Question

1. given \\( \triangle xps \cong \triangle dnf \\), find the values of \\( x \\) and \\( y \\).
2. given \\( \triangle mtw \cong \triangle bgk \\), find the values of \\( x \\) and \\( y \\).

Explanation:

Step1: Use the property of congruent triangles (corresponding sides and angles are equal) for problem 7

Since \(\triangle XPS\cong\triangle DNF\), then \(XS = NF\) and \(XP=DN\).
For \(XS = NF\):
\(17x + 3=57\)
\(17x=57 - 3\)
\(17x=54\)
\(x = 3\)
For \(XP=DN\):
\(4y-3 = 51\)
\(4y=51 + 3\)
\(4y=54\)
\(y = 13.5\)

Step2: Use the property of congruent triangles (corresponding angles are equal) for problem 8

Since \(\triangle MTW\cong\triangle BGK\), then \(\angle T=\angle G\) and \(\angle M=\angle B\).
For \(\angle T=\angle G\):
\(4x-3 = 45\)
\(4x=45 + 3\)
\(4x=48\)
\(x = 12\)
For \(\angle M=\angle B\):
In \(\triangle MTW\), \(\angle M=180^{\circ}-\angle T-\angle W\). Since \(\angle W = 41^{\circ}\) and \(\angle T = 45^{\circ}\), \(\angle M=180-(45 + 41)=94^{\circ}\).
So \(11y+6=94\)
\(11y=94 - 6\)
\(11y=88\)
\(y = 8\)

Answer:

For problem 7: \(x = 3\), \(y = 13.5\)
For problem 8: \(x = 12\), \(y = 8\)