Question

if \\( \triangle bts \cong \triangle ghd, bs = 25, ts = 14, bt = 31, \\)
\\( gd = 4x - 11, m\angle s = 56^{\circ}, m\angle b = 21^{\circ}, \\) and
\\( m\angle h = (7y + 5)^{\circ} \\), find the values of \\( x \\) and \\( y \\).
identify the correct equation to help solve for \\( y \\).
\\( 4x - 11 = 25 \\)
\\( 4x - 11 = 14 \\)
\\( 7y + 5 = 103 \\)
\\( 7y + 5 = 56 \\)
\\( 7y + 5 = 21 \\)

Explanation:

Step1: Find the measure of \(\angle T\)

In \(\triangle BTS\), using the angle - sum property of a triangle (\(m\angle B+m\angle S + m\angle T=180^{\circ}\)).
Substitute \(m\angle B = 21^{\circ}\) and \(m\angle S=56^{\circ}\) into the formula:
\(m\angle T=180-(21 + 56)=103^{\circ}\)

Step2: Use the property of congruent triangles

Since \(\triangle BTS\cong\triangle GHD\), corresponding angles are equal. So, \(m\angle H=m\angle T\).
Given \(m\angle H=(7y + 5)^{\circ}\) and \(m\angle T = 103^{\circ}\), we get the equation \(7y+5 = 103\).

Answer:

\(7y + 5=103\)