Question

How can you tell that triangles are congruent without knowing the lengths of all sides and the measures of all angles?

the _____ congruency theorems can all be used

Answer 1

We can say that two triangles are congruent using these 5 rules:

1: SSS or the side, side, side rule, that means that we have two triangles with all three sides equal.

2. SAS or side, angle, side states that in the two triangles, two sides and the included angle are equal.

3. ASA or angle, side, angle states that in two triangles, the two angles and the included side are equal.

4. AAS or angle, angle, side states that we have two triangles where two angles and the non-included side are equal.

5. HL or the Hypotenuse Leg rule states that, if the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, then the two triangles are congruent.

Answer 2

• SAS
• ASA
• AAS
• HL
• SSA (conditionally — see below)

Explanation:

Of the 5 triangle congruence theorems, the only one that cannot be used is SSS, which requires you know all three sides.

If you don't know all three sides, you must know at least one angle, at least one side, and at least one additional angle or side (see below).

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If the sides you know (in relation to the angle you know) are SSA, then the first side (opposite the angle) must be known to be the longest, or congruence is not assured. This is why SSA is not usually listed among the congruence theorems. (However, HL is a version of this, as the right angle will be opposite the longest side.)