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L4 – Special Angles and Lines Name __________________________________ 4.1 – Angles Jigsaw Activity Per _____ Date __________________________

Geometry Q1: L4 – Special Angles and Lines Handouts

We now proceed to investigate some important types of angles through a “Jigsaw” activity. Each

member in your 4-person “home” group will be assigned one of the following four sections that

introduces a new angle concept; Adjacent Angles, Vertical Angles, Complementary and

Supplementary Angles, and Parallel and Perpendicular lines. You will then investigate your angle

type with “experts” in other home groups that have been assigned the same concept. Once you’ve

completed your investigation you will return to your home group and teach your fellow group

members about your angle type, and help them to complete the Frayer Model handouts in Lesson

2.2 as a visual organizer. Be sure to include in your booklets all the new vocabulary and theorems

that your fellow group members teach you. Good luck!

1. Adjacent Angles and the Angle Addition Postulate

Next door neighbors, seat partners, table partners, lane 1 and lane 2….

What does adjacent mean? Who is adjacent to you? It’s the person who lives right next door, or

your classmate in the seat right next to you. It could be the runner in the next lane over from

where you are, on either side. Adjacent angles have important properties – let’s explore:

These are adjacent angles:

These are not:

Adjacent angles (#VOC) are two angles that share a common vertex and a common ray, and do

not overlap. The two angles in Figure 4 overlap, in Figure 5 they only share a common ray, and

in Figure 6 they only share the vertex.

Fig. 1 Fig. 3 Fig. 2

Fig. 4 Fig. 6 Fig. 5

21

2

1



The Angle Addition Postulate (#THM): If BCD and DCE are adjacent, then

m BCE m BCD m DCE provided BCE 180.

For example:

Therefore, 55 37 92m BCE

On scratch paper, create some adjacent angles of your own choosing and use a protractor to

verify the Angle Addition Postulate. Be sure to choose angles whose sum is less than 180.

2. Vertical Angles

X marks the spot!

Just as fun as looking for hidden treasure, look for vertical angles at the “X”. Intersecting lines

form an X-shape, and vertical angles are on opposite sides of the X from one another. Use patty

paper to show that as you move the lines around the vertical/opposite angles remain congruent.

Two angles are called congruent angles (#VOC) if and only if their measures are the same.

Vertical angles (#VOC) are non-adjacent angles formed by two intersecting lines. Two pairs of

vertical angles can be found at the intersection of two lines.

Pair 1: 1 and 3

Pair 2: 2 and 4

Given an angle, its vertical angle is determined by its

opposite rays.

37°55°

B

C

E

D

1 2

3

4



More than two pairs of vertical angles can be found at the same vertex, depending on how many

intersecting lines share the same intersection.

Examples:

1 and 4

2 and 5

3 and 6

(1 and 2) and (4 and 5)

(2 and 3) and (5 and 6)

(1 and 2 and 3) and (4 and 5 and 6)

How many more vertical angles can you find in the diagram above? Name them.

3. Complementary and Supplementary Angles

Enhancing, improving, and to make better – typical words that might replace complement or

supplement. But how do you improve an angle?

Complementary angles (#VOC) are two angles whose measures sum to 90, or together they

create a right angle (#VOC). On diagrams, you can determine right angles by the small box used

to denote the angle (e.g. MLK below). Right angles are necessary in many facets in life…corners

of Grandma’s rooms, your body when standing in perfect posture, and so forth. Caution: angles

do not need to be adjacent to be complementary, their measures just need to sum to 90. Locate

some right angles in your classroom. Can you find any complementary angles in your classroom?

Name them.

mMLJ + mJLK = mMLK = 90, so

angles MLJ and JLK are complementary.

6 54

321

M

L K

J



Investigation: On a separate sheet of paper, use patty paper and a protractor to create an angle with

measurement 35. On another patty paper create an angle with measurement 55. Place the two

angles together, so that they share a common ray and become adjacent angles. Use your protractor

to verify that the resultant larger angle is 90.

Now, use your protractor to create a right angle. Draw a line that cuts this angle. Measure the two

resulting complementary angles. Do their measures sum to 90?

Supplementary angles (#VOC) are two angles whose measures sum to 180, or together they

create a straight angle. We learned about straight angles in the very first Geometry lesson, and

you’ll see its importance now and throughout our unit! A special type of supplementary angles is

called a linear pair (#VOC), which are two adjacent supplementary angles (as seen below).

Caution: angles do not need to be adjacent to be supplementary.

mMNP + mPNO = mMNO = 180

Linear Pair Postulate (#THM) If two angles are linear pairs, their measures sum to 180.

Can you find any supplementary angles in your classroom? How about on Grandma’s house

plans? (Note: Grandma’s house plans can be found in L1 – Foundations, page 3.)

Investigation: Draw a line on a piece of paper. Draw a ray with endpoint somewhere on the line.

Use a protractor to measure the two resulting angles. Do they sum to 180?

Use patty paper and a protractor to create an angle with measurement 65. On another patty

paper create an angle with measurement 115. Place the two angles together, so that they share a

common ray and become adjacent angles. Do they now form a line?

