Question

17.
7 points
include x value and the measure of each angle, as follows:
x:
b:
c:
d:
in \\(\triangle bcd\\), if \\(\overline{bc}=\overline{bd}), \\(m\angle b = 13x - 35\\), \\(m\angle c=5x - 19\\), and \\(m\angle d = 2x+14\\), find \\(x\\) and the measure of each angle.
your answer

Explanation:

Step1: Use the property of isosceles - triangle

In \(\triangle BCD\), since \(BC = BD\), then \(\angle C=\angle D\). So we set up the equation \(5x - 19=2x + 14\).
\[

$$\begin{align*} 5x-2x&=14 + 19\\ 3x&=33\\ x& = 11 \end{align*}$$

\]

Step2: Find the measure of \(\angle C\) and \(\angle D\)

Substitute \(x = 11\) into the expressions for \(\angle C\) and \(\angle D\).
\(\angle C=5x-19=5\times11 - 19=55 - 19 = 36^{\circ}\)
\(\angle D=2x + 14=2\times11+14=22 + 14=36^{\circ}\)

Step3: Find the measure of \(\angle B\)

Use the angle - sum property of a triangle (\(\angle B+\angle C+\angle D = 180^{\circ}\)).
\(\angle B=180^{\circ}-\angle C-\angle D\)
Substitute \(\angle C = 36^{\circ}\) and \(\angle D = 36^{\circ}\) into the formula.
\(\angle B=180-(36 + 36)=108^{\circ}\)

Answer:

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