Question

How can you find out of a triangle is congruent?
I have a test on this and have been struggling.

Answer 1

Answer:

  any of AAS, ASA, HL, SAS, SSS

Step-by-step explanation:

You want to know how to tell triangles are congruent.

Congruence theorems

Triangles are congruent if they can be overlaid on each other exactly. For that to be the case, all corresponding sides and all corresponding angles must have exactly the same measures. Because angle measures and side lengths are related, you don't have to prove all of this to show congruence. Proof of congruence of certain subsets of sides and angles is sufficient.

The congruence theorems are summarized by the acronyms ...

  AAS, ASA, HL, SAS, SSS . . . . . . listed in alphabetical order

In these acronyms, A stands for Angle; S stands for Side; H and L are Hypotenuse and Leg (of a right triangle), respectively. The sequence of A and S is important.

The one (A, S) sequence that is missing from this list is SSA. The HL theorem is essentially SSA applied to a right triangle (only).

Sides and Angles

The attachment shows the markings associated with each of the congruence theorems. Sides (or angles) are congruent when they have the same mark.

In the sorts of geometries you find in problems, there are angle and side relations that you are expected to know from your study of geometry:

• vertical angles are congruent
• a line segment is congruent with itself
• alternate interior and alternate exterior angles are congruent where a transversal crosses parallel lines
• sides (and base angles) of an isosceles triangle are congruent
• a median divides the side of a triangle into two congruent parts
• a segment midpoint divides the segment into two congruent parts
• the sum of angles in a triangle is 180°

Knowing these relations can help you identify congruent parts of triangles even when they are not marked as such.

Order

While two corresponding angles and a corresponding side are sufficient to demonstrate congruence (ASA or AAS), you need to make sure the side has the same relation to the angles in both triangles. To claim AAS congruence, it must be opposite the angle with the same measure or marking in both triangles. Of course, for ASA congruence, the side must lie between the angles.

Please refer to the attachment for ways to show triangle congruence.

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Additional comment

Congruence by "SSA" (HL) works when the given angle is the largest in the triangle, which is to say the first "S" represents the longest side. That is only guaranteed if you know angle measures (90°, in the case of a right triangle).

In most diagrams where you're trying to prove congruence, angle measures are not given, so you cannot claim congruence this way. The appearance of a diagram is not sufficient proof of anything. (They are usually not drawn to scale, and sometimes represent geometry that is actually impossible to draw to scale.)

You may see other right triangle congruence abbreviations: HA, LL, AL (or LA), where the "A" is one of the acute angles. These are right-triangle versions of AAS, SAS, ASA, respectively. You may find it works better to stick with the names shown in the attachment.

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