Chapter 9, Problem 11CT

To determine

To find: the value of $m\angle ABC$ .

Answer to Problem 11CT

The measure of angle is $m\angle ABC=100$ .

Explanation of Solution

Given information:

$\overline{DA}$ and $\overline{DB}$ are tangent to the circle.

If $\overline{AB}\cong \overline{BC}$ and $marcBC=80$

Calculation:

Consider the following diagram:

If the chords are congruent, the arcs made by them are also congruent.

If $\overline{AB}\cong \overline{BC}$ , then $arcAB\cong arcBC$

If $marcBC=80$ , then $marcAB=80$

Then,

  $\begin{array}{c}marcAC=360-\left(marcAB+marcBC\right)\\ =360-\left(80+80\right)\\ =200\end{array}$

The measure of inscribed angle is half the measure of its intercepted arc.

Thus,

  $\begin{array}{c}m\angle ABC=\frac{1}{2}marcAC\\ =\frac{1}{2}\cdot 200\\ =100\end{array}$

Therefore, the measure of angle is $m\angle ABC=100$ .