Chapter 10.3, Problem 15PSB

To determine

To Prove : $\angle A\cong \angle C$

Answer to Problem 15PSB

1

Explanation of Solution

Given information : $\text{Circle with centre Q, B is mid-point of }\stackrel{⌢}{\text{AC}}.$

  

  $\begin{array}{l}\text{As}\text{B is mid-point of }\stackrel{⌢}{\text{AC}}\\ \Rightarrow \stackrel{⌢}{BC}\cong \stackrel{⌢}{AB}\\ \Rightarrow \stackrel{⌢}{BC}\cong \stackrel{⌢}{AB}[\text{Equal arcs intercepts equal chords}]\\ \text{Now, In }\Delta \text{QCB and }\Delta \text{QAB}\\ QC\cong QA\text{[Radii of circle]}\\ \text{QB}\cong QB[\text{Common}]\\ CB\cong AB[\text{Proved above}]\\ bySSScongruency\\ \Delta QCB\cong \Delta QAB\\ \Rightarrow \angle A\cong \angle C\text{(by CPCT)}\end{array}$