Chapter 10.7, Problem 3PSA

a.

To determine

Find angle $\angle C$ .

Answer to Problem 3PSA

  $\angle C=70°$

Explanation of Solution

Given information:

  

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

Calculation:

It is given ,

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

If a quadrilateral is inscribed in a circle its opposite angles are supplementary.

  $\begin{array}{l}\angle A+\angle C=180°\\ 110°+\angle C=180°\\ \angle C=180°-110°\\ \angle C=70°\end{array}$

Hence, $\angle C=70°$

b.

To determine

Find chord $BC$ .

Answer to Problem 3PSA

  $\stackrel{}{\overbrace{BC}}=110°$

Explanation of Solution

Given information:

  

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

Calculation:

It is given ,

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

If the chord of a circle is congruent then corresponding arcs are congruent.

  $\begin{array}{l}\stackrel{}{\overbrace{BC}}=\stackrel{}{\overbrace{CD}}\left[\stackrel{}{\overbrace{arc}}\right]\\ \stackrel{}{\overbrace{BCD}}=2\left(m\angle A\right)\\ \stackrel{}{\overbrace{BC}}=\stackrel{}{\overbrace{CD}}=2\left(110°\right)\\ 2\stackrel{}{\overbrace{\left(BC\right)}}=2\left(110°\right)\\ \stackrel{}{\overbrace{BC}}=110°\end{array}$

Hence, $\stackrel{}{\overbrace{BC}}=110°$

c.

To determine

Find angle $\angle B$ .

Answer to Problem 3PSA

  $\angle B=85°$

Explanation of Solution

Given information:

  

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

Calculation:

It is given ,

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

If a quadrilateral is inscribed in a circle its opposite angles are supplementary.

  $\begin{array}{l}\angle D+\angle B=180°\\ 95°+\angle B=180°\\ \angle B=180°-95°\\ \angle B=85°\end{array}$

Hence, $\angle B=85°$

d.

To determine

Find chord $\stackrel{}{\overbrace{AB}}$ .

Answer to Problem 3PSA

  $\stackrel{}{\overbrace{AB}}=80°$

Explanation of Solution

Given information:

  

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

Calculation:

It is given ,

  $\angle A=110°,\overline{BC}\cong \overline{CD},\angle D=95°$

If the chord of a circle is congruent then corresponding arcs are congruent.

  $\begin{array}{l}\stackrel{}{\overbrace{ABC}}=2\left(m\angle D\right)\\ \stackrel{}{\overbrace{AB}}+\stackrel{}{\overbrace{BC}}=2\left(95°\right)\\ \stackrel{}{\overbrace{AB}}+110°=190°\\ \stackrel{}{\overbrace{AB}}=80°\end{array}$

Hence, $\stackrel{}{\overbrace{AB}}=80°$