Question

1. in \\( \triangle r s t \\), if \\( \overline{r t} \cong \overline{s t} \\), \\( m \angle r=(9 x+2)^{\circ}, m \angle s=(13 x-18)^{\circ} \\), and \\( m \angle t=(17 x+1)^{\circ} \\), find \\( x \\) and the measure of each angle.
2. 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-35 \\), find \\( x \\) and the measure of each side.
3. in \\( \triangle w x y \\), if \\( \overline{w x} \cong \overline{w y}, m \angle w=(x+6)^{\circ}, m \angle x=(5 x-12)^{\circ} \\), and \\( m \angle y=(7 x-48)^{\circ} \\), find \\( x \\) and the measure of each angle.

a triangle is equilateral if and only if it is equiangular!

• if \\( m \angle a=m \angle b=m \angle c \\), then
• if \\( a b=b c=a c \\), then

find each missing measure.
11.
12.

1. if \\( \triangle p q r \\) is an equilateral triangle, solve for \\( x \\) and \\( y \\).

Explanation:

Step1: Use the property of equilateral triangle

In an equilateral triangle, all angles are equal to \(60^{\circ}\) and all sides are equal. For \(\triangle PQR\) being equilateral, we first equate the side - length expressions.
Set \(14y - 59=9y + 1\)
\(14y-9y=1 + 59\)
\(5y=60\)
\(y = 12\)
Also, set \(9y + 1=11y-23\) (checking consistency). Substitute \(y = 12\):
Left - hand side: \(9\times12+1=108 + 1=109\)
Right - hand side: \(11\times12-23=132-23 = 109\)
Since all sides are equal, for the angle \((7x - 3)^{\circ}\), and because each angle of an equilateral triangle is \(60^{\circ}\)

Step2: Solve for \(x\)

Set \(7x-3=60\)
\(7x=60 + 3\)
\(7x=63\)
\(x = 9\)

Answer:

\(x = 9\), \(y=12\)