Question

In the diagram, use only what you know about supplementary angles to write expressions for $m\angle 1$, $m\angle 2$, and $m\angle 3$ in terms of x. Then use the expressions to show that the vertical angles have equal measures.

Short answer

When two lines intersect at one point, two pairs of vertically opposite angles are formed and they have equal measure.

Step-by-step solution

Step 1. Given Information.

The figure is shown below.

Step 2. Calculation.

Since,supplementaryangles are those whose sum of the angles is $180°$. That is,

$m\angle 1+m\angle 2=180°$ …… (1)

And,

$x{}^{\circ }+m\angle 3=180°$ …… (2)

Equate equation (1) and (2),

$m\angle 1+m\angle 2=x{}^{\circ }+m\angle 3$ …… (3)

But $m\angle 1=m\angle 3$, since vertically opposite angles have equal measure, therefore, the equation becomes,

$m\angle 2=x°$

And,

$m\angle 1=m\angle 3$

Thus, when two lines intersect at one point, two pairs of vertically opposite angles are formed and they have equal measure.