Question

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

\\( x= \\)
\\( m \angle b= \\)
\\( \circ \\)
\\( m \angle c= \\)
\\( \circ \\)
\\( m \angle d= \\)
\\( \circ \\)

Explanation:

Step1: Identify Isosceles Triangle Angles

Since \(\overline{BC} \cong \overline{BD}\), \(\triangle BCD\) is isosceles with \(\angle C \cong \angle D\). So \(m\angle C = m\angle D\).

$$ 5x - 19 = 2x + 14 $$

Step2: Solve for \(x\)

Subtract \(2x\) from both sides:

$$ 3x - 19 = 14 $$

Add 19 to both sides:

$$ 3x = 33 $$

Divide by 3:

$$ x = 11 $$

Step3: Find \(m\angle B\)

Substitute \(x = 11\) into \(m\angle B = (13x - 35)^\circ\):

$$ m\angle B = 13(11) - 35 = 143 - 35 = 108^\circ $$

Step4: Find \(m\angle C\)

Substitute \(x = 11\) into \(m\angle C = (5x - 19)^\circ\):

$$ m\angle C = 5(11) - 19 = 55 - 19 = 36^\circ $$

Step5: Find \(m\angle D\)

Since \(m\angle C = m\angle D\), \(m\angle D = 36^\circ\) (or substitute \(x = 11\) into \(m\angle D = (2x + 14)^\circ\): \(2(11) + 14 = 22 + 14 = 36^\circ\)).

Answer:

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