Question

1. in \\( \triangle rst \\), if \\( \overline { r t } \cong \overline { s t } \\), \\( m \angle r = ( 9 x + 2 ) ^ { \circ } \\), \\( m \angle s = ( 1 3 x - 1 8 ) ^ { \circ } \\), and \\( m \angle t = ( 1 7 x + 1 ) ^ { \circ } \\), find \\( x \\) and the measure of each angle.

\\( x = \\)
\\( m \angle r = \\)
\\( m \angle s = \\)
\\( m \angle t = \\)

1. in \\( \triangle d e f \\), if \\( \angle d \cong \angle e \\), \\( d e = x + 4 \\), \\( e f = 4 x - 8 \\), and \\( d f = 7 x - 3 5 \\), find \\( x \\) and the measure of each side.

\\( x = \\)
\\( d e = \\)
\\( e f = \\)
\\( d f = \\)

Explanation:

Step1: Use the property of isosceles triangle

Since \(\overline{RT}\cong\overline{ST}\) in \(\triangle RST\), then \(\angle R=\angle S\). So \(9x + 2=13x-18\).

Step2: Solve the equation for \(x\)

Subtract \(9x\) from both sides: \(2 = 4x-18\).
Add \(18\) to both sides: \(20=4x\).
Divide both sides by \(4\): \(x = 5\).

Step3: Find the measure of each angle

For \(\angle R\): \(m\angle R=(9x + 2)^{\circ}\), substitute \(x = 5\), \(m\angle R=(9\times5+2)^{\circ}=47^{\circ}\).
For \(\angle S\): \(m\angle S=(13x-18)^{\circ}\), substitute \(x = 5\), \(m\angle S=(13\times5 - 18)^{\circ}=47^{\circ}\).
For \(\angle T\): \(m\angle T=(17x + 1)^{\circ}\), substitute \(x = 5\), \(m\angle T=(17\times5+1)^{\circ}=86^{\circ}\).

Step1: Use the property of isosceles triangle

Since \(\angle D\cong\angle E\) in \(\triangle DEF\), then \(DF=EF\). So \(7x-35 = 4x-8\).

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(3x-35=-8\).
Add \(35\) to both sides: \(3x=27\).
Divide both sides by \(3\): \(x = 9\).

Step3: Find the measure of each side

For \(DE\): \(DE=x + 4\), substitute \(x = 9\), \(DE=9 + 4=13\).
For \(EF\): \(EF=4x-8\), substitute \(x = 9\), \(EF=4\times9-8=28\).
For \(DF\): \(DF=7x-35\), substitute \(x = 9\), \(DF=7\times9-35=28\).

Answer:

\(x = 5\)
\(m\angle R=47^{\circ}\)
\(m\angle S = 47^{\circ}\)
\(m\angle T=86^{\circ}\)