Geometry Notes
Name ______________________________
2.7 Proving Lines are Parallel
Date ________________ Period ______
Objectives
Use the angles formed by a transversal to prove two lines are parallel.
THEOREM
HYPOTHESIS
Converse of the Corresponding Angles Postulate
CONCLUSION
𝑚∥𝑛
corresp. ∠𝑠 ≅ → 2 𝑙𝑖𝑛𝑒𝑠 ∥
Converse of the Alternate Interior Angles Postulate
𝑚∥𝑛
Alt. Int. ∠𝑠 ≅ → 2 𝑙𝑖𝑛𝑒𝑠 ∥
Converse of the Alternate Exterior Angles Postulate
𝑚∥𝑛
Alt. Ext. ∠𝑠 ≅ →2 𝑙𝑖𝑛𝑒𝑠 ∥
Converse of the Same-Side Interior Angles Postulate
Same Side Int. ∠𝑠 𝑠𝑢𝑝𝑝. →2 𝑙𝑖𝑛𝑒𝑠 ∥
Example 1
Use the given information and the theorems you have learned to show that ℓ || m.
4  8
Example 2
Use the given information and the theorems you have learned to show that r || s.
4  8
𝑚∥𝑛
Example 3
Use the given information and the theorems you have learned to show that ℓ || m.
m1 = m3
Example 4
Use the given information and the theorems you have learned to show that r || s.
m2 = 58° & m3 = 122°
Example 5
Find the value of x that makes l ‖ m.
m2 = (20x + 12)° & m7 = (25x – 3)°
X = _________
What rule supports this conclusion? _________________________________
Example 6
Find the value of x that makes l ‖ m.
m3 = (4x – 80)° & m5 = (3x + 50)°
X = _________
What rule supports this conclusion? _________________________________
Example 7
Use the diagram and the given information to determine which lines, if any, are parallel. Give the Theorem that
supports your answer.
a.
b.
c.
d.
e.
f.
∠2 ≅ ∠10, so ___________, by ______________________________
∠15 ≅ ∠10, so ___________, by _____________________________
∠15 ≅ ∠4, so ___________, by _________________________________
∠6 ≅ ∠11, so ___________, by _________________________________
𝑚∠6 + ∠7 + 180°, so _________, by ___________________________
∠9 ≅ ∠14, so ___________, by _________________________________
Example 8
Given: ℓ || m, 1  3
Prove: p || r
Statements
Reasons
Example 9
Given: 1  4, 3 and 4 are supplementary.
Prove: ℓ || m
Statements
Reasons