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

Since \( AB \) is a diameter of circle \( O \) and \( \angle C \) is inscribed in the circle, by Thales' theorem, \( \angle ACB = 90^\circ \) (an angle inscribed in a semicircle is a right angle).

Step2: Use triangle angle sum property

In \( \triangle ACB \), we know \( \angle A = 40^\circ \) and \( \angle ACB = 90^\circ \). The sum of angles in a triangle is \( 180^\circ \). Let \( \angle B = x \). Then:

$$ \angle A + \angle ACB + \angle B = 180^\circ $$

Substitute the known values:

$$ 40^\circ + 90^\circ + x = 180^\circ $$

Step3: Solve for \( x \)

Simplify the left - hand side:

$$ 130^\circ + x = 180^\circ $$

Subtract \( 130^\circ \) from both sides:

$$ x = 180^\circ - 130^\circ = 50^\circ $$

Answer:

\( 50^\circ \)