Chapter 9, Problem 12CT

To determine

To find: the value of angle BCA .

Answer to Problem 12CT

The angle $m\angle BCA$ is 35.

Explanation of Solution

Given:

The circle is shown below:

Sides DA and DB are tangents. Angle $m\angle D$ is 110.

Calculation:

Consider triangle ADB ,

  $\begin{array}{l}DB=DA\left(\text{Tangents}\right)\\ \therefore \angle DBA=\angle DAB\left(\text{opposite}\text{angles}\text{of}\text{equal}\text{sides}\right)\end{array}$

Hence,

  $\begin{array}{l}\angle D+\angle DBA+\angle DAB=180\left(\text{Sum}\text{of}\text{angles}\text{of}\text{triangle}\right)\\ 110+\angle DBA+\angle DBA=180\\ 2\angle DBA=70\\ \angle DBA=35\end{array}$

For circle with alternate segment as shown below:

Alternate segment theorem states that:

  $\begin{array}{l}\angle a=\angle b\\ \angle d=\angle c\end{array}$

For given circle,

Hence, according to Alternate segment theorem:

  $m\angle DBA=m\angle BCA=35$

Hence, the angle $m\angle BCA$ is 35.