Question

1. in \\( \triangle w x y \\), if \\( \angle x \cong \angle y, w x=9 x-11, x y=4 x+1 \\), and \\( w y=7 x-3 \\), find \\( x \\) and the measure of each angle.

\\( x= \\)
\\( w x= \\)
\\( x y= \\)
\\( w y= \\)

Explanation:

Step1: Identify Isosceles Triangle Property

In $\triangle WXY$, since $\angle X \cong \angle Y$, the triangle is isosceles with $WX = WY$ (sides opposite equal angles are equal).
So, set $WX = WY$: $9x - 11 = 7x - 3$.

Step2: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 9x - 7x - 11 = -3 \)
Simplify: \( 2x - 11 = -3 \)
Add 11 to both sides: \( 2x = -3 + 11 \)
Simplify: \( 2x = 8 \)
Divide by 2: \( x = \frac{8}{2} = 4 \).

Step3: Find \( WX \)

Substitute \( x = 4 \) into \( WX = 9x - 11 \):
\( WX = 9(4) - 11 = 36 - 11 = 25 \).

Step4: Find \( XY \)

Substitute \( x = 4 \) into \( XY = 4x + 1 \):
\( XY = 4(4) + 1 = 16 + 1 = 17 \).

Step5: Find \( WY \)

Substitute \( x = 4 \) into \( WY = 7x - 3 \):
\( WY = 7(4) - 3 = 28 - 3 = 25 \).

Answer:

\( x = 4 \)
\( WX = 25 \)
\( XY = 17 \)
\( WY = 25 \)