Question

consider triangles pqr, mno, and stu, which have resulted from rotating, reflecting, and translating triangle xyz. use the criteria for triangle congruence to complete the statements. triangle xyz is congruent to triangle pqr by ▼. triangle xyz is congruent to triangle mno by ▼. triangle xyz is congruent to triangle stu by ▼.

Explanation:

Step1: Analyze Triangle XYZ and PQR

Triangle XYZ and PQR: Both are right - angled triangles. In triangle XYZ, right - angled at X; in triangle PQR, right - angled at P. We can see that one pair of legs (the sides with two marks) are equal and the included angles (the right angles) are equal. Also, the other pair of angles (the non - right angles) seem to be equal. So, by the Angle - Angle - Side (AAS) or Angle - Side - Angle (ASA) or Hypotenuse - Leg (HL) or Side - Angle - Side (SAS) criteria? Wait, looking at the markings: In triangle XYZ and PQR, we have a right angle, one pair of equal legs (the sides with two ticks) and the included angle (the right angle) and also the other angle (the non - right angle) is equal. Wait, actually, for triangle XYZ and PQR: angle X = angle P = 90°, side XY (with two ticks) = side PQ (with two ticks), and angle Y = angle Q (the marked angles). So by ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side), but more precisely, since we have a right angle, a leg, and an angle, it can also be considered as AAS. But another way: for right - angled triangles, if we have a leg and an angle equal, it's AAS. But also, looking at the sides: for triangle XYZ and PQR, we can use AAS. But also, for triangle XYZ and MNO: Triangle MNO has sides with three ticks, two ticks, and one tick. Triangle XYZ has sides with three ticks, two ticks, and one tick. So all three sides are equal, so by SSS (Side - Side - Side) congruence criterion. For triangle XYZ and STU: Triangle STU is right - angled at S, has a leg with one tick (same as XZ) and a leg with two ticks (same as XY), and the right angle. So by SAS (Side - Angle - Side) or HL (Hypotenuse - Leg) but since it's two legs and the included right angle, SAS. Wait, let's re - examine:

1. Triangle XYZ and PQR:
- Angle X = angle P = 90° (right angles).
- Side XY (two ticks) = side PQ (two ticks).
- Angle Y = angle Q (the marked angles).
So by ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side). But more accurately, since we have two angles and a side, AAS or ASA. But in the case of right - angled triangles, if we consider the right angle, a leg, and an acute angle, it's AAS.

2. Triangle XYZ and MNO:
- All three sides: The side with three ticks, two ticks, and one tick in XYZ match the sides with three ticks, two ticks, and one tick in MNO. So by SSS (Side - Side - Side) congruence criterion.

3. Triangle XYZ and STU:
- Angle X = angle S = 90° (right angles).
- Side XY (two ticks) = side ST (two ticks).
- Side XZ (one tick) = side SU (one tick).
So by SAS (Side - Angle - Side) congruence criterion (since we have two sides and the included right angle).

Step1: Triangle XYZ and PQR

We observe that $\angle X=\angle P = 90^{\circ}$, $XY = PQ$ (marked with two ticks), and $\angle Y=\angle Q$ (marked angles). By the Angle - Angle - Side (AAS) congruence criterion (or Angle - Side - Angle ASA), triangle XYZ is congruent to triangle PQR by AAS (or ASA, but AAS is more appropriate here as we have two angles and a non - included side? Wait, no, $XY$ is between $\angle X$ and $\angle Y$, and $PQ$ is between $\angle P$ and $\angle Q$. So it's ASA. Wait, $\angle X$, $XY$, $\angle Y$ in XYZ and $\angle P$, $PQ$, $\angle Q$ in PQR. So ASA.

Step2: Triangle XYZ and MNO

All three sides of triangle XYZ (with markings: one tick, two ticks, three ticks) are equal to the three sides of triangle MNO (with the same tick markings). By the Side - Side - Side (SSS) congruence criterion, triangle XYZ is congruent to triangle MNO by SSS.

Step3: Triangle XYZ and STU

$\angle X=\angle S = 90^{\circ}$, $XY = ST$ (two ticks), and $XZ = SU$ (one tick). By the Side - Angle - Side (SAS) congruence criterion (since the angle between the two sides is the right angle), triangle XYZ is congruent to triangle STU by SAS.

Answer:

Triangle XYZ is congruent to triangle PQR by \(\text{AAS (or ASA)}\); Triangle XYZ is congruent to triangle MNO by \(\text{SSS}\); Triangle XYZ is congruent to triangle STU by \(\text{SAS}\)