Question

b. find ( mangle a ). show your work.
because segment ( a b ) is congruent to segment ( b c ), we know that ( angle c cong angle b ) by the
isosceles triangle theorem.
( mangle a+mangle b+mangle c=180 ) (triangle angle-sum theorem)
( mangle a + 2mangle c=) (definition of congruent)
( mangle a+2(quad)=) (substitute.)
( mangle a+quad=) (multiply.)
( mangle a=) (solve.)

Explanation:

Step1: Triangle Angle - Sum Theorem

The sum of the interior angles of a triangle is \(180^{\circ}\), so \(m\angle A + m\angle B + m\angle C=180\).

Step2: Substitute using congruent angles

Since \(AB = BC\), by the Isosceles Triangle Theorem \(\angle C\cong\angle B\). Let \(m\angle C = m\angle B = 40^{\circ}\) (assuming a value for illustration, as the original problem may have a figure with \(\angle C\) value. If we follow the fill - in - the - blank structure and assume \(\angle C\) value is given in a previous part or figure).
Then \(m\angle A+2m\angle C = 180\) (because \(m\angle B=m\angle C\)).

Step3: Substitute the value of \(\angle C\)

If \(m\angle C = 40^{\circ}\), then \(m\angle A+2(40)=180\).

Step4: Multiply

\(m\angle A + 80=180\).

Step5: Solve for \(m\angle A\)

Subtract \(80\) from both sides: \(m\angle A=180 - 80=100^{\circ}\).

Answer:

If \(m\angle C = 40^{\circ}\) (assuming a value for \(\angle C\) based on typical problem setups where this information is available in a figure or prior part), then \(m\angle A = 100^{\circ}\)