We are given a circle with a triangle inscribed in it. One angle of the triangle is given as $95^\circ$. The angles are labeled $a$, $b$, and $c$. Also, it is indicated that two sides of the triangle are equal in length, which means the triangle is isosceles. We need to find the measures of angles $a$, $b$, and $c$. Angle $a$ is given as $95^\circ$, which seems to be an error given the figure. We will assume the $95^{\circ}$ angle is angle $a$. The side opposite angle $a$ is the diameter of the circle, therefore $a$ is a right angle, and $a = 90^\circ$.

1. Problem Description

We are given a circle with a triangle inscribed in it. One angle of the triangle is given as $95^\circ$. The angles are labeled $a$, $b$, and $c$. Also, it is indicated that two sides of the triangle are equal in length, which means the triangle is isosceles. We need to find the measures of angles $a$, $b$, and $c$. Angle $a$ is given as $95^\circ$, which seems to be an error given the figure. We will assume the $95^{\circ}$ angle is angle $a$. The side opposite angle $a$ is the diameter of the circle, therefore $a$ is a right angle, and $a = 90^\circ$.

2. Solution Steps

Since the side opposite angle $a$ is the diameter of the circle, the angle $a$ is a right angle. Therefore, $a = 90^\circ$.

The sum of angles in a triangle is $180^\circ$.

$a + b + c = 180^\circ$

Since $a = 90^\circ$, we have

$90^\circ + b + c = 180^\circ$

$b + c = 180^\circ - 90^\circ$

$b + c = 90^\circ$

The two sides of the triangle are equal. This means that the angles opposite these sides are equal, so $b = c$.

Therefore, we can substitute $b$ for $c$:

$b + b = 90^\circ$

$2b = 90^\circ$

$b = \frac{90^\circ}{2}$

$b = 45^\circ$

Since $b = c$, then $c = 45^\circ$.

3. Final Answer

$a = 90^\circ$

$b = 45^\circ$

$c = 45^\circ$