Question

2 multiple choice 1 point determine if the triangles can be proved congruent, if possible, by sss, sas, asa, aas, or hl. sas congruence hl congruence aas congruence asa congruence not congruent sss congruence

Explanation:

Step1: Analyze the diagram

The figure is a parallelogram with a diagonal, creating two triangles. The markings show two pairs of sides are equal (the sides of the parallelogram) and the diagonal is common to both triangles. Wait, no, wait—wait, the markings: the top and bottom sides have two ticks, left and right sides have one tick? Wait, no, looking at the diagram, the diagonal is shared. Wait, actually, in the parallelogram, opposite sides are equal. So the two triangles formed by the diagonal: let's check the sides. The diagonal is a common side (so one side equal). Then, the two sides of the parallelogram: one pair with two ticks, one pair with one tick? Wait, no, maybe I misread. Wait, the correct approach: in the diagram, the two triangles share the diagonal (so that's a common side, equal). Then, the other two sides: the top and bottom sides (marked with two ticks) are equal, and the left and right sides (marked with one tick) are equal? Wait, no, maybe the markings are: the top side and bottom side have two ticks (so they are equal), left and right sides have one tick (equal). Then the diagonal is common. Wait, but SSS would be if all three sides are equal. Wait, no, maybe the triangles have three sides equal: the two sides of the parallelogram (so two sides) and the diagonal (common, so third side). So SSS? Wait, no, the original selected option was SAS, but maybe I made a mistake. Wait, no, let's re-examine. Wait, the diagram: the parallelogram has a diagonal. So triangle 1: sides with one tick, two ticks, and diagonal. Triangle 2: sides with one tick, two ticks, and diagonal. So all three sides equal (SSS). Wait, but the options include SSS. Wait, maybe the initial selection was wrong. Wait, no, let's check the congruence criteria. SSS: three sides equal. SAS: two sides and included angle. ASA: two angles and included side. AAS: two angles and non-included side. HL: right triangles, hypotenuse and leg. In a parallelogram, opposite sides are equal, and the diagonal is common. So the two triangles formed by the diagonal have three sides equal: side 1 (equal), side 2 (equal), diagonal (common, so equal). So SSS congruence. Wait, but the original selected option was SAS. Maybe I misread the markings. Wait, maybe the markings are: one side with two ticks, one side with one tick, and the included angle? No, the diagonal is the included side? Wait, no, let's look again. The diagram: the parallelogram has a diagonal. The top and bottom sides have two ticks (so they are congruent), left and right sides have one tick (congruent). The diagonal is common. So the two triangles: each has sides: (two ticks), (one tick), and diagonal. So all three sides equal (SSS). So the correct answer should be SSS Congruence. Wait, but the initial selected option was SAS. Maybe the markings are different. Wait, maybe the two sides with ticks and the included angle. Wait, no, the diagonal is the side between the two sides. Wait, maybe the triangles have two sides equal and the included angle equal. Wait, in a parallelogram, opposite sides are equal and alternate interior angles are equal. Wait, maybe the correct criteria is SSS. Let's confirm: SSS requires three sides. In the parallelogram, AB = CD, AD = BC, and AC is common. So triangle ABC and triangle CDA: AB=CD, BC=AD, AC=CA. So SSS congruence. So the correct answer is SSS Congruence.

Step2: Evaluate the options

- SAS: requires two sides and included angle. But we have three sides equal, so SSS is better.
- HL: not right triangles (no right angle marked).
- AAS: no angles given.
- ASA: no angles given.
- Not Congruent: incorrect, since sides are equal.
- SSS: correct, as three sides are equal (two sides of parallelogram, common diagonal).

Answer:

SSS Congruence (the option with "SSS Congruence" text)