Question

question 1
congruent triangles - practice
∠x ≅ ∠ select
∠y ≅ ∠ select
∠z ≅ ∠ select
segment xy ≅ segment select
segment xz ≅ segment select
segment yz ≅ segment select
△wyz ≅ △ select
(there are two triangles in the image, one with vertices x, y, z and angles 30°, 60°, and a side of length 4.3, and another with vertices a, b, c and angles 30°, 60°, a right angle at a, and a side of length 4.3)

Explanation:

Step1: Analyze Angle X

Looking at the triangles, $\angle X$ in the first triangle and $\angle A$ in the second triangle (since $\angle A$ is a right angle? Wait, no, let's check the angles. Wait, the first triangle has $\angle X$ with some angle, and the second triangle has $\angle A$? Wait, no, the options for $\angle X \cong$ is $\angle A$ (since $\angle A$ is a right angle? Wait, the first triangle: angle at X, and the second triangle: angle at A. Wait, the first triangle: side XZ is 4.3, angle at Z is 30°, angle at X: let's calculate. In a triangle, sum of angles is 180°. So first triangle: angles are 30°, 60°, so the third angle (at X) is 90°? Wait, the second triangle: angle at A is 90° (right angle), angle at B is 60°, angle at C is 30°. So $\angle X \cong \angle A$ (both right angles? Wait, the option is $\angle X \cong \angle A$ (since $\angle A$ is a right angle, and $\angle X$ is also a right angle? Wait, the first triangle: angle at X: 180 - 30 - 60 = 90°, so $\angle X$ is 90°, and $\angle A$ is 90° (right angle). So $\angle X \cong \angle A$.

Step2: Analyze Angle Y

$\angle Y$ in the first triangle: angle at Y is 60°, and $\angle B$ in the second triangle is 60°, so $\angle Y \cong \angle B$.

Step3: Analyze Angle Z

$\angle Z$ in the first triangle is 30°, and $\angle C$ in the second triangle is 30°, so $\angle Z \cong \angle C$.

Step4: Analyze Segment XY

Segment XY: in the first triangle, side XY? Wait, the first triangle has side XZ = 4.3, and the second triangle has side AC = 4.3? Wait, no, the second triangle has side AC = 4.3? Wait, the first triangle: side XZ is 4.3, and the second triangle: side AC is 4.3. Wait, segment XY: wait, the first triangle's side XY? Wait, no, the first triangle: sides: XZ = 4.3, YZ = 5? Wait, no, the first triangle: XZ = 4.3, angle at Z = 30°, angle at Y = 60°, angle at X = 90°. The second triangle: AC = 4.3, BC = 5, angle at C = 30°, angle at B = 60°, angle at A = 90°. So segment XY: wait, no, segment XY in the first triangle and segment AB in the second? Wait, no, the option for segment XY $\cong$ is segment AB? Wait, no, the first triangle: segment XY? Wait, maybe I misread. Wait, the first triangle: vertices X, Y, Z. The second: A, B, C. So $\angle X \cong \angle A$, $\angle Y \cong \angle B$, $\angle Z \cong \angle C$. Then segment XY: in triangle XYZ, XY is opposite angle Z (30°), and in triangle ABC, AB is opposite angle C (30°). Wait, but the length: in triangle ABC, AB: let's see, AC = 4.3 (adjacent to 30°), BC = 5 (hypotenuse), AB =? Wait, no, maybe it's ASA or SAS. Wait, the first triangle: side XZ = 4.3, angle at X (90°), angle at Z (30°). The second triangle: side AC = 4.3, angle at A (90°), angle at C (30°). So by ASA, triangles XYZ and ABC are congruent? Wait, but the last part is $\triangle WYZ \cong \triangle$? Wait, maybe a typo, but the first angle: $\angle X \cong \angle A$ (so $\angle X \cong \angle A$), $\angle Y \cong \angle B$, $\angle Z \cong \angle C$. Then segment XY $\cong$ segment AB, segment XZ $\cong$ segment AC, segment YZ $\cong$ segment BC. Then $\triangle WYZ \cong \triangle$? Wait, maybe the last triangle is $\triangle ABC$? Wait, no, the options for the last congruence: $\triangle WYZ \cong \triangle ABC$? Wait, but let's go step by step.

First, $\angle X \cong \angle A$ (so choose $\angle A$ for $\angle X \cong$).

Then $\angle Y \cong \angle B$ (so choose $\angle B$ for $\angle Y \cong$).

Then $\angle Z \cong \angle C$ (so choose $\angle C$ for $\angle Z \cong$).

Then segment XY $\cong$ segment AB (so choose segment AB for segment XY $\cong$).

Segment XZ $\cong$ segment AC (so choose segment AC for segment XZ $\cong$).

Segment YZ $\cong$ segment BC (so choose segment BC for segment YZ $\cong$).

Then $\triangle WYZ \cong \triangle ABC$? Wait, maybe the first triangle is XYZ, and the second is ABC, so $\triangle XYZ \cong \triangle ABC$, but the last is $\triangle WYZ$? Maybe a typo, but assuming the last triangle is $\triangle ABC$, but let's focus on the angle congruences first.

For $\angle X \cong$: the option is $\angle A$ (since $\angle X$ and $\angle A$ are both right angles, 90°).

For $\angle Y \cong$: $\angle B$ (both 60°).

For $\angle Z \cong$: $\angle C$ (both 30°).

For segment XY $\cong$: segment AB (since they are corresponding sides).

For segment XZ $\cong$: segment AC (corresponding sides, both 4.3).

For segment YZ $\cong$: segment BC (corresponding sides, both 5).

Then $\triangle WYZ \cong \triangle ABC$? Wait, maybe the first triangle is XYZ, so $\triangle XYZ \cong \triangle ABC$, and then $\triangle WYZ$? Maybe a typo, but the key is the angle congruences.

So first, $\angle X \cong \angle A$ (so the first angle choice is $\angle A$).

Answer:

For $\angle X \cong$: $\angle A$ (so the correct choice is $\angle A$).

For $\angle Y \cong$: $\angle B$ (correct choice is $\angle B$).

For $\angle Z \cong$: $\angle C$ (correct choice is $\angle C$).

For segment XY $\cong$: segment AB (correct choice is segment AB).

For segment XZ $\cong$: segment AC (correct choice is segment AC).

For segment YZ $\cong$: segment BC (correct choice is segment BC).

For $\triangle WYZ \cong$: $\triangle ABC$ (assuming the second triangle is ABC).

But since the question is about the first angle: $\angle X \cong \angle A$, so the answer for $\angle X \cong$ is $\angle A$.