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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-2 Properties
of Parallel Lines
3-2 Properties of Parallel Lines
Teks Focus
TEKS (6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a
transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.
TEKS (1)(G) Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Additional TEKS (1)(F), (5)(A)
Vocabulary
• Justify – explain with logical reasoning. You can justify a mathematical argument.
• Argument – a set of statements put forth to show the truth or falsehood of a mathematical claim
ESSENTIAL UNDERSTANDING
The special angle pairs formed by parallel lines and a transversal are congruent, supplementary, or both.
take note
Postulate 3-1 Same-Side Interior Angles Postulate
Postulate
If …
Then …
ℓǁm
m∠4 + m∠5 = 180
m∠3 + m∠6 = 180
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
take note
Theorem 3-1 Alternate Interior Angles Theorem
Theorem
If …
Then …
ℓǁm
∠4 ≅ ∠6
∠3 ≅ ∠5
If a transversal intersects two parallel lines, then alternate interior angles are congruent.
For a proof of Theorem 3-1, see Problem 3.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-2 Properties
of Parallel Lines
take note
Theorem 3-2 Corresponding Angles Theorem
Theorem
If …
ℓǁm
Then …
∠1 ≅
∠5
∠2 ≅
∠6
∠3 ≅
∠7
∠4 ≅
∠8
If a transversal intersects two parallel lines, then corresponding angles are
congruent.
You will prove Theorem 3-2 in Exercise 16.
Theorem 3-3 Alternate Exterior Angles Theorem
Theorem
If …
Then …
ℓǁm
If a transversal intersects two parallel lines, then alternate exterior angles are
congruent.
∠1 ≅
∠7
∠2 ≅
∠8
You will prove Theorem 3-3 in the Got It for Problem 4.
Problem 1
Investigating Patterns in Geometric Relationships
A Choose from a variety of tools (such as a compass, a protractor, or a ruler) to investigate the relationships
among angles formed by parallel lines cut by a transversal. Explain your choice.
Use the lines on a sheet of lined paper to trace out two parallel lines. Draw a point on each line. Then draw a transversal
through these two points. Label the angles as shown.
A protractor can help you investigate relationships because you can use a protractor to measure angles.
Plan
What information will help
you observe relationships
among the angles?
You can observe relationships
based on the measurements of
the angles.
B Measure the eight angles. Observe any patterns. Based on your measurements, make two conjectures about how
certain angles are related.
m∠1 = m∠4 = m∠5 = m∠8, so ∠1 ≅ ∠4 ≅ ∠5 ≅ ∠8.
m∠2 = m∠3 = m∠6 = m∠7, so ∠2 ≅ ∠3 ≅ ∠6 ≅ ∠7.
Observe that ∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ ∠7, and ∠4 ≅ ∠8.
Conjecture 1: Corresponding angles are congruent.
Observe that m∠3 + m∠5 = 180 and m∠4 + m∠6 = 180.
Conjecture 2: Same-side interior angles are supplementary.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-2 Properties
of Parallel Lines
Problem 2
TEKS Process Standard (1)(G)
Identifying Supplementary Angles
The measure of ∠3 is 55. Which angles are supplementary to ∠3? How do you know?
Think
How do you find angles
supplementary to a given
angle?
d
Look for angles whose measure
is the difference of 180 and the
given angle measure.
By definition, a straight angle measures 180.
If m∠a + m∠b = 180, then ∠a and ∠b are supplementary by the definition of supplementary angles.
180 − 55 = 125, so any angle x, where m∠x = 125, is supplementary to ∠3.
m∠4 = 125 by the definition of a straight angle.
m∠8 = 125 by the Same-Side Interior Angles Postulate.
m∠6 = m∠8 by the Vertical Angles Theorem, so m∠6 = 125.
m∠2 = m∠4 by the Vertical Angles Theorem, so m∠2 = 125.
Problem 3
Proof Proving the Alternate Interior Angles Theorem
Given: c ǁ d
Plan
Prove: ∠1 ≅ ∠7
Show that m∠6 + m∠7 = 180
and that m∠6 + m∠1 = 180. So
m∠6 + m∠7 = m∠6 + m∠1.
Subtract m∠6 from each side to
prove that ∠1 and ∠7 are
congruent.
Statements
Reasons
1) c ǁ d
1) Given
2) m∠6 + m∠7 = 180
2) Linear Pair Postulate
3) m∠6 + m∠1 = 180
3) Same-Side Interior Angles Postulate
4) m∠6 + m∠7 = m∠6 + m∠1 4) Transitive Property of Equality
5) m∠7 = m∠1
5) Subtraction Property of Equality
6) ∠7 ≅ ∠1
6) Definition of congruent angles
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-2 Properties
of Parallel Lines
Problem 4
Proof Proving an Angle Relationship
Given: a ǁ b
Prove: ∠1 and ∠8 are supplementary.
Know
•a∥b
From the diagram you know
• ∠1 and ∠5 are corresponding
• ∠5 and ∠8 form a linear pair
Need
∠1 and ∠8 are supplementary, or m∠1 + m∠8 = 180.
Plan
Show that ∠1 ≅ ∠5 and that m∠5 + m∠8 = 180. Then substitute m∠1 for m∠5 to prove that ∠1 and ∠8 are supplementary.
