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Prove two lines are parallel from 3-to-1 angle ratios

Equal alternate interior angles prove parallel lines

If a transversal cuts two lines and a pair of alternate interior angles are equal, the two lines are parallel. A supplementary angle pair with a $3 : 1$ ratio measures $13 5^{\circ}$ and $4 5^{\circ}$ .

Problem

A transversal meets lines $a$ and $b$ at points $A$ and $B$ . At $A$ , adjacent angles $A_{1}$ and $A_{2}$ form a straight line and $A_{1} = 3 A_{2}$ . At $B$ , adjacent angles $B_{1}$ and $B_{2}$ also form a straight line and $B_{1} = 3 B_{2}$ . Angles $A_{2}$ and $B_{2}$ are alternate interior angles. Prove that $a$ and $b$ are parallel.

Answer

a ∥ b

The alternate interior position is part of the given configuration; equal measures then prove parallelism.

Step-by-step solution

1. Find the small angle at A

Since $A_{1}$ and $A_{2}$ are supplementary, $A_{1} + A_{2} = 18 0^{\circ}$ . With $A_{1} = 3 A_{2}$ , we get $4 A_{2} = 18 0^{\circ}$ , so $A_{2} = 4 5^{\circ}$ .

2. Repeat at B

The same argument gives $B_{1} + B_{2} = 18 0^{\circ}$ and $B_{1} = 3 B_{2}$ . Thus $B_{2} = 4 5^{\circ}$ .

3. Apply the converse theorem

The alternate interior angles $A_{2}$ and $B_{2}$ are equal. By the converse of the alternate interior angles theorem, $a ∥ b$ .

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