Question

1. y =?

7 given \\( \triangle phs \cong \triangle cnf \\), find the values of x, y, and z.

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle PHS\cong\triangle CNF\), corresponding angles are equal. \(\angle P=\angle F\), \(\angle H=\angle N\), \(\angle S=\angle C\).
We know that the sum of angles in a triangle is \(180^{\circ}\).
First, find \(x\):
\(\angle H=\angle N = 115^{\circ}\)
Set up the equation \(6x - 29=115\)
\(6x=115 + 29\)
\(6x=144\)
\(x = 24\)

Step2: Find \(y\)

\(\angle P=\angle F\), \(\angle P = 36^{\circ}\), so \(3y-1=36\)
\(3y=36 + 1\)
\(3y=37\)
\(y=\frac{37}{3}\approx12.33\)

Step3: Find \(z\)

\(\angle S=\angle C\)
First, find \(\angle S\): \(\angle S=180^{\circ}-\angle P-\angle H=180 - 36-115 = 29^{\circ}\)
Set up the equation \(4z-32 = 29\)
\(4z=29+32\)
\(4z=61\)
\(z=\frac{61}{4}=15.25\)

Answer:

\(x = 24\), \(y=\frac{37}{3}\), \(z=\frac{61}{4}\)