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. \(\angle T=(4x - 3)^{\circ}\), \(\angle K = 41^{\circ}\), \(\angle B=(11y + 6)^{\circ}\), \(\angle M\) corresponds to \(\angle G = 45^{\circ}\).
In \(\triangle MTW\), using the angle - sum property of a triangle (\(\angle M+\angle T+\angle W=180^{\circ}\)).
Since \(\angle M = 45^{\circ}\), \(\angle W = 41^{\circ}\), then \(\angle T=180-(45 + 41)=94^{\circ}\).
Set up the equation for \(x\): \(4x-3 = 94\).

Step2: Solve for \(x\)

Add \(3\) to both sides of the equation \(4x-3 = 94\): \(4x=94 + 3\), so \(4x=97\).
Divide both sides by \(4\): \(x=\frac{97}{4}=24.25\).

Step3: Solve for \(y\)

Since \(\angle B=\angle M = 45^{\circ}\), set up the equation \(11y+6 = 45\).
Subtract \(6\) from both sides: \(11y=45 - 6\), so \(11y=39\).
Divide both sides by \(11\): \(y=\frac{39}{11}\approx3.55\).

Answer:

\(x = 24.25\), \(y=\frac{39}{11}\)