Question

find the value of x and y, given △abc ≅ △def.
x = 8
y = 29

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle ABC\cong\triangle DEF\), corresponding angles are equal. So \(\angle A=\angle D\) and \(\angle C=\angle F\).
We know that \(\angle A = 87^{\circ}\), \(\angle D=(5x + 2)^{\circ}\), \(\angle C = 3y^{\circ}\), and \(\angle F = 87^{\circ}\) (because the sum of angles in a triangle is \(180^{\circ}\), and \(\angle B=\angle E = 42^{\circ}\), so \(\angle C=\angle F=180^{\circ}-42^{\circ}-87^{\circ} = 51^{\circ}\)).
First, solve for \(x\):
Set up the equation \(87=5x + 2\).
Subtract \(2\) from both sides: \(87-2=5x\), so \(85 = 5x\).
Divide both sides by \(5\): \(x=\frac{85}{5}=17\).

Step2: Solve for \(y\)

We know that \(3y = 87\) (since \(\angle C=\angle F = 87^{\circ}\) is wrong, correct: \(\angle C=\angle F\), and \(\angle A=\angle D = 87^{\circ}\), \(\angle B=\angle E=42^{\circ}\), so \(\angle C=\angle F=180-(87 + 42)=51^{\circ}\). Wait, no, correct: since \(\triangle ABC\cong\triangle DEF\), \(\angle A=\angle D\), \(\angle B=\angle E\), \(\angle C=\angle F\).
\(\angle A = 87^{\circ}=\angle D=(5x + 2)^{\circ}\), so \(5x+2=87\), \(5x=85\), \(x = 17\).
\(\angle C=3y^{\circ}\), \(\angle F\): in \(\triangle DEF\), \(\angle D = 87^{\circ}\), \(\angle E=42^{\circ}\), so \(\angle F=180-(87 + 42)=51^{\circ}\). Also \(\angle C=\angle F\), so \(3y=51\).
Divide both sides by \(3\): \(y=\frac{51}{3}=17\).

Answer:

\(x = 17\), \(y = 17\)