4. Parallel and Perpendicular Lines

To cross, or not to cross, that is the question…

Parallel lines are lines that do not intersect. From a single line and a point not on the line, there

can be only one parallel line that passes through the given point.

M O

P

N



We can use matching arrows on each line to signify

that the two lines are parallel, or we can use the

notation:

A line that crosses two lines is called a

transversal.

is a transversal of the parallel lines

.

How many examples of pairs of parallel lines can you find in your classroom? How about on

Grandma’s house plan?

Perpendicular lines are lines that intersect to form right angles (90 angles). Given a single line

and a point not on the line, there can be only one line passing through the point that is perpendicular

to the given line. The box drawn in ABC denotes a 90, or right angle.

We can also use the notation: .

How many examples of perpendicular lines can you find in your classroom? How about on

Grandma’s house plans?

M N

O Q

E

D

A B

C

B C

A

L4 – Special Angles and Lines Name __________________________________ 4.1 Angles Jigsaw Activity Per _____ Date __________________________


Fill in the following Frayer Model for each new definition in your group, and then for homework

complete the remaining: adjacent angles, vertical angles, complementary angles, supplementary

angles, parallel lines, perpendicular lines.

DEFINITION/NOTATION

NON-EXAMPLES

CHARACTERISTICS

EXAMPLES



DEFINITION/NOTATION

NON-EXAMPLES

CHARACTERISTICS

EXAMPLES



Angle Jigsaw Activity Participation Scoring Rubric (12 pts possible)

Exceeds

Proficiency (4)

Meets

Proficiency (3)

Approaches

Proficiency (2)

Does Not Meet

(0)

Focus on Work Consistently

stays focused on

in-class work.

Very self-

directed. Ensures

work is

completed on

time.

Focuses on in-

class work.

Moderately self-

directed.

Focuses on in-

class work

sometimes.

Often needs to

be directed by

teachers or

peers.

Does not focus

on in-class work.

Needs constant

reminders.

Leaves it to

others to ensure

work is

completed on

time.

Contributions Routinely

provides useful

ideas in

discussion.

Leader and/or

contributor in

effort.

Usually provides

useful ideas in

discussion.

Sometimes

provides useful

ideas in

discussion.

Does the

minimum

required.

Rarely provides

useful ideas in

discussion. May

refuse

participation.

Collaboration Almost always

listens to, shares

with, and

supports the

efforts of others.

Allows others to

feel safe and

comfortable.

Usually listens

to, shares with,

and supports the

efforts of others.

Often listens to,

shares with, and

supports the

efforts of others,

but is not

actively listening

or responding.

Rarely listens to,

shares with, and

supports the

efforts of others.

Often disrupts or

discourages

others’ attempts

to participate.

L4 – Special Angles and Lines Name __________________________________ 4.2 Angles Practice/Homework Per _____ Date __________________________


Practice: Determine if the pair of angles (∠1 and ∠2) is adjacent or not adjacent.

1.

2.

3.

4.

5.

6.

7.

8.

Adjacent Not adjacent










Find the indicated angle measure.

9.

10.

11.

12.

13.

14.

15.

16.

m1=35, m2=42.

Find mJLM.

m1=40, m2=46.

Find mNOQ.

mABC=87, mABD=119.

Find mCBD.

mEFG=128, mHFG=42.

Find mHFE.

mRUS=48, mTUS=52.

Find mRUT.

mILK=82, mILJ=39.

Find mJLK.

mVWY=112, mYWX=26.

Find mVWX.

mMQN=62, mPQO=67.

Find mNQO.



17. List all possible vertical angle pairs:

18. List all possible vertical angle pairs:

G

I

H

B

E

D

A

C

F

87 6

5

432

1 I

A

E

B

F

C

G

D

H



For each, find the value of x.

19.

20.

21.

22.

23.

24.

25.

26.

x68°

22°

x

103°24°x

x49°56°

86° x

31°

x

119° x

123° x



27. Given the figure below, name two unique pairs of complementary angles.

28. Given the figure below, name an angle supplementary to ∠EJF. Name a different angle

supplementary to ∠EJF.

For each, find the value of x.

29. 30.

31. 32.

B D

E

F

G

A

C

JI F

E

H

D

G



33. Create a pair of adjacent angles and label them ∠ABC and ∠BCD.

34. Create a pair of non-adjacent angles that share a vertex, and label them ∠LMN and ∠OMP.

L4 – Special Angles and Lines Name __________________________________ 4.3 What’s Your Angle? Per _____ Date _________________________


Follow the directions below:

i. Determine if each of the angles below is acute (<90 #VOC) or obtuse (between 90 and

180 #VOC) before using the protractor.

ii. Use a protractor to measure each angle below.

iii. Use a straightedge to extend the two rays into two lines that intersect at the vertex shown,

and without the use of a protractor determine the measure of each of the three new angles

thus created. Check your answers using a protractor.

1

2

3

4

5

6

7

8

9

L4 – Special Angles and Lines Name __________________________________ 4.4 Parallel Lines with a Transversal Per _____ Date _________________________


When two parallel lines are intersected by a third line, called a transversal, then 8 angles are

formed. It appears that many pairs of angles are congruent (i.e. have equal measure), and we can

intuitively verify which angle pairs are congruent using rigid motion transformations.

