4.1 Angles Formed by Intersecting Lines

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Name
Class
4.1
Date
Angles Formed by Intersecting
Lines
Essential Question: How can you find the measures of angles formed by intersecting lines?
Explore 1
Resource
Locker
Exploring Angle Pairs Formed by Intersecting Lines
When two lines intersect, like the blades of a pair of scissors, a number of angle pairs are formed.
You can find relationships between the measures of the angles in each pair.
A
Using a straightedge, draw a pair of intersecting lines like the open scissors. Label the angles
formed as 1, 2, 3, and 4.
Possible answer:
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4
B
1
3
2
Use a protractor to find each measure.
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Angle
Measure of Angle
m∠1
120°
m∠2
60°
m∠3
120°
m∠4
60°
m∠1 + m∠2
180°
m∠2 + m∠3
180°
m∠3 + m∠4
180°
m∠1 + m∠4
180°
163
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You have been measuring vertical angles and linear pairs of angles. When two lines intersect, the angles that are
opposite each other are vertical angles. Recall that a linear pair is a pair of adjacent angles whose non-common sides
are opposite rays. So, when two lines intersect, the angles that are on the same side of a line form a linear pair.
Reflect
1.
Name a pair of vertical angles and a linear pair of angles in your diagram in Step A.
Possible answers: vertical angles: ∠1 and ∠3 or ∠2 and ∠4 ; linear pairs:
∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, or ∠1 and ∠4
2.
Make a conjecture about the measures of a pair of vertical angles.
Vertical angles have the same measure and so are congruent.
3.
Use the Linear Pair Theorem to tell what you know about the measures of angles that form a linear pair.
The angles in a linear pair are supplementary, so the measures of the angles add
up to 180°.
Explore 2
Proving the Vertical Angles Theorem
The conjecture from the Explore about vertical angles can be proven so it can be stated as a theorem.
The Vertical Angles Theorem
If two angles are vertical angles, then the angles are congruent.
4
1
3
2
∠1 ≅ ∠3 and ∠2 ≅ ∠4
Follow the steps to write a Plan for Proof and a flow proof to prove the Vertical Angles
Theorem.
Given: ∠1 and ∠3 are vertical angles.
Prove: ∠1 ≅ ∠3
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4
1
3
2
164
© Houghton Mifflin Harcourt Publishing Company
You have written proofs in two-column and paragraph proof formats. Another type of proof is called aflow proof. A
flow proof uses boxes and arrows to show the structure of the proof. The steps in a flow proof move from left to right
or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the
box. You can use a flow proof to prove the Vertical Angles Theorem.
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A
Complete the final steps of a Plan for Proof:
Because ∠1 and ∠2 are a linear pair and ∠2 and ∠3 are a linear pair, these pairs of angles
are supplementary. This means that m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180°. By the
Transitive Property, m∠1 + m∠2 = m∠2 + m∠3. Next:
Subtract m∠2 from both sides to conclude that m∠1 = m∠3. So, ∠1 ≅ ∠3.
B
Use the Plan for Proof to complete the flow proof. Begin with what you know is true from
the Given or the diagram. Use arrows to show the path of the reasoning. Fill in the missing
statement or reason in each step.
∠1 and ∠2 are a
linear pair.
Given
(see diagram)
∠1 and ∠2 are
supplementary .
Linear Pair
Theorem
m∠1 + m∠2 = 180°
Def. of
supplementary angles
∠1 and ∠3 are
vertical angles.
Given
∠2 and ∠3 are a
linear pair.
Given
(see diagram)
∠1 ≅ ∠3
© Houghton Mifflin Harcourt Publishing Company
Def. of
congruence
∠2 and ∠3 are
supplementary.
Linear Pair
Theorem
m∠2 + m∠3 = 180°
Def. of
supplementary angles
m∠1 = m∠3
m∠1 + m∠2 = m∠2 + m∠3
Subtraction
Property of
Equality
Transitive Property
of Equality
Reflect
4.
Discussion Using the other pair of angles in the diagram, ∠2 and ∠4, would a proof that ∠2 ≅ ∠4 also
show that the Vertical Angles Theorem is true? Explain why or why not.
Yes, it does not matter which pair of vertical angles is used in the proof. Similar statements
and reasons could be used for either pair of vertical angles.
5.
Draw two intersecting lines to form vertical angles. Label your lines and tell which angles are congruent.
Measure the angles to check that they are congruent.
A
D
E
C
Possible answer:
B
By the Vertical Angles Theorem, ∠AEC ≅ ∠DEB and ∠AED ≅ ∠CEB. Checking by
measuring, m∠AEC = m∠DEB = 45° and m∠AED = m∠CEB = 135°.
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Using Vertical Angles
Explain 1
You can use the Vertical Angles Theorem to find missing angle measures in situations involving intersecting lines.
Example 1 Cross braces help keep the deck posts straight. Find the measure of
each angle.
146°
5

