Question

1. if \\( \triangle bts \cong \triangle ghd \\), \\( b s = 25 \\), \\( t s = 14 \\), \\( b t = 31 \\), \\( g d = 4 x - 11 \\), \\( m \angle s = 56 ^ { \circ } \\), \\( m \angle b = 21 ^ { \circ } \\), and \\( m \angle h = ( 7 y + 5 ) ^ { \circ } \\), find the values of \\( x \\) and \\( y \\).

Explanation:

Step1: Use the property of congruent triangles (corresponding sides are equal)

Since \(\triangle BTS\cong\triangle GHD\), then \(TS = GD\). Given \(TS = 14\) and \(GD=4x - 11\), we have the equation \(4x-11 = 14\).

$$
LATEXBLOCK0
$$

Step2: Use the property of congruent triangles (corresponding angles are equal)

The sum of angles in a triangle is \(180^{\circ}\). In \(\triangle BTS\), \(m\angle T=180-(m\angle B + m\angle S)=180-(21 + 56)=103^{\circ}\). Since \(\triangle BTS\cong\triangle GHD\), then \(m\angle H=m\angle T\). Given \(m\angle H=(7y + 5)^{\circ}\), we have the equation \(7y+5 = 103\).

$$
LATEXBLOCK1
$$

Answer:

\(x = 6.25\) and \(y = 14\)