Question

1. given $\triangle abccong\triangle def$, find the values of $x$ and $y$

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 \(\angle A = 51^{\circ}\), \(\angle D=(5x + 2)^{\circ}\), then \(5x+2 = 51\).

Step2: Solve the equation for \(x\)

Subtract \(2\) from both sides of the equation \(5x+2 = 51\):
\(5x=51 - 2\)
\(5x=49\)
Divide both sides by \(5\): \(x=\frac{49}{5}=9.8\)

Step3: Find the third - angle of \(\triangle DEF\)

In \(\triangle DEF\), using the angle - sum property of a triangle (\(\angle D+\angle E+\angle F = 180^{\circ}\)). We know \(\angle D = 51^{\circ}\), \(\angle E = 42^{\circ}\), then \(\angle F=180-(51 + 42)=87^{\circ}\)
Since \(\angle C=\angle F\) and \(\angle C = 3y^{\circ}\), so \(3y=87\)

Step4: Solve the equation for \(y\)

Divide both sides of \(3y = 87\) by \(3\): \(y=\frac{87}{3}=29\)

Answer:

\(x = 9.8\) and \(y = 29\)