1. Identify which angle pairs you believe must be congruent, and the rigid motion

transformations that appear to justify your conjectures.

Pairs of Angles that you believe

must be Congruent

Transformation or Facts

Used to Justify your Result

2. List all the angles you believe are congruent to angle 8.

k

j

j || k

8765

43

21



3. With a protractor, measure all eight angles formed by this pair of parallel lines and the

transversal, and record them in the box to the right. Round off to the nearest degree.

What do you notice? Are your measurement results consistent with your earlier conjectures?

Which pairs of angles are congruent? Are there any angle pairs that are supplementary? List as

many unique pairs of congruent and supplementary angles as you can find (for example, pair 1

and 2 would be the same as pair 2 and 1).

Congruent Supplementary

m1 = m2 =

m3 = m4 =

m5 = m6 =

m7 = m8 =

k

j

j || k

8765

43

21



4. The angle pairs that we just identified have special names that are related to their position

formed by the two parallel lines and the transversal.

Use the Vocabulary Bank at the bottom to fill in the blanks with the vocabulary word that we use

to identify these angles, based on their definitions, and draw an example of the angle pair in the

space provided. (#VOC) Be sure to include each in your Vocabulary Booklet.

__________________________ are a pair of non-adjacent angles, one interior and one exterior,

that both lie on the same side of the transversal.

Draw:

__________________________ are a pair of non-adjacent angles between parallel lines and on

opposite sides of the transversal.

Draw:

__________________________ are a pair of non-adjacent angles outside of the parallel lines

and on opposite sides of the transversal.

Draw:

__________________________ are a pair of angles between parallel lines and on the same side

of the transversal.

Draw:

__________________________ are a pair of angles outside of the parallel lines and on the same

side of the transversal.

Draw:

Vocabulary Bank (#VOC):

Alternate exterior angles Corresponding angles Same-side interior angles

Alternate interior angles Same-side exterior angles



We just learned that a pair of parallel lines cut by a transversal will give special angle pairs that

are either complementary or supplementary. Now we can determine if two lines cut by a

transversal are parallel, based on measurement.

Determine if the following pairs of lines are parallel, non-parallel, or cannot be determined with

the given information.

A. B.

C. D.

Ticket to continue: Explain how you can determine whether two seemingly parallel lines are

indeed parallel, and which are not.

L4 – Special Angles and Lines Name __________________________________ 4.5 The Parallel Postulate Per _____ Date _________________________


There are five foundational postulates of Euclidean Geometry (the kind of geometry that we are

learning about that deals with lines and points in a plane). Each one is presented below. You will

need to include each/all five in your Theorem Booklet. Let’s investigate each of these postulates.

Write about, draw an example of, and/or show what you can discover about each of the five

postulates. Your demonstration can be in the form of explanation, examples, and/or non-

examples that illustrate each postulate. Consider if the postulates would still be true if instead of

locating the points on a plane they were located on a sphere, such as Earth (assuming it was a

perfect sphere.)

1. A straight line segment can be drawn joining any two points. (#THM)

2. Any straight line segment can be extended indefinitely in a straight line. (#THM) Note:

this implies that a line segment can be extended to any finite length.

3. Given any straight line segment, a circle can be drawn having the segment as radius and

one endpoint as center. (#THM)

4. All right angles are congruent. (#THM)

5. If a straight line falling on two straight lines make the interior angles on the same side

less than two right angles, the two straight lines, if produced indefinitely, meet on that

side on which the angles are less than the two right angles. (#THM)

This is equivalent to: Given any straight line and a point not on it, there exists one and

only one straight line which passes through that point and does not intersect the first line.

L4 – Special Angles and Lines Name __________________________________ 4.7 Mine Match Per _____ Date _________________________


With colored markers or highlighters, shade the following information on the house plan above:

1. With blue, color four right angles.

2. With red/pink, color three pairs of parallel lines (be sure to add arrows/tick marks to

denote unique pairs).

3. With green, color three pairs of adjacent angles.

4. With yellow, color one pair of vertical angles.

5. With purple, color a pair of supplementary angles.

6. Why is it impossible to find a pair of complementary angles in the house plan?

Grandmas’ House

Plans Master Bath Dining

Room

Bath Kitchen

Short

Hallway Storage

Master

Bedroom

Main Hallway

Living

Room

Bedroom

Porch

Ramp

L4 – Special Angles and Lines Name __________________________________ 4.7 Mine Match Per _____ Date _________________________


Using the two diagrams below, you and your partner will take turns calling out an example of

one of the vocabulary terms listed in the bank below. The other partner needs to determine

which vocabulary term describes the example. Keep taking turns until time runs out or you

finish the word bank.

Word Bank:

Alternate exterior angles Linear Pair Same-side exterior angles

Alternate interior angles Parallel Lines Same-side exterior angles

Complementary angles Perpendicular Lines Straight angle

Corresponding angles Right angle Supplementary angles

Reflections: Which terms were the most difficult to identify? For which vocabulary word(s)

were you only able to find a single example?

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