Question

1. in \\( \triangle b c d \\), if \\( \overline{b c} \cong \overline{b d}, m \angle b=(13 x-35) \\), \\( m \angle c=(5 x-19) \\), and \\( m \angle d=(2 x+14) \\). find \\( x \\) and the measure of each angle.

Explanation:

Step1: Use the property of congruent triangles

Since \( \overline{BC}\cong\overline{BD}\), then \( \angle C=\angle D\) (angles opposite congruent sides are congruent). So, \(5x - 19=2x+14\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(5x-2x - 19=2x-2x + 14\), which gives \(3x-19 = 14\).
Add \(19\) to both sides: \(3x-19 + 19=14 + 19\), so \(3x=33\).
Divide both sides by \(3\): \(x=\frac{33}{3}=11\).

Step3: Find the measure of each angle

For \( \angle B\): Substitute \(x = 11\) into \(m\angle B=(13x - 35)\). Then \(m\angle B=13\times11-35=143 - 35 = 108^{\circ}\).
For \( \angle C\): Substitute \(x = 11\) into \(m\angle C=(5x - 19)\). Then \(m\angle C=5\times11-19=55 - 19 = 36^{\circ}\).
For \( \angle D\): Substitute \(x = 11\) into \(m\angle D=(2x + 14)\). Then \(m\angle D=2\times11+14=22 + 14 = 36^{\circ}\).

Answer:

\(x = 11\), \(m\angle B = 108^{\circ}\), \(m\angle C = 36^{\circ}\), \(m\angle D = 36^{\circ}\)