To find the measure of angle PBA, we need to consider the properties of angles formed by a diameter in a circle.

Property: An angle inscribed in a semicircle is a right angle.

Since AB is a diameter of the circle, angle PAB is inscribed in a semicircle and therefore a right angle. We are given that angle PAB measures 55 degrees, which means angle PBA is also 55 degrees.

Therefore, angle PBA = 55 degrees.



To find the measure of angle BPQ, we can use the properties of angles formed by intersecting chords in a circle.

Property: When two chords intersect in a circle, the product of the measures of their intercepted arcs is equal.

Let's consider the intercepted arcs PA and AB. Since angle PAB is a right angle, the intercepted arcs PA and AB must be complementary. In other words, the sum of their measures is 90 degrees.

We are given that angle PAB measures 55 degrees, so the intercepted arc PA measures 90 - 55 = 35 degrees.

Now, let's consider the intercepted arcs AB and BQ. Since angle PBQ is 25 degrees, the intercepted arc AB must also measure 25 degrees.

Using the property mentioned earlier, we can set up the following equation:

Intercepted arc PA * Intercepted arc AB = Intercepted arc AB * Intercepted arc BQ

35 * 25 = 25 * Intercepted arc BQ

875 = 25 * Intercepted arc BQ

Intercepted arc BQ = 875 / 25 = 35 degrees

Therefore, angle BPQ = intercepted arc BQ = 35 degrees.



To find the measure of angle BAR, we can use the properties of angles formed by intersecting chords in a circle.

Property: The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the intercepted arcs.

In this case, the intercepted arcs are PA and AB. We already found that the intercepted arc PA measures 35 degrees.

The intercepted arc AB can be calculated as:

Intercepted arc AB = 180 - angle ABR

Intercepted arc AB = 180 - 50 = 130 degrees

Now, we can find the measure of angle BAR:

Angle BAR = (Intercepted arc PA + Intercepted arc AB) / 2

Angle BAR = (35 + 130) / 2

Angle BAR = 165 / 2

Angle BAR = 82.5 degrees

Therefore, angle BAR ≈ 82.5 degrees.