Question

angle c is inscribed in circle o.
\\(\overline{ab}\\) is a diameter of circle o.
what is the measure of \\(\angle b\\)?
\\(\square\\) °

Explanation:

Step1: Identify the triangle type

Since \( \overline{AB} \) is a diameter of circle \( O \) and \( \angle C \) is inscribed in the circle, by the Thales' theorem, \( \angle C = 90^\circ \) (a triangle inscribed in a semicircle is a right triangle). So \( \triangle ABC \) is a right triangle with \( \angle C = 90^\circ \).

Step2: Use triangle angle sum

In a triangle, the sum of interior angles is \( 180^\circ \). We know \( \angle C = 90^\circ \) and \( \angle BAC = 46^\circ \) (from the diagram). Let \( \angle B = x \). Then:

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

Substitute the known values:

$$ 46^\circ + x + 90^\circ = 180^\circ $$

Step3: Solve for \( x \)

Simplify the equation:

$$ x + 136^\circ = 180^\circ $$

Subtract \( 136^\circ \) from both sides:

$$ x = 180^\circ - 136^\circ = 44^\circ $$

Answer:

\( 44 \)