Statements
Reasons
1) a ǁ b
1) Given
2) ∠1 ≅ ∠5
2) If lines are ǁ, then corresp. ∠
s are ≅.
3) m∠1 = m∠5
3) Congruent ∠
s have equal measures.
4) ∠5 and ∠8 are supplementary. 4) ∠
s that form a linear pair are suppl.
5) m∠5 + m∠8 = 180
5) Def. of suppl. ∠
s
6) m∠1 + m∠8 = 180
6) Substitution Property
7) ∠1 and ∠8 are supplementary. 7) Def. of suppl. ∠
s
Problem 5
TEKS Process Standard (1)(F)
Finding Measures of Angles
What are the measures of ∠3 and ∠4? Which theorem or postulate justifies each answer?
Think
How do ∠3 and ∠4 relate to
the given 105° angle?
d
∠3 and the given angle are
alternate interior angles. ∠4
and the given angle are sameside interior angles.
Since p ǁ q, m∠3 = 105 by the Alternate Interior Angles Theorem.
Since ℓ ǁ m, m∠4 + 105 = 180 by the Same-Side Interior Angles Postulate. So m∠4 = 180 − 105 = 75.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-2 Properties
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Problem 6
TEKS Process Standard (1)(F)
Finding an Angle Measure
Algebra What is the value of y?
Think
What do you know from the
diagram?
You have one pair of parallel
lines. The 80° angle and the
angle formed by the 40° and y°
angles are same-side interior
angles.
By the Angle Addition Postulate, y + 40 is the measure of an interior angle.
(y + 40) + 80 = 180 Same-side interior ∠
s of ǁ lines are suppl.
y + 120 =180
Simplify.
y = 60
Subtract 120 from each side.
PRACTICE and APPLICATION EXERCISES
Identify all the numbered angles that are congruent to the given angle. Justify your answers.
Scan page for a Virtual
Nerd™ tutorial video.
1.
For additional
2.
support when completing your
homework, go to
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d
3.
d
4. Justify Mathematical Arguments (1)(G) Supply the missing reasons in the two-column proof.
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Given: a ǁ b, c ǁ d
Prove: ∠1 ≅ ∠3
d
Statements
Reasons
1) a ǁ b
1) Given
2) ∠3 and ∠2 are supplementary. 2) _?_
3) c ǁ d
3) Given
4) ∠1 and ∠2 are supplementary. 4) b. _?_
5) ∠1 ≅ ∠3
5) c. _?_
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-2 Properties
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Proof 5. Write a two-column proof for Exercise 4 that does not use ∠2.
Find m∠1 and m∠2. Justify each answer.
6.
7.
8.
Find the value of x. Then find the measure of each labeled angle.
9.
10.
11.
Find the values of the variables.
12.
13.
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d
14.
d
15. Analyze Mathematical Relationships (1)(F) People in ancient Rome played a game called terni lapilli. The exact rules of this game are not known. Etchings on
floors and walls in Rome suggest that the game required a grid of two intersecting pairs of parallel lines, similar to the modern game tick-tack-toe. The measure of one
of the angles formed by the intersecting lines is 90°. Find the measure of each of the other 15 angles. Justify your answers.
Proof 16. Write a two-column proof to prove the Corresponding Angles Theorem (Theorem 3-2).
Given: ℓ ǁ m
Prove: ∠2 ≅ ∠6
Proof 17. Write a two-column proof.
Given: a ǁ b, ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
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18. Apply Mathematics (1)(A) Campers often use a “bear bag” at night to avoid attracting animals to their food supply. In the bear bag system at the right, a camper
pulls one end of the rope to raise and lower the food bag.
a. Suppose a camper pulls the rope taut between the two parallel trees, as shown. What is m∠1?
b. Are ∠1 and the given angle alternate interior angles, same-side interior angles, or corresponding angles?
19. Create Representations to Communicate Mathematical Ideas (1)(E) Draw two parallel lines. Then draw a transversal so that pairs of same-side interior angles
have the same measure. Make a conjecture about the relationship between the transversal and the parallel lines.
20. Select Tools to Solve Problems (1)(C) In the diagram, two parallel lines are cut by a transversal.
a. Choose from a variety of tools (such as a ruler, a protractor, a compass, or geometry software) to investigate the relationships among the angles formed. Explain
your choice.
b. Make a conjecture about how same-side exterior angles are related. Explain your reasoning.
Use the diagram at the right for Exercises 21 and 22.
21. Analyze Mathematical Relationships (1)(F) Suppose the measures of ∠1 and ∠2 are in a 4:11 ratio. Find their measures. (Diagram is not to scale.)
22. Explain Mathematical Ideas (1)(G) The diagram contains contradictory information. What is it? Why is it contradictory?
TEXAS Test Practice
23. ∠1 and ∠2 are same-side interior angles formed by two parallel lines and a transversal. If m∠1 = 115, what is m∠2?
24. The measure of an angle is two times the measure of its complement. What is the measure of the angle?
25. ∠1 and ∠2 are vertical angles. If m∠1 = 4x and m∠2 = 56, what is the value of x?
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