Question

29 which triangle congruence theorem can be used to prove the triangles belo a sss b sas c asa d aas

Explanation:

Step1: Identify given info

Both triangles are right-angled (right angles marked), share a common hypotenuse (marked with one tick, so equal length), and the hypotenuse is common (vertical side? Wait, no—wait, the two triangles: one right angle, hypotenuse with tick, and the other right angle, same hypotenuse? Wait, no, the vertical sides? Wait, no, let's re-examine. The triangles: each has a right angle, the hypotenuse (the slanted side) is marked with one tick (so equal), and the vertical sides? Wait, no, the two triangles: one is upper right, one is lower left. The right angles are at the top and bottom. The hypotenuse (slanted side) is common? Wait, no, the slanted side has a tick, so they are equal. Also, the right angles are equal (both 90°), and the other angle? Wait, no—wait, the two triangles: let's list the congruent parts.

Wait, the triangles: each has a right angle (∠ = 90°), the hypotenuse is equal (marked with tick), and the vertical side? Wait, no, maybe the two triangles have: right angle, hypotenuse equal, and one leg? Wait, no, the AAS theorem: two angles and a non-included side. Wait, let's recall the congruence theorems:

- SSS: all three sides equal.
- SAS: two sides and included angle.
- ASA: two angles and included side.
- AAS: two angles and a non-included side.

Looking at the diagram: both triangles are right-angled (so ∠ = 90°), they share the hypotenuse (so hypotenuse is equal), and the other angle: since they are vertical angles? Wait, no, the two triangles: let's see, the slanted side is common? Wait, no, the slanted side has a tick, so it's equal. The right angles are equal. Then, the other angle: since the triangles are formed by a transversal, maybe the angles at the base? Wait, no, maybe the two triangles have: right angle, hypotenuse equal, and one angle (the acute angle) equal. Wait, but AAS: two angles and a non-included side. Let's check:

- Right angle (90°) in both.
- The hypotenuse is equal (side).
- The other angle: since the triangles are congruent in shape, the acute angles would be equal. So two angles (right angle and another angle) and the hypotenuse (non-included side) equal. So AAS? Wait, but wait, the options: D is AAS. Wait, but let's confirm.

Wait, maybe I made a mistake. Wait, the two triangles: each has a right angle, the hypotenuse is equal (marked), and the vertical side? No, the vertical sides: are they equal? Wait, no, the diagram: the upper triangle has a vertical side (left side) and the lower triangle has a vertical side (right side)? Wait, no, the two triangles: one is upper right, with right angle at top, vertical side on left, hypotenuse slanting right. The lower triangle is lower left, right angle at bottom, vertical side on right, hypotenuse slanting left. The hypotenuse (slanting side) has a tick, so they are equal. The right angles are equal (90°). Then, the other angle: the angle at the top (upper triangle) and the angle at the bottom (lower triangle) are equal because they are vertical angles? Wait, no, vertical angles are equal, but here, maybe the two triangles have: right angle, hypotenuse equal, and one acute angle equal (AAS). So AAS theorem. So the answer is D (AAS).

Wait, but let's check again. SSS: need three sides. SAS: two sides and included angle. ASA: two angles and included side. AAS: two angles and non-included side. In this case, right angle (angle 1), hypotenuse (side), and another angle (angle 2), so two angles (right angle and angle 2) and the hypotenuse (non-included side) equal. So AAS.

Step2: Eliminate other options

- SSS: We only know one side (hypotenuse) is equal, not all three. Eliminate A.
- SAS: Need two sides and included angle. We have right angle, hypotenuse, but is the included angle? The right angle is between the vertical side and the horizontal side, but we don't know the vertical sides are equal. So SAS not applicable. Eliminate B.
- ASA: Need two angles and included side. The included side would be between two angles, but we don't have that. Eliminate C.
- AAS: Two angles (right angle and another angle) and the hypotenuse (non-included side) equal. This fits. So D is correct.

Answer:

D. AAS