Section 5.2Proving That Lines are

Parallel

By Jeremy Cummings, Tarek Khalil, and Jai Redkar

The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle.

The Exterior Angle Inequality Theorem

The Logic Behind This Theorem:∠1+∠2=180

∠2+∠3+∠4=180∠1+ ∠2 =∠2+∠3+∠4

∠1=∠3+∠4∠1>∠3∠1>∠4

The Exterior Angle Inequality Theorem Sample Problem

How this would be done:1. x <62 because of the exterior angle inequality theorem- 62°

is the exterior angle and x is the remote interior2. x > 0 because every angle in a triangle is greater than 03. So, the answer is 0<x<62

62°x

Find the retrictions on x.

The Exterior Angle Inequality Theorem Practice Problem

125°5x-525°

Write an inequality that states the restrictions on x:Do the problem and then continue to see work and answer.

Work and Answer25< 5x-5 <125

25<5x-5 5x-5<125 30<5x 5x<130 6<x x<26

6<x<26

If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel.

◦ Short Form: Alt. int. ∠' s ≅ => ∥ lines

Identifying Parallel Lines Theorem 31

Given: ∠ 1≅ ∠2 Prove: y∥z

If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel.

◦ Short Form: Alt. ext. ∠' s ≅ => ∥ lines

Identifying Parallel Lines Theorem 32

Given: ∠ 1≅ ∠2 Prove: y∥z

Identifying Parallel Lines Theorem 33

If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel.

◦ Short Form: Corr. ∠' s ≅ => ∥ lines

Given: ∠ 1≅ ∠2 Prove: y∥z

Identifying Parallel Lines Theorem 34

If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel.

◦ Short Form: Same side int. ∠' s suppl. =>∥lines

Given: ∠1 suppl ∠2Prove: y∥z

Identifying Parallel Lines Theorem 35 If two lines are cut by a transversal such that two exterior angles

on the same side of the transversal are supplementary, the lines are parallel.

◦ Short Form: Same side ext. ∠' s suppl. =>∥lines

Given: ∠1 suppl ∠2Prove: y∥z

If two coplanar lines are perpendicular to a third line, they are parallel.

Theorem 36

Given: x⊥z and y⊥zProve: x∥y

Transversal t cuts lines k and n. m ∠ 1 = (148 - 3x)° and m ∠ 2 = (5x + 10)°. Find the value of x that makes k ∥ n.

Sample Problem Dealing With Theorems

k n

t1 2

How to do this:1. In order for k ∥ n, ∠ 1 has to be suppl. to

∠ 2 because of the theorem “Same side int. ∠'s suppl. =>∥lines.”

2. So, m ∠ 1 = (148 - 3x)° + m ∠ 2 = (5x + 10)°=180 because suppl. angles =180°.

3. Through algebra, 148-3x+5x+10=180 2x+158=180 2x=22 x=11

Given: BD bisects∠ABC BC≅CDProve: CD∥BA

Statements Reasons1. Given2. Given

3. ∠ABD ≅ ∠ CBD 3. If a ray bisects an ∠, then it divides the ∠ into 2 ≅ ∠’s.4. ∠ CDB≅ ∠ CBD

5. ∠ CDB ≅ ∠ABD

5. Transitive6. Alt. int. ∠’s ≅ =>∥ lines

Practice ProblemWrite a 2- column proof and then continue to see the correct steps.

“9-1: Proving Lines Parallel.” Ekcsk12.org. Edwards-Knox Central School, n.d. Web. 18 Jan. 2011.

“Perpindicular and Parallel Lines.” edHelper.com. edHelper.com, n.d. Web. 18 Jan. 2011.

Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challange . New Edition. Evanston, Illinois:

McDougal, Littell and Company, 2004. 216-18. Print.

Works Cited