Question

in the diagram below, \\( \overline { a b } \cong \overline { b c } \\), and \\( m \angle b c d = 114 ^ { \circ } \\). find \\( m \angle b \\).
you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

Explanation:

Step1: Find the measure of ∠ACB

Since ∠BCD and ∠ACB are supplementary angles (they form a linear pair), we use the formula \(m\angle ACB=180^{\circ}-m\angle BCD\).
\(m\angle ACB = 180^{\circ}- 114^{\circ}=66^{\circ}\)

Step2: Use the property of isosceles triangle

Because \(AB\cong BC\), \(\triangle ABC\) is isosceles with \(AB = BC\). In an isosceles triangle, the base - angles are equal, so \(m\angle A=m\angle ACB = 66^{\circ}\)

Step3: Apply the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle B=x\). Then \(m\angle A + m\angle B+m\angle ACB=180^{\circ}\)
Substitute \(m\angle A = 66^{\circ}\) and \(m\angle ACB = 66^{\circ}\) into the equation: \(66^{\circ}+x + 66^{\circ}=180^{\circ}\)
Simplify the left - hand side: \(x+132^{\circ}=180^{\circ}\)
Solve for \(x\): \(x=m\angle B=180^{\circ}-132^{\circ}\)

Answer:

\(48^{\circ}\)