Question

1. what theorem would be used to prove the triangles below congruent? image of two triangles with markings options: sas, ssa, aaa, sss

Explanation:

Step1: Analyze triangle markings

Triangles \( \triangle UVW \) and \( \triangle XYZ \) have: one pair of sides with single tick marks (equal), one pair with double tick marks (equal), and a right angle (marked by the small square, so equal angles). Wait, no—wait, the angles at \( W \) and \( Z \) are marked as equal (right angles? Or congruent angles), and two sides: one side with single tick, one with double tick. Wait, actually, let's check the congruence criteria. SSS: three sides. SAS: two sides and included angle. Let's see: the sides with double ticks are equal, the sides with single ticks are equal, and the included angle (the angle between the double - ticked side and single - ticked side) – wait, no, the right angle? Wait, the triangles have a right angle? Wait, \( \angle V \) and \( \angle X \) are right angles? Wait, no, \( \angle V \) and \( \angle Y \) (wait, no, \( V \) and \( X \) – wait, the triangles: \( UVW \) has \( \angle V \) as right angle? \( XYZ \) has \( \angle X \) as right angle? Wait, no, looking at the diagram: \( UV \) and \( XY \) are vertical sides (right angles at \( V \) and \( X \)? Wait, no, \( V \) and \( Y \) – no, the bottom triangle: \( X \) and \( Y \), \( X \) is a right angle? Wait, maybe the angles at \( W \) and \( Z \) are the included angles. Wait, the sides: one side with double ticks (so \( VW = YZ \)), one side with single ticks ( \( UW = XZ \) ), and the included angle between them? Wait, no, the angle between the double - ticked side and single - ticked side? Wait, no, let's re - examine. The triangles: \( \triangle UVW \) and \( \triangle XYZ \). Let's list the equal parts:

- Side \( UW = XZ \) (single tick marks)
- Side \( VW = YZ \) (double tick marks)
- Angle \( \angle VWU=\angle YZX \) (the angles at \( W \) and \( Z \), marked as equal, maybe right angles? Or congruent angles). Wait, but for SAS, we need two sides and the included angle. Wait, no—wait, the sides with double ticks: \( VW \) and \( YZ \), the sides with single ticks: \( UW \) and \( XZ \), and the angle between \( VW \) and \( UW \) (at \( W \)) and between \( YZ \) and \( XZ \) (at \( Z \)) – are those angles equal? If so, then SAS. But wait, another way: SSS? No, because we have two sides and an angle. Wait, no, wait the answer options: SSS is an option. Wait, no, let's check the markings again. The two triangles have three pairs of equal sides? Wait, \( UW = XZ \) (single tick), \( VW = YZ \) (double tick), and what about the third side? Wait, \( UV \) and \( XY \) – are they equal? Wait, maybe I misread. Wait, the triangles are right triangles? Wait, \( \angle V \) and \( \angle X \) are right angles? So \( \triangle UVW \) and \( \triangle XYZ \) are right triangles. Then, by HL (hypotenuse - leg), but HL is a special case of SAS. Wait, but the options are SAS, SSA, AAA, SSS. Wait, SSS: three sides equal. Let's see: if \( UW = XZ \) (single), \( VW = YZ \) (double), and \( UV = XY \) (the vertical sides). Wait, maybe the three sides: \( UW = XZ \), \( VW = YZ \), and \( UV = XY \). So that would be SSS? Wait, no, wait the diagram: the vertical sides ( \( UV \) and \( XY \)) – are they marked? No, but the single - ticked sides ( \( UW \) and \( XZ \) ), double - ticked sides ( \( VW \) and \( YZ \) ), and the vertical sides ( \( UV \) and \( XY \)) – maybe all three sides. Wait, but the answer is SSS? Wait, no, wait let's check the congruence criteria again.

Wait, no, let's look at the options. The correct congruence criterion here: SSS (Side - Side - Side) would be if three sides are equal. Let's see: the sides with single ticks are equal, sides with double ticks are equal, and the third sides (the vertical sides) are equal (since the triangles are drawn with the same height? Or maybe the vertical sides are equal). So if all three sides are equal, then SSS. But wait, maybe I made a mistake. Wait, no, let's think again. The triangles have:

- \( UW = XZ \) (single tick)
- \( VW = YZ \) (double tick)
- \( UV = XY \) (the vertical sides, not marked but maybe equal as per the diagram's symmetry)

So three sides equal, so SSS. Wait, but the options include SSS. Wait, but let's check the other options. SAS: two sides and included angle. SSA: not a valid congruence criterion. AAA: similar, not congruent. So the correct answer is SSS? Wait, no, wait the diagram: maybe the included angle. Wait, no, let's re - examine the diagram. The triangles are drawn such that:

- The sides with double ticks ( \( VW \) and \( YZ \)) are equal.
- The sides with single ticks ( \( UW \) and \( XZ \)) are equal.
- The included angle between the double - ticked side and single - ticked side: wait, no, the angle at \( W \) and \( Z \) – if the triangles are right - angled, then by HL (which is a form of SAS, since in a right triangle, hypotenuse and leg: hypotenuse is one side, leg is another, and included angle is 90 degrees). But the options have SAS? Wait, no, maybe I messed up. Wait, let's check the congruence criteria again.

SSS: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

SAS: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

Looking at the diagram:

- The sides with double tick marks (let's say \( VW \) and \( YZ \)) are equal.
- The sides with single tick marks ( \( UW \) and \( XZ \)) are equal.
- The third sides ( \( UV \) and \( XY \)): are they equal? From the diagram, since the triangles are drawn in a way that the vertical sides ( \( UV \) and \( XY \)) seem to be equal (same length, as per the diagram's symmetry). So three sides are equal, so SSS.

Step2: Eliminate other options

- SSA: Not a valid congruence criterion for proving triangle congruence.
- AAA: Proves similarity, not congruence.
- SAS: We need two sides and the included angle. But in our case, if we consider the three sides, SSS is more appropriate. The sides with single ticks, double ticks, and the third side (the vertical side) are equal.

Answer:

SSS (corresponding to the option with "SSS" text. Since the options are: SAS, SSA, AAA, SSS, the correct option is the one labeled "SSS". So the answer is the option with "SSS" (the last option in the list: "SSS").