Question

1. given \\( \triangle m t w \cong \triangle b g k \\), find the values of \\( x \\) and \\( y \\).

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle MTW\cong\triangle BGK\), corresponding angles are equal. \(\angle T=\angle G = 45^{\circ}\), \(\angle W=\angle K = 41^{\circ}\), \(\angle M=\angle B\).
For \(\angle T\):
\(4x - 3=45\)

Step2: Solve for \(x\)

Add \(3\) to both sides of the equation \(4x - 3=45\):
\(4x=45 + 3\)
\(4x=48\)
Divide both sides by \(4\):
\(x=\frac{48}{4}=12\)

Step3: Find \(\angle M\) (or \(\angle B\))

In \(\triangle MTW\), using the angle - sum property of a triangle (\(\angle M+\angle T+\angle W = 180^{\circ}\)).
\(\angle M=180-(45 + 41)=180 - 86=94^{\circ}\)
Since \(\angle M=\angle B\), then \(11y+6 = 94\)

Step4: Solve for \(y\)

Subtract \(6\) from both sides of the equation \(11y+6 = 94\):
\(11y=94 - 6\)
\(11y=88\)
Divide both sides by \(11\):
\(y=\frac{88}{11}=8\)

Answer:

\(x = 12\) and \(y = 8\)