Download common core 3.2 gem

Name Class 3-2
Date Angles Formed by Parallel Lines
and Transversals
Going Deeper
Essential question: How can you prove and use theorems about angles formed by
transversals that intersect parallel lines?
1
G-CO.3.9
Engage
Introducing Transversals
Recall that a transversal is a line that intersects two coplanar
lines at two different points. In the figure, line t is a transversal.
The table summarizes the names of angle pairs formed by
a transversal.
t
1
2
p
4 3
5 6
8
q
7
Angle Pair
Example
Corresponding angles lie on the same side of the transversal and on
the same sides of the intersected lines.
∠1 and ∠5
Same-side interior angles lie on the same side of the transversal and
between the intersected lines.
∠3 and ∠6
Alternate interior angles are nonadjacent angles that lie on opposite
sides of the transversal between the intersected lines.
∠3 and ∠5
Alternate exterior angles are angles that lie on opposite sides of the
transversal outside the intersected lines.
∠2 and ∠8
© Houghton Mifflin Harcourt Publishing Company
The following postulate is the starting point for proving theorems about parallel lines that
are intersected by a transversal.
Same-Side Interior Angles Postulate
t
If two parallel lines are cut by a transversal, then the pairs
of same-side interior angles are supplementary.
1
3
4
Given p ǁ q, ∠4 and ∠5 are supplementary.
Given p ǁ q, ∠3 and ∠6 are supplementary.
5
8
2
6
7
p
q
REFLECT
1a. Explain how you can find m∠3 in the postulate diagram if p ǁ q and m∠6 = 61°.
1b. In the postulate diagram, suppose p ǁ q and line t is perpendicular to line p.
Can you conclude that line t is perpendicular to line q ? Explain.
Chapter 3
95
Lesson 2
2
G-CO.3.9
proof
Alternate Interior Angles Theorem
t
If two parallel lines are cut by a transversal, then the
pairs of alternate interior angles have the same measure.
1
4
5
Given: p ǁ q
Prove: m∠3 = m∠5
8
2
p
3
6
q
7
Complete the proof by writing the missing reasons. Choose from the following reasons.
You may use a reason more than once.
Same-Side Interior Angles Postulate
Subtraction Property of Equality
Given
Definition of supplementary angles
Substitution Property of Equality
Statements
Linear Pair Theorem
Reasons
1. p ǁ q
1.
2. ∠3 and ∠6 are supplementary.
2.
3. m∠3 + m∠6 = 180°
3.
4. ∠5 and ∠6 are a linear pair.
4.
5. ∠5 and ∠6 are supplementary.
5.
6. m∠5 + m∠6 = 180°
6.
7. m∠3 + m∠6 = m∠5 + m∠6
7.
8. m∠3 = m∠5
8.
© Houghton Mifflin Harcourt Publishing Company
REFLECT
2a. Suppose m∠4 = 57° in the above figure. Describe two different ways to
determine m∠6.
2b. In the above figure, explain why ∠1, ∠3, ∠5, and ∠7 all have the same measure.
2c. In the above figure, is it possible for all eight angles to have the same measure?
If so, what is that measure?
Chapter 3
96
Lesson 2
3
G-CO.3.9
proof
Corresponding Angles Theorem
t
If two parallel lines are cut by a transversal, then the
pairs of corresponding angles have the same measure.
1
4
Given: p ǁ q
Prove: m∠1 = m∠5
2
p
3
5 6
8 7
q
Complete the proof by writing the missing reasons.
Statements
Reasons
1. p ǁ q
1.
2. m∠3 = m∠5
2.
3. m∠1 = m∠3
3.
4. m∠1 = m∠5
4.
REFLECT
© Houghton Mifflin Harcourt Publishing Company
3a. Explain how you can you prove the Corresponding Angles Theorem using the
Same-Side Interior Angles Postulate and a linear pair of angles.
Many postulates and theorems are written in the form “If p, then q.” The converse of
such a statement has the form “If q, then p.” The converse of a postulate or theorem may
or may not be true. The converse of the Same-Side Interior Angles Postulate is accepted
as true, and this makes it possible to prove that the converses of the previous theorems
are true.
Converse of the Same-Side Interior Angles Postulate
If two lines are cut by a transversal so that a pair of same-side interior
angles are supplementary, then the lines are parallel.
Converse of the Alternate Interior Angles Theorem
If two lines are cut by a transversal so that a pair of alternate interior
angles have the same measure, then the lines are parallel.
Converse of the Corresponding Angles Theorem
If two lines are cut by a transversal so that a pair of corresponding
angles have the same measure, then the lines are parallel.
Chapter 3
97
Lesson 2
A paragraph proof is another way of presenting a mathematical argument. As in a
two-column proof, the argument must flow logically and every statement should have a reason.
4
G-CO.3.9
proof
Equal-Measure Linear Pair Theorem
If two intersecting lines form a linear pair of angles with equal measures, then the lines are perpendicular.
1
2
Given: m∠1 = m∠2
Prove: ℓ ⊥ m
m
Complete the following paragraph proof.
