Question

1. (\triangle mnpcong\triangle tus)

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle MNP\cong\triangle TUS\), corresponding sides and angles are equal. So \(MP = TU\) and \(\angle P=\angle U\).

First, for the angles:
We know that in \(\triangle MNP\), \(\angle NMP = 142^{\circ}\), \(\angle P=24^{\circ}\). Using the angle - sum property of a triangle (\(\angle N+\angle M+\angle P = 180^{\circ}\)), but we can also use the congruence. Since \(\angle U=(2x - 50)^{\circ}\) and \(\angle P = 24^{\circ}\), then \(2x-50=24\).

Step2: Solve the equation for \(x\)

Add \(50\) to both sides of the equation \(2x-50 = 24\):
\(2x=24 + 50\)
\(2x=74\)
Divide both sides by \(2\): \(x=\frac{74}{2}=37\)

Step3: Use the side - length congruence

Since \(MP=(2x - y)\) and \(TU = 13\) (because \(\triangle MNP\cong\triangle TUS\)), substitute \(x = 37\) into \(2x-y=13\)
\(2\times37-y=13\)
\(74-y=13\)

Step4: Solve the equation for \(y\)

Subtract \(74\) from both sides: \(-y=13 - 74\)
\(-y=-61\)
Multiply both sides by \(- 1\): \(y = 61\)

Answer:

\(x = 37\) and \(y=61\)