Geometry22
Name: ____________________________________________________ Per: ______ Date: ____________
3-2 Properties of Parallel Lines
Target: To prove theorems about parallel lines.
To use properties of parallel lines to find angle measures.
Postulate 3-1 Same-Side Interior Angles Postulate
Postulate
If…
Then…
If a transversal intersects two
parallel lines, then same-side
interior angles are
supplementary.
4 and 5 are supplementary
3 and 6 are supplementary
We can use this postulate to PROVE the other angle pair THEOREMS.
ALL of these theorems are saying:
IF the lines are PARALLEL, THEN the special angle relationships are true.
Theorem
Theorem 3-1 Alternate Interior Angles Theorem
If…
If a transversal intersects two
parallel lines, then alternate
interior angles are congruent.
Theorem
Theorem 3-2 Corresponding Angles Theorem
If…
If a transversal intersects two
parallel lines, then
corresponding angles are
congruent.
Theorem
Theorem 3-3 Alternate Exterior Angles Theorem
If…
If a transversal intersects two
parallel lines, then alternate
exterior angles are
congruent.
Then…
4  6
3  5
Then…
1  5
2  6
3  7
4  8
Then…
1  7
2  8
Use the diagram below to write a 2 column proof of the Alternate Interior Angles Theorem:
Given: l m
Prove: 4  6
STATEMENTS
REASONS
1.
1.
Given
2. 3 and 6 are same-side interior angles
2.
4. 3 and 6 are supplementary angles
4.
5.
5. Definition of Supplementary Angles
6. 4 and 3 form a linear pair
6.
7.
7. Linear Pair Postulate
8.
8.
9. m4  m3  m3  m6
9.
10.
10. Subtraction Property of Equality
11.
11.
Use the diagram below to write a 2 column proof of the Corresponding Angles Theorem:
Given: l m
Prove: 6  2
STATEMENTS
REASONS
1.
1.
Given
2. 3 and 6 are same-side interior angles
2.
3. 3 and 6 are supplementary angles
3.
4. 2 and 3 form a linear pair
4.
5.
5. Linear Pair Postulate
6. 6  2
6.
Use the diagram below to write a 2 column proof of the Alternate Exterior Angles Theorem:
Prove: 1  7
Given: a b
STATEMENTS
REASONS
1.
1.
Given
2. 1 and 5 are corresponding angles
2.
3. 1  5
3.
4. 5 and 7 are vertical angles
4.
5.
5. Vertical Angles Theorem
6.
6. Transitive Property of Congruence
Now we can use these theorems as REASONS in proofs dealing with
parallel lines…
Prove: 1 and 8 are supplementary
Given: a b
STATEMENTS
REASONS
1.
1. Given
2.
2. Definition of corresponding angles
3.
3. Corresponding Angles Theorem
4. m1  m5
4.
5. 5 and 8 form a linear pair
5.
6.
6.
7.
7. Definition of supplementary angles
8. m1 m8  180
8.
9.
9.
Notice: these arrows
…Or to find the missing angle measures in a diagram…
mean the 2 lines are
parallel to each
Find the measure of each numbered angle. Justify each answer with a theorem or postulate.
other.
m7 =_______ because of ____________________theorem
m6 =_______ because it is ____________________angle to 7
m8 = ______ because it is ____________________angle to 6
m4 = ______ because it is ____________________angle to 8
m5 = ______ because it is ____________________angle to 8
m3 = ______ because it is ____________________angle to 5
m2 = ______ because it is ____________________angle to 5
m1 = ______ because it is ____________________angle to 2
Directions: Refer to the figure to the right and answer the following questions:
1) If 𝑚∠5 = 102, find 𝑚∠8.
8
7
2) If 𝑚∠1 = 75, find 𝑚∠2.
3) If 𝑚∠1 = 75, find 𝑚∠4.
2
4
3
1
5
4) If 𝑚∠3 = (3𝑥 − 5) and the 𝑚∠2 = (4𝑥 − 29), find 𝑥.
5) If 𝑚∠1 = (3𝑥 + 10) and the 𝑚∠2 = (4𝑥 − 5), find 𝑥.
6