Question

in exercises 3 and 4, find the values of x and y.

1. ( \triangle xyz cong \triangle rst )
2. ( abcd cong efgh )

Explanation:

Problem 3: $\boldsymbol{\triangle XYZ \cong \triangle RST}$

Step1: Find $x$ (Angle Congruence)

Since $\triangle XYZ \cong \triangle RST$, corresponding angles are equal. $\angle Y = \angle S$, so $2x - 20 = 30$.
Solve: $2x = 30 + 20$ → $2x = 50$ → $x = \frac{50}{2} = 25$.

Step2: Find $y$ (Side Congruence)

Corresponding sides $XZ = RS$, so $15 = 3y + 9$.
Solve: $3y = 15 - 9$ → $3y = 6$ → $y = \frac{6}{3} = 2$.

Problem 4: $\boldsymbol{ABCD \cong EFGH}$ (Assuming it’s a trapezoid, so consecutive angles are supplementary or corresponding angles congruent)

Step1: Find $x$ (Angle Congruence)

In congruent figures, $\angle A = \angle E$? Wait, no—wait, $ABCD$ and $EFGH$: $\angle A = (60 + 8x)^\circ$, $\angle E = 108^\circ$? Wait, no, maybe $\angle A + \angle B + \angle C + \angle D = 360^\circ$ (quadrilateral), but since congruent, $\angle A = \angle E$, $\angle C = \angle G$? Wait, $\angle C = 62^\circ$, $\angle E = 108^\circ$—wait, maybe it’s a trapezoid with $AB \parallel CD$, so $\angle A + \angle D = 180^\circ$, but no, congruent: $\angle A = \angle E$? Wait, the diagram: $ABCD$ has $\angle A = (60 + 8x)^\circ$, $\angle C = 62^\circ$; $EFGH$ has $\angle E = 108^\circ$, $\angle G = (8y - 3x)^\circ$? Wait, maybe $\angle A + \angle C = 180^\circ$ (if it’s an isosceles trapezoid), but no—wait, $\angle A = 108^\circ$ (since $EFGH$ has $\angle E = 108^\circ$ and $ABCD \cong EFGH$). So $60 + 8x = 108$.
Solve: $8x = 108 - 60$ → $8x = 48$ → $x = \frac{48}{8} = 6$.

Step2: Find $y$ (Angle Congruence)

$\angle C = \angle G$, so $62 = 8y - 3x$. Substitute $x = 6$:
$62 = 8y - 3(6)$ → $62 = 8y - 18$ → $8y = 62 + 18$ → $8y = 80$ → $y = \frac{80}{8} = 10$.

Answer:

s:

• Problem 3: $x = \boldsymbol{25}$, $y = \boldsymbol{2}$
• Problem 4: $x = \boldsymbol{6}$, $y = \boldsymbol{10}$ (assuming angle congruence as above)