Take Note: Theorem 3-8

Theorem

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

If …

m ⊥ t m up tack t  and n ⊥ t n up tack t

Then …

m || n m vertical linevertical line n

Proof

Proof of Theorem 3-8

Given: In a plane, r ⊥ t r up tack t  and s ⊥ t . s up tack t .

Prove: r || s r vertical linevertical line s

Proof: ∠ 1 angle 1  and ∠ 2 angle 2  are right angles by the definition of perpendicular. So, ∠ 1 ≅ ∠ 2 . angle 1 approximately equal to angle 2 .  Since corresponding angles are congruent, r || s . r vertical linevertical line s .

Problem 1 Solving a Problem With Parallel Lines

Carpentry A carpenter plans to install molding on the sides and the top of a doorway. The carpenter cuts the ends of the top piece and one end of each of the side pieces at 45 ° 45 degrees  angles as shown. Will the side pieces of molding be parallel? Explain.

Yes, the sides are parallel. When the pieces fit together, they form 45 ° + 45 ° , 45 degrees , plus , 45 degrees comma  or 90 ° , 90 degrees comma  angles. So, each side is perpendicular to the top. If two lines (the sides) are perpendicular to the same line (the top), then they are parallel to each other.

✓ Got It?

1. Can you assemble the pieces below to form a picture frame with opposite sides parallel? Explain.