Question

angle c is inscribed in circle o,
ab is a diameter of circle o,
what is the measure of ∠b?
□°

Explanation:

Step1: Recall Thales' theorem

Thales' theorem states that if \( AB \) is a diameter of a circle and \( C \) is a point on the circle (not coinciding with \( A \) or \( B \)), then \( \angle ACB = 90^\circ \). So triangle \( ABC \) is a right - triangle with \( \angle C = 90^\circ \).

Step2: Use the angle - sum property of a triangle

In a triangle, the sum of the interior angles is \( 180^\circ \). For triangle \( ABC \), we know that \( \angle A=45^\circ \) and \( \angle C = 90^\circ \). Let \( \angle B=x \). Then, by the angle - sum property of a triangle (\( \angle A+\angle B+\angle C = 180^\circ \)), we substitute the known values: \( 45^\circ+x + 90^\circ=180^\circ \).
Simplify the left - hand side: \( 135^\circ+x=180^\circ \).
Subtract \( 135^\circ \) from both sides: \( x=180^\circ - 135^\circ=45^\circ \).

Answer:

\( 45 \)