Question

5.2. intro to triangle congruence
1.
in the diagram, \\( \triangle a b c \cong \triangle e d c \\). find
the values of \\( x \\) and \\( y \\).
\\( x = \\);\\( y = \\)

Explanation:

Step1: Find x using congruent sides

Since $\triangle ABC \cong \triangle EDC$, corresponding sides are equal. So, $AB = ED$. Given $AB = 3x - 10$ and $ED = 20$.
Set up the equation: $3x - 10 = 20$
Add 10 to both sides: $3x = 20 + 10 = 30$
Divide by 3: $x = \frac{30}{3} = 10$

Step2: Find y using congruent angles

Since $\triangle ABC \cong \triangle EDC$, corresponding angles are equal. The right angles are equal, and angle at D is $36.87^\circ$, so angle at A's corresponding angle? Wait, angle at C in $\triangle ABC$ (angle y) corresponds to angle at C in $\triangle EDC$? Wait, no, in right triangles, the other acute angle: in $\triangle EDC$, angle at D is $36.87^\circ$, so angle at C (angle ECD) is $90 - 36.87 = 53.13^\circ$? Wait, no, wait $\triangle ABC$ has a right angle at A, so angle at B? Wait, no, $\triangle ABC \cong \triangle EDC$, so angle at D (36.87°) corresponds to angle at B? Wait, no, let's check the correspondence. $\triangle ABC \cong \triangle EDC$, so vertex A corresponds to E, B to D, C to C? Wait, no, the notation $\triangle ABC \cong \triangle EDC$ means A corresponds to E, B to D, C to C. So angle at B (in $\triangle ABC$) corresponds to angle at D (in $\triangle EDC$), and angle at A (right angle) corresponds to angle at E (right angle). So in $\triangle ABC$, angle at C (y) is equal to angle at C in $\triangle EDC$? Wait, no, in $\triangle EDC$, angles are 90°, 36.87°, so the other angle is 90 - 36.87 = 53.13°? Wait, but maybe angle y is equal to angle D? No, wait, let's see: $\triangle ABC$ has right angle at A, so angle at B: in $\triangle ABC$, angle at A is 90°, angle at B is y? Wait, no, the diagram: $\triangle ABC$ has right angle at A, side AB is horizontal, AC is vertical (length 15), and BC is the hypotenuse. $\triangle EDC$ has right angle at E, side DE is horizontal (20), EC is vertical (15, since AC = EC as congruent triangles), and DC is hypotenuse. So angle at D is 36.87°, so angle at B (in $\triangle ABC$) corresponds to angle at D (36.87°)? No, wait, $\triangle ABC \cong \triangle EDC$, so angle at B (angle ABC) corresponds to angle at D (angle EDC), which is 36.87°? No, wait, no, in $\triangle ABC$, angle at A is 90°, angle at C is y, angle at B is 90 - y. In $\triangle EDC$, angle at E is 90°, angle at D is 36.87°, angle at C is 90 - 36.87 = 53.13°. But since $\triangle ABC \cong \triangle EDC$, angle at C (y) in $\triangle ABC$ corresponds to angle at C in $\triangle EDC$? Wait, no, the correspondence is A-E, B-D, C-C. So angle at B (ABC) corresponds to angle at D (EDC), which is 36.87°? No, wait, maybe I got the correspondence wrong. Let's use the sides: AB = ED (20), AC = EC (15), BC = DC. So in $\triangle ABC$, right-angled at A, legs AB=20, AC=15. In $\triangle EDC$, right-angled at E, legs ED=20, EC=15. So angle at D (EDC) is arctan(EC/ED) = arctan(15/20) = arctan(3/4) ≈ 36.87°, which matches. Then angle at B (ABC) is equal to angle at D (EDC) = 36.87°? No, wait, in $\triangle ABC$, angle at B: tan(angle B) = AC/AB = 15/20 = 3/4, so angle B is 36.87°, and angle at C (y) is 90 - 36.87 = 53.13°? Wait, no, wait the problem says angle y is at C in $\triangle ABC$. Wait, in $\triangle ABC$, right-angled at A, so angles are 90° at A, y at C, and angle at B is 90 - y. In $\triangle EDC$, right-angled at E, angles are 90° at E, 36.87° at D, and angle at C is 90 - 36.87 = 53.13°. Since $\triangle ABC \cong \triangle EDC$, angle at C (y) in $\triangle ABC$ corresponds to angle at C in $\triangle EDC$, which is 53.13°? Wait, but let's check: if $\triangle ABC \cong \triangle EDC$, then angle at A (90°) corresponds to angle at E (90°), angle at B corresponds to angle at D (36.87°), and angle at C corresponds to angle at C (so y is equal to angle at C in $\triangle EDC$, which is 90 - 36.87 = 53.13°). Alternatively, since in $\triangle ABC$, angle at B: tan(angle B) = AC/AB = 15/20 = 3/4, so angle B = 36.87°, so angle at C (y) = 90 - 36.87 = 53.13°. So y = 53.13°? Wait, but let's confirm with congruence: $\triangle ABC \cong \triangle EDC$, so angle at C (in $\triangle ABC$) corresponds to angle at C (in $\triangle EDC$). In $\triangle EDC$, angle at C is 90° - 36.87° = 53.13°, so y = 53.13°? Wait, but maybe I made a mistake. Wait, the problem says "Find the values of x and y". We found x=10. For y: since $\triangle ABC \cong \triangle EDC$, angle at D (36.87°) corresponds to angle at B, and angle at C (y) corresponds to angle at C in $\triangle EDC$, which is 90° - 36.87° = 53.13°, or maybe angle y is equal to angle at D? No, no, let's re-express the congruence: $\triangle ABC \cong \triangle EDC$ means A↔E, B↔D, C↔C. So angle at A (∠A) = ∠E (90°), ∠B = ∠D (36.87°), ∠C = ∠C. Wait, but in $\triangle ABC$, ∠A is 90°, ∠B is 36.87°, so ∠C (y) is 180 - 90 - 36.87 = 53.13°. So y = 53.13°? Wait, but maybe the problem considers angle y as the angle at C, which is 53.13°, or maybe 36.87°? Wait, no, let's check the diagram: angle y is at C, between AC and BC. In $\triangle ABC$, AC is vertical (15), AB is horizontal (20), so BC is the hypotenuse. So angle at C: adjacent side is AC=15, opposite side is AB=20? Wait, no, in right triangle, angle at C: adjacent to angle C is AC, opposite is AB. So tan(angle C) = AB/AC = 20/15 = 4/3, so angle C = arctan(4/3) ≈ 53.13°, which is 90 - 36.87. So that's correct. So y = 53.13°? Wait, but maybe the problem expects it as 53.13 or 53.1? Or maybe 36.87? Wait, no, let's go back. Since $\triangle ABC \cong \triangle EDC$, angle at D (36.87°) is equal to angle at B, so angle at C (y) is equal to angle at C in $\triangle EDC$, which is 90 - 36.87 = 53.13°. So y = 53.13° (or 53.1 if rounded, but 36.87 + 53.13 = 90, so that's correct).

Answer:

x = 10; y = 53.13 (or 53.1 or 53, depending on rounding, but 53.13 is precise as arctan(4/3) is 53.13°)

Wait, but let's confirm the angle: in $\triangle EDC$, right-angled at E, angle at D is 36.87°, so angle at C is 90 - 36.87 = 53.13°, which is equal to angle at C in $\triangle ABC$ (y), so y = 53.13°. And x = 10.