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 \(\overline{AB}\cong\overline{BC}\), \(\triangle ABC\) is an isosceles triangle with \(\angle A=\angle ACB\). So \(m\angle A = 66^{\circ}\)

Step3: Calculate \(m\angle B\)

In \(\triangle ABC\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). Let \(m\angle B=x\). Then \(x + m\angle A+m\angle ACB=180^{\circ}\). Substitute \(m\angle A = 66^{\circ}\) and \(m\angle ACB = 66^{\circ}\) into the equation: \(x+66^{\circ}+66^{\circ}=180^{\circ}\). Simplify the left - hand side: \(x + 132^{\circ}=180^{\circ}\). Solve for \(x\) using the subtraction property of equality: \(x=180^{\circ}-132^{\circ}\)

Answer:

\(48^{\circ}\)