It is given that ∠1 and ∠2 form a linear pair. Therefore, ∠1 and ∠2 are supplementary
by the
. By the definition of supplementary angles,
m∠1 + m∠2 = 180°. It is also given that m∠1 = m∠2. So, m∠1 + m∠1 = 180° by the
. Simplifying gives 2m∠1 = 180° and m∠1 = 90°
by the Division Property of Equality. Therefore, ∠1 is a right angle and ℓ ⊥ m by the
.
REFLECT
4a. State the converse of the Equal-Measure Linear Pair Theorem shown above.
Is the converse true?
In Exercises 1−2, complete each proof by writing the missing statements or reasons.
1. I f two parallel lines are cut by a transversal, then the
pairs of alternate exterior angles have the same measure.
t
1
4
iven: p ǁ q
G
Prove: m∠1 = m∠7
5
8
Statements
q
Reasons
1. p ǁ q
1.
2. m∠1 = m∠5
2.
3. m∠5 = m∠7
3.
4. m∠1 = m∠7
4.
Chapter 3
p
3
6
7
2
98
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
practice
2. P
rove the Converse of the Alternate Interior
Angles Theorem.
t
1
4
iven: m∠3 = m∠5
G
Prove: p ǁ q
5
8
Statements
p
6
7
q
Reasons
1. m∠3 = m∠5
1.
2. ∠5 and ∠6 are a linear pair.
2. Definition of linear pair
3.
3. Linear Pair Theorem
4. m∠5 + m∠6 = 180°
4.
5. m∠3 + m∠6 = 180°
5.
6. ∠3 and ∠6 are supplementary.
6.
7. p ǁ q
7.
m
3. Complete the paragraph proof.
p
1
iven: ℓ ǁ m and p ǁ q
G
Prove: m∠1 = m∠2
© Houghton Mifflin Harcourt Publishing Company
2
3
2
3
q
It is given that p ǁ q, so m∠1 = m∠3 by the
.
It is also given that ℓ ǁ m, so m∠3 = m∠2 by the
.
Therefore, m∠1 = m∠2 by the
.
4. T
he figure shows a given line m, a given point P, and the construction of a line ℓ
that is parallel to line m. Explain why line ℓ is parallel to line m.
P
m
Chapter 3
99
Lesson 2
5. Can you use the information in the figure to conclude that p ǁ q? Why or why not?
71˚
1
p
2 3
4 109˚
5
6
q
© Houghton Mifflin Harcourt Publishing Company
Chapter 3
100
Lesson 2
3-2
Name ________________________________________ Date __________________ Class__________________
Name LESSON
Practice
Class Date Angles Formed
by Parallel Lines and Transversals
Additional
Practice
3-2
Find each angle measure.
1. m∠1 _______________________
2. m∠2 _______________________
3. m∠ABC _______________________
4. m∠DEF _______________________
Complete the two-column proof to show that same-side exterior angles
are supplementary.
5. Given: p || q
Prove: m∠1 + m∠3 = 180°
Proof:
© Houghton Mifflin Harcourt Publishing Company
Statements
Reasons
1. p || q
1. Given
2. a. _______________________
2. Lin. Pair Thm.
3. ∠1 ≅ ∠2
3. b. _______________________
4. c. _______________________
4. Def. of ≅ ∠s
5. d. _______________________
5. e. _______________________
6. Ocean waves move in parallel lines toward the shore.
The figure shows Sandy Beaches windsurfing across
several waves. For this exercise, think of Sandy’s wake
as a line. m∠1 = (2x + 2y)° and m∠2 = (2x + y)°.
Find x and y.
x = _________
y = _________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Chapter 3
101
16
Lesson 2
Holt McDougal Geometry
Name ________________________________________ Date __________________ Class__________________
Problem
Solving
Problem Solving
LESSON
3-2
Angles Formed by Parallel Lines and Transversals
Find each value. Name the postulate or theorem that you used to find the values.
2. In the diagram, roads a and b are parallel.
1. In the diagram of movie theater seats,
the incline of the floor, f, is parallel to
the seats, s.
If m∠1 = 68°, what is x?
What is the measure of ∠PQR?
_________________________________________
________________________________________
3. In the diagram of the gate, the horizontal bars
are parallel and the vertical bars are parallel.
Find x and y.
_________________________________________
_________________________________________
_________________________________________
Use the diagram of a staircase railing for Exercises 4 and 5. AG || CJ and AD || FJ .
Choose the best answer.
© Houghton Mifflin Harcourt Publishing Company
4. Which is a true statement about the measure of ∠DCJ?
A It equals 30°, by the Alternate Interior Angles Theorem.
B It equals 30°, by the Corresponding Angles Postulate.
C It equals 50°, by the Alternate Interior Angles Theorem.
D It equals 50°, by the Corresponding Angles Postulate.
5. Which is a true statement about the value of n?
F It equals 25°, by the Alternate Interior Angles Theorem.
G It equals 25°, by the Same-Side Interior Angles Theorem.
H It equals 35°, by the Alternate Interior Angles Theorem.
J It equals 35°, by the Same-Side Interior Angles Theorem.
Chapter
3 Copyright © by Holt McDougal. Additions and changes102
Original
content
to the original content are the responsibility of the instructor.
100
Lesson 2
Holt McDougal Geometry