Question

1. given ( \triangle mtvcong\triangle bgk ), find the values of ( x ) and ( y ).

Explanation:

Step1: Use triangle angle - sum property

In \(\triangle MTW\), the sum of angles is \(180^{\circ}\). Let \(\angle T=(4x - 3)^{\circ}\), \(\angle M = x^{\circ}\), \(\angle W=41^{\circ}\). Then \(x+(4x - 3)+41 = 180\).
Simplify the left - hand side: \(x+4x-3 + 41=5x + 38\). So, \(5x+38 = 180\).
Subtract 38 from both sides: \(5x=180 - 38=142\).
Divide both sides by 5: \(x = 28.4\) (This seems wrong. Let's use the congruent triangle property).

Since \(\triangle MTW\cong\triangle BGK\), \(\angle T=\angle G = 45^{\circ}\) (corresponding angles of congruent triangles).
Set \(4x-3 = 45\).
Add 3 to both sides: \(4x=45 + 3=48\).
Divide both sides by 4: \(x=\frac{48}{4}=12\).

Step2: Find \(y\)

Also, \(\angle M=\angle B\). In \(\triangle MTW\), \(\angle M=180-(45 + 41)=94^{\circ}\).
Since \(\angle B=(11y + 6)^{\circ}\), set \(11y+6 = 94\).
Subtract 6 from both sides: \(11y=94 - 6=88\).
Divide both sides by 11: \(y=\frac{88}{11}=8\).

Answer:

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