Question

Short answer

(a)

The final answer is $35,70and70.$

(b)

The final answer is $3k\text{.}$

Step-by-step solution

a.Step 1. Given information:

Diameter figure is shown below.

$m\angle A=35$

Step 2. Concept used:

We use basic geometric and angle concept

Step 3. Applying the concept:

In $\Delta AOB$.

$OA=OB \therefore \left(\text{Radius}\right)$

If two sides are equal in triangle, then angle opposite to these sides are also equal.

$\begin{array}{l}m\angle B=m\angle A\\ m\angle B=n\end{array}$

And

$\begin{array}{l}m\angle B+m\angle A+\angle AOB=180\\ n+n+\angle AOB=180\\ \angle AOB=180-2n\end{array}$

Since, AOC is a straight line therefore,

$\begin{array}{l}\angle AOB+\angle BOC=180\\ \left(180-2n\right)+\angle BOC=180\\ \angle BOC=2n\end{array}$

Measure of arc BC is the central angle.

$\begin{array}{l}m\stackrel{⏜}{BC}=m\angle BOC\\ m\stackrel{⏜}{BC}=2n\end{array}$

Therefore, the answer is$\left(180-2n\right),2n\text{and}2n$.

b.Step 1. Given information:

$m\stackrel{⏜}{BC}=6k$

Step 2. Concept used:

We use basic geometric and angle concept

Step 3. Applying the concept:

Measure of arc BC is the central angle.

$\begin{array}{l}m\stackrel{⏜}{BC}=m\angle BOC\\ m\angle BOC=6k\end{array}$

Since, AOC is a straight line therefore,

$\begin{array}{l}\angle AOB+\angle BOC=180\\ \angle AOB+6k=180\\ \angle AOB=180-6k\end{array}$

In$\Delta AOB$.

$OA=OB\therefore \left(\text{Radius}\right)$

If two sides are equal in triangle, then angle opposite to these sides are also equal.

$m\angle B=m\angle A$

And

$\begin{array}{l}m\angle B+m\angle A+\angle AOB=180\\ m\angle A+m\angle A+\left(180-6k\right)=180\\ 2\angle A=6k\\ \angle A=3k\end{array}$

Therefore, the answer is $3k$.