 Complete the congruence statement. Name the congruence shortcut used.
Q U
ADQUA
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Example 2 Complete the congruence statement. Name the congruence shortcut used.
RMQ
M R
Q
N
P
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Example 3 Determine if the following are congruent. Name the congruence shortcut used.
A
B
C
F
E
DABC FED
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Example 4
Given: B ≅C, D ≅F; M is the midpoint of DF.Prove: ∆BDM ≅∆CFM
B
D M
C
F
Statements Reasons
1.
2.
3. Def of Midpoint
4.
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Geometry 1Unit 4
4.5 Using Congruent Triangles
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Warm-up State which postulate or theorem you can
use to prove that the triangles are congruent.
Then, write the congruence statement.
C
G
H
S
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Example 1
Given: NO is parallel to MP, MN is parallel to PO
Prove MN = OP(Prove Δ MNO Δ OPM)
Mark the given information first
Then, mark the deduced information
PO
N M
Statements Reasons1.NO||MP, MN|| PO 1. Given
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
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Example 2
Given: HJ || KL, JK || HLProve: LHJ ≅ JKL
H J
L K
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Example 3
Given: MS || TR, MS ≅ TR
Prove: A is the midpoint of MT.
S
M
A
T
R
Statements Reasons
1.
2.
3.
4.
5.
6.
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Geometry 1Unit 4
4.6 Isosceles, Equilateral, and Right Triangles
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Warm-Up 1
Find the measure of each angle.
90°
90°
30°
60°a
b
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Warm-Up 2
Find the measure of each angle.
110
15090
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Base Angles Theorem◦ If two sides of a triangle are congruent, then the
angles opposite them are also congruent. Converse of the Base Angles theorem
◦ If two angles of a triangle are congruent, then the sides opposite them are also congruent
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Example 1
35°
x
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Example 2
15°
b
a
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Example 3
Find each missing measure
63°
10 cmm n
p
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Equilateral Triangles If a triangle is equilateral, then it is
equiangular.
If a triangle is equiangular, then it is equilateral.
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Hypotenuse-Leg (HL) If the hypotenuse and the leg of a right
triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.
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Example 4 Find the value of x
12 in
2x in
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Example 5 Find the value of x and y.
yx
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Example 6 Find the value of x and y.
75°
x°
y°
x°
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Geometry 1Unit 4
4.7 Triangles and Coordinate Proof
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Warm-up
What is the midpoint formula?
What is the distance formula?
What are some postulates and theorems you have learned about triangles this chapter?
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Vocabulary Coordinate Proof
◦ A proof involving placing geometric figures on a coordinate plane.
◦ Uses the midpoint formula, distance formula, postulates and theorems to prove statements about the figure
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Placing Figures in a Coordinate Plane Complete the activity on p. 243 individually. Compare your results to those of your
partners. What did you learn?
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Example 1 A right triangle has legs of 3 units and 4
units. Place the triangle on a coordinate grid.
Label the vertices, then find the length of the hypotenuse.
3
4
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Example 2 In the diagram, ΔABO ≅ ΔCBO. Find the coordinates of point B.
C(10,0)
A(0,10)
O(0,0)
B
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Example 3 Write a plan to prove that OU bisects TOV.
V(3,5)
U(0,5)
T(-3,5)
O(0,0)
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Example 4 Find the coordinates of P.
P
N(h,0)
M(0,k)
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Constructions review Duplicate the given triangle. Write the steps that you used to construct
the new triangle