Chapter 12, Problem 16PTTS

To determine

To determine& calculate: The types of angles and the measurement of the angles: $m\angle 1$ , $m\angle 2$ , $m\angle 4$ , $m\angle 5$ , $m\angle 6$ , $m\angle 7$ and $m\angle 8$

Answer to Problem 16PTTS

The measurement of the angles are, $m\angle 1={56}^{\circ }$ , $m\angle 2={124}^{\circ }$ , $m\angle 4={56}^{\circ }$ , $m\angle 5={56}^{\circ }$ , $m\angle 6={124}^{\circ }$ , $m\angle 7={124}^{\circ }$ and $m\angle 8={56}^{\circ }$ .

Thus, congruent angles are: $m\angle 3\cong m\angle 2\cong m\angle 6\cong m\angle 7={124}^{\circ }$ .

There are no complementary angles of $\angle 3$ and supplementary angles of $\angle 3$ are $\angle 1,\angle 4,\angle 5$ and $\angle 8$ .

Explanation of Solution

Given information : $m\angle ={124}^{\circ }$

Concept used: If the sum of two angles right angle or ${90}^{\circ }$ is known as complementary angle, and the sum of two angles are ${180}^{\circ }$ then it known as supplementary angles.

Congruent angles are two or more angles that have the same measurement.

$\angle 1+\angle 2={180}^{\circ }$ (Linear pair angles) , $\angle 1=\angle 4$ (vertically opposite angle), $\angle 2=\angle 3$ (vertically opposite angle), $\angle 5=\angle 8$ (vertically opposite angles) , $\angle 6=\angle 7$ (Vertically opposite angles), $\angle 1=\angle 5$ (Corresponding angles), $\angle 5+\angle 6={180}^{\circ }$ (Linear pair angles) and $\angle 4=\angle 8$ (Corresponding angles).

Calculation :

  $m\angle 3=m\angle 2={124}^{\circ }$ (Vertically opposite angles)

  $\begin{array}{l}m\angle 2+m\angle 1={180}^{\circ }\\ \Rightarrow {124}^{\circ }+m\angle 1={180}^{\circ }\\ \Rightarrow m\angle 1={180}^{\circ }-{124}^{\circ }\\ \Rightarrow m\angle 1={56}^{\circ }\end{array}$

  $m\angle 1=m\angle 4={56}^{\circ }$ (Vertically opposite angles)

  $m\angle 1=m\angle 5={56}^{\circ }$ (Corresponding angles)

  $m\angle 5=m\angle 8={56}^{\circ }$ (Vertically opposite angles)

  $\begin{array}{l}m\angle 5+m\angle 6={180}^{\circ }\\ \Rightarrow {56}^{\circ }+m\angle 6={180}^{\circ }\\ \Rightarrow m\angle 6={180}^{\circ }-{56}^{\circ }\\ \Rightarrow m\angle 6={124}^{\circ }\end{array}$

$m\angle 6=m\angle 7={124}^{\circ }$ (Vertically opposite angles)

So, congruent angles are: $m\angle 3\cong m\angle 2\cong m\angle 6\cong m\angle 7={124}^{\circ }$ .

There are no complementary angles of $\angle 3$ and supplementary angles of $\angle 3$ are $\angle 1,\angle 4,\angle 5$ and $\angle 8$ .