Question

example 2
ex2: which of the following congruency theorems can be used to prove the two triangles are congruent?
sss sas aas asa hl
rewatch

Explanation:

Step1: Analyze Triangle Markings

First, identify the marked sides and angles. The first triangle has two marks on one side, one mark on another, and an angle. The second triangle (rotated) has two marks on a side, one mark on another, and an angle.

Step2: Match with Congruence Theorems

- SSS: Needs all three sides equal. Here, we have two sides (one with two marks, one with one mark) and an angle. Not SSS.
- SAS: Requires two sides and the included angle. Wait, but let's check the angles. Wait, the angle is between the sides? Wait, no—wait, the triangles are rotated. Wait, the two - marked sides, one - marked sides, and the included angle? Wait, no, actually, let's look at AAS. Wait, AAS is two angles and a non - included side, but here we have two sides (one pair with two marks, one pair with one mark) and a pair of angles. Wait, no, let's re - examine. Wait, the triangles: one triangle has a side with two ticks, a side with one tick, and an angle. The other (rotated) has a side with two ticks, a side with one tick, and an angle. When we rotate the second triangle, the angle is between the two - ticked side and one - ticked side? Wait, no, actually, AAS: if two angles and a non - included side are equal. Wait, no, let's think again. Wait, the markings: let's denote the sides. Let the side with two ticks be \( S_2 \), one tick be \( S_1 \), and the angle be \( \angle \). In the first triangle, we have \( S_2 \), \( \angle \), \( S_1 \). In the second (rotated) triangle, when we align, we have \( S_2 \), \( \angle \), \( S_1 \) but in AAS? Wait, no, AAS is two angles and a side. Wait, maybe I made a mistake. Wait, no—wait, the correct theorem here: AAS (Angle - Angle - Side) or SAS? Wait, no, let's check the sides and angles. The two triangles: one has a side with two ticks (so equal to the other's two - ticked side), a side with one tick (equal to the other's one - ticked side), and an angle. Wait, actually, when we look at the angles, the angle is between the two - ticked side and one - ticked side? No, wait, the angle is not between them. Wait, the triangles are congruent by AAS? No, wait, let's recall: AAS is when two angles and a non - included side are congruent. But here, we have two sides (one pair with two ticks, one pair with one tick) and a pair of angles. Wait, no, maybe it's SAS? Wait, no, SAS is two sides and the included angle. Wait, maybe I messed up. Wait, let's re - express. The first triangle: side \( a \) (two ticks), side \( b \) (one tick), angle \( \theta \). The second triangle (rotated): side \( a \) (two ticks), side \( b \) (one tick), angle \( \theta \). When we rotate the second triangle, the angle \( \theta \) is not between \( a \) and \( b \) in the same way? Wait, no, actually, the correct theorem here is AAS? No, wait, no—wait, the answer is AAS? Wait, no, let's check the options again. Wait, the options are SSS, SAS, AAS, ASA, HL. HL is for right triangles, these are not right triangles. SSS: need three sides. SAS: two sides and included angle. ASA: two angles and included side. AAS: two angles and non - included side. Let's look at the triangles: the two - ticked sides are equal (\( S_2 = S_2 \)), one - ticked sides are equal (\( S_1 = S_1 \)), and the angles are equal. Wait, but the angle is not between \( S_1 \) and \( S_2 \) in both? Wait, no, when we rotate the second triangle, the angle is adjacent to \( S_2 \) and \( S_1 \) but in a way that it's AAS. Wait, actually, the correct theorem here is AAS. Wait, no, let's think of the triangles: if we have two sides (one pair with two ticks, one pair with one tick) and a pair of angles, and the angle is not between the two sides, then it's AAS. Wait, but let's confirm. The key is: AAS (Angle - Angle - Side) congruence: if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. In these triangles, we have two sides (one with two ticks, one with one tick) and a pair of angles. Wait, maybe I was wrong earlier. Wait, no, let's look at the markings again. The first triangle: side with two ticks, side with one tick, and an angle. The second triangle (rotated) has side with two ticks, side with one tick, and an angle. When we rotate the second triangle, the angle is in the same relative position as a non - included angle with respect to the two sides. So the congruence theorem here is AAS. Wait, but let's check the options. The options include AAS. So the correct theorem is AAS.

Answer:

AAS (assuming the option is labeled as "AAS" in the list, e.g., if the options are like: SSS, SAS, AAS, ASA, HL, then the answer is the option with "AAS")