Question

4.2 central and inscribed angles
triangle bcd is isosceles and ( bccong bd ).
the measure of arc ( bc=)

Explanation:

Step1: Find the measure of arc \( CD \)

The measure of an arc is equal to the measure of its central angle. Given that the measure of arc \( CD \) is \( 100^{\circ}\).

Step2: Use the property of the sum of arcs in a circle

The sum of the measures of arcs in a circle is \( 360^{\circ}\). Let the measure of arc \( BC\) be \(x\) and the measure of arc \( BD\) be \(y\). Since \(BC\cong BD\) (given \(BC\cong BD\) in \(\triangle BCD\)), then arc \(BC\cong\) arc \(BD\) (in a circle, congruent chords subtend congruent arcs), so \(x = y\).
We know that \(x + y+100^{\circ}=360^{\circ}\). Substitute \(y=x\) into the equation: \(2x+100^{\circ}=360^{\circ}\).

Step3: Solve the equation for \(x\)

Subtract \(100^{\circ}\) from both sides of the equation \(2x+100^{\circ}=360^{\circ}\):
\(2x=360^{\circ}- 100^{\circ}=260^{\circ}\).
Divide both sides by 2: \(x=\frac{260^{\circ}}{2}=130^{\circ}\).

Answer:

\(130^{\circ}\)