7
∠6
Because vertical angles are congruent, m

6
∠5 and ∠7
From Part A, m
∠6 = 146°.
∠6 = 146°. Because ∠5 and ∠6 form a linear pair , they are
supplementary and m
∠5 = 180° - 146° = 34° . m∠
also forms a linear pair with
∠6, or because it is a
7
= 34° because ∠ 7
vertical angle
with ∠5.
Your Turn
6.
The measures of two vertical angles are 58° and (3x + 4) °. Find the value ofx.
58 = 3x + 4
18 = x
7.
The measures of two vertical angles are given by the expressions (x + 3) ° and(2x - 7) °. Find the value
of x. What is the measure of each angle?
x + 3 = 2x - 7
x + 10 = 2x
10 = x
The measure of each angle is (x + 3)° = (10 + 3)° = 13°.
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54 = 3x
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Using Supplementary and Complementary Angles
Explain 2
Recall what you know about complementary and supplementary angles. Complementary angles are two angles
whose measures have a sum of 90°. Supplementary angles are two angles whose measures have a sum of 180°. You
have seen that two angles that form a linear pair are supplementary.
Example 2 Use the diagram below to find the missing angle measures. Explain your
reasoning.
C
B
D
50°
F
A
 Find the measures of
E
∠AFC and ∠AFB.
∠AFC and ∠CFD are a linear pair formed by an intersecting line and ray,
‹ ›
→
−
‾ , so they are supplementary and the sum of their measures is 180°. By the
AD and FC
diagram, m∠CFD = 90°, so m∠AFC = 180° - 90° = 90° and ∠AFC is also a right angle.
Because together they form the right angle
∠AFC, ∠AFB and ∠BFC are complementary and
the sum of their measures is 90°. So, m∠AFB = 90° - m∠BFC = 90° - 50° = 40°.
 Find the measures of
∠DFE and ∠AFE.
∠BFA and ∠DFE are formed by two
so the angles are
vertical
intersecting lines
and are opposite each other,
angles. So, the angles are congruent. From Part A
m∠AFB = 40°, so m∠DFE = 40° also.
© Houghton Mifflin Harcourt Publishing Company
Because ∠BFA and ∠AFE form a linear pair, the angles are supplementary and the sum
of their measures is
180° . So, m∠AFE = 180° - m∠BFA = 180° - 40° = 140° .
Reflect
8.
In Part A, what do you notice about right angles ∠AFC and ∠CFD? Make a conjecture about right angles.
Possible answer: Both angles have measure 90°. Conjecture: All right angles are
congruent.
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Your Turn
You can represent the measures of an angle and its complement as x° and (90 - x)°.
Similarly, you can represent the measures of an angle and its supplement as x° and
(180 - x)°. Use these expressions to find the measures of the angles described.
9.
The measure of an angle is equal to
the measure of its complement.
10. The measure of an angle is twice
the measure of its supplement.
x = 2(180 - x)
x = 90 - x
2x = 90
x = 360 - 2x
x = 45; so, 90 - x = 45
3x = 360
x = 120; so, 180 - x = 60
The measure of the angle is 45°
the measure of its complement is 45°.
The measure of the angle is 120°
the measure of its supplement is 60°.
Elaborate
11. Describe how proving a theorem is different than solving a problem and describe how they are the same.
Possible answer: A theorem is a general statement and can be used to justify steps in solving
problems, which are specific cases of algebraic and geometric relationships. Both proofs and
problem solving use a logical sequence of steps to justify a conclusion or to find a solution.
12. Discussion The proof of the Vertical Angles Theorem in the lesson includes a Plan for Proof. How are a
Plan for Proof and the proof itself the same and how are they different?
Possible answer: A plan for a proof is less formal than a proof. Both start from the given
information and reach the final conclusion, but a formal proof presents every logical step
in detail, while a plan describes only the key logical steps.
13. Draw two intersecting lines. Label points on the lines and tell what angles you
know are congruent and which are supplementary.
Possible answer:
K
L
J
H
Vertical angles are congruent: ∠GLK ≅ ∠JLH and ∠GLJ ≅ ∠KLH
Angles in a linear pair are supplementary: ∠GLJ and ∠GLK; ∠GLJ and ∠JLH; ∠JLH and
∠HLK; and ∠HLK and ∠GLK
© Houghton Mifflin Harcourt Publishing Company
G
14. Essential Question Check-In If you know that the measure of one angle in a linear pair is 75°, how can
you find the measure of the other angle?
The angles in a linear pair are supplementary, so subtract the known measure from 180°:
180 - 75 = 105, so the measure of the other angle is 105°.
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