Chapter 4.6, Problem 32PPS

To determine

The measure of angle ABC.

Answer to Problem 32PPS

  $m\angle ABC=22$

Explanation of Solution

Given:

The given figure is:

  

Calculation:

According to the Isosceles Triangle Theorem, if the two sides of the triangle are congruent, then the angles opposite to these sides are also congruent.

Therefore,

  $\begin{array}{l}\angle CAD\cong \angle ACD;\\ m\angle CAD=m\angle ACD\end{array}$

Sum of all angles of a triangle is 180.

So,

  $\begin{array}{l}m\angle D+m\angle ACD+m\angle CAD=180\\ 92+m\angle CAD+m\angle CAD=180\\ 92+2\left(m\angle CAD\right)=180\\ 2\left(m\angle CAD\right)=180-92\\ 2\left(m\angle CAD\right)=88\\ m\angle CAD=\frac{88}{2}\\ m\angle CAD=44\end{array}$

Now, the measure of an exterior angle in a triangle is sum of the measures of two remote interior angles.

  $\begin{array}{l}m\angle ACB=m\angle CAD+m\angle D\\ m\angle ACB=44+92\\ m\angle ACB=136\end{array}$

As it is given, line AC is congruent to CB;

Therefore,

  $\begin{array}{l}\angle CAB\cong \angle ABC;\\ m\angle CAB=m\angle ABC\end{array}$

Sum of all angles of a triangle is 180.

So,

  $\begin{array}{l}m\angle ACB+m\angle ABC+m\angle CAB=180\\ 136+m\angle ABC+m\angle ABC=180\\ 2\left(m\angle ABC\right)=180-136\\ 2\left(m\angle ABC\right)=44\\ m\angle ABC=\frac{44}{2}\\ m\angle ABC=22\end{array}$