Question

The figure below shows a straight line AB intersected by another straight line t:

A segment AB is intersected by a transversal labeled t. Angles 1 and 3 and 2 and 4 are vertically opposite angles formed by the transversal on the segment.

Write a paragraph to prove that the measure of angle 1 is equal to the measure of angle 3.

Answer 1

When two straight lines intersect, the pairs of nonadjacent angles in opposite posi-tions are known as vertical angles.

If a segment AB is intersected by a transversal labeled t, then ∠1 and ∠3 and ∠2 and ∠4 are vertically angles formed by the transversal t on the segment AB.

Angles ∠1 and ∠2 can be described as adjacent and supplementary angles, so

$m\angle 1+m\angle 2=180^{\circ}$.

Angles ∠3 and ∠2 can be also described as adjacent and supplementary angles, so

$m\angle 3+m\angle 2=180^{\circ}$.

Subtract from the first equation the second equation:

$m\angle 1+m\angle 2-(m\angle 3+m\angle 2)=180^{\circ}-180^{\circ},\\ m\angle 1-m\angle 3=0,\\ m\angle 1=m\angle 3$.

Similarly you can prove that $m\angle 2=m\angle 4$.