Chapter 12.1, Problem 31E

To determine

To write:The expressions for $m\angle 1,m\angle 2,m\angle 3$ in terms of $x$ .

Answer to Problem 31E

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

Explanation of Solution

Given: Angle measures $m\angle 1,m\angle 2,m\angle 3,x^{\circ }$

Calculation:

Since, supplementary angles are those whose sum of the angles is ${180}^{\circ }$ . That is $m\angle 1+m\angle 2={180}^{\circ }$ ……………. (1)

and $x^{\circ }+m\angle 3={180}^{\circ }$ ……………..(2)

Equate equation (1) and (2),

  $m\angle 1+m\angle 2$

  $=$

  $x^{\circ }+m\angle 3$ …………….. (3)

  $\text{But }m\angle 1=m\angle 3$ , since vertically opposite angles have equal measure

Therefore, the equation becomes

  $\Rightarrow m\angle 2=x^{\circ }$

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.