Some More Criteria for Congruence of Triangles

Theorem 4 - SSS Congruence Rule

SSS Triangle Congruence

What if your parents were remodeling their kitchen so that measurements between the sink, refrigerator, and oven are in the picture at the left, below. Your neighbor’s kitchen has the measurements on the right, below. Are the two triangles congruent? After completing this Concept, you'll be able to determine whether or not two triangles are congruent given only their side lengths.

[Figure 1]

Side-Side-Side (SSS) Triangle Congruence Theorem: If three sides in one triangle are congruent to three sides in another triangle, then the triangles are congruent.

Now, we only need to show that all three sides in a triangle are congruent to the three sides in another triangle. We no longer have to show 3 sets of angles are congruent and 3 sets of sides are congruent in order to say that the two triangles are congruent.

Example A

Write a triangle congruence statement based on the diagram below:

[Figure 2]

From the tic marks, we know AB¯¯¯¯¯¯¯¯≅LM¯¯¯¯¯¯¯¯¯,AC¯¯¯¯¯¯¯¯≅LK¯¯¯¯¯¯¯¯,BC¯¯¯¯¯¯¯¯≅MK¯¯¯¯¯¯¯¯¯¯. Using the SSS Congruence rule we know the two triangles are congruent. Lining up the corresponding sides, we have △ABC≅△LMK.

Don’t forget ORDER MATTERS when writing triangle congruence statements. Here, we lined up the sides with one tic mark, then the sides with two tic marks, and finally the sides with three tic marks.

Example B

Write a two-column proof to show that the two triangles are congruent.

Given: AB¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯

[Figure 3]

C is the midpoint of AE¯¯¯¯¯¯¯¯ and DB¯¯¯¯¯¯¯¯.

Prove: △ACB≅△ECD

Statement	Reason
1. AB¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯ C is the midpoint of AE¯¯¯¯¯¯¯¯ and DB¯¯¯¯¯¯¯¯	Given
2. AC¯¯¯¯¯¯¯¯≅CE¯¯¯¯¯¯¯¯,BC¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯	Definition of a midpoint
3. △ACB≅△ECD	SSS Congruence rule

Make sure that you clearly state the three sets of congruent sides BEFORE stating that the triangles are congruent.

Feel free to mark the picture with the information you are given as well as information that you can infer (vertical angles, information from parallel lines, midpoints, angle bisectors, right angles).

Concept Problem Revisited

From what we have learned in this section, the two triangles are not congruent because the distance from the fridge to the stove in your house is 4 feet and in your neighbor’s it is 4.5 ft. The SSS Congruence rule tells us that all three sides have to be congruent.

Vocabulary

Two figures are congruent if they have exactly the same size and shape. By definition, two triangles are congruent if the three corresponding angles and sides are congruent. The symbol ≅ means congruent. There are shortcuts for proving that triangles are congruent. The SSS Triangle Congruence Rule states that if three sides in one triangle are congruent to three sides in another triangle, then the triangles are congruent.

Theorem 5 - RHS Congruence Rule

HL Triangle Congruence

What if you were given two right triangles and provided with only the measure of their hypotenuses and one of their legs? How could you determine if the two right triangles were congruent? After completing this Concept, you'll be able to use the Hypotenuse-Leg (HL) shortcut to prove right triangles are congruent.

Guidance

Recall that a right triangle has exactly one right angle. The two sides adjacent to the right angle are called legs and the side opposite the right angle is called the hypotenuse.

[Figure 13]

The Pythagorean Theorem says, for any right triangle, (leg)2+(leg)2=(hypotenuse)2. What this means is that if you are given two sides of a right triangle, you can always find the third. Therefore, if you know that two sides of a right triangle are congruent to two sides of another right triangle, you can conclude that third sides are also congruent.

RHS Congruence Theorem: If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, then the two triangles are congruent.

The markings in the picture are enough to say △ABC≅△XYZ.

[Figure 14]

Notice that this theorem is only used with a hypotenuse and a leg. If you know that the two legs of a right triangle are congruent to two legs of another triangle, the two triangles would be congruent by SAS, because the right angle would be between them.

Example A

What information would you need to prove that these two triangles are congruent using the RHS Congruence Theorem?

[Figure 15]

For RHS Congruence condition, we need the hypotenuses to be congruent. So, AC¯¯¯¯¯¯¯¯≅MN¯¯¯¯¯¯¯¯¯¯.

Example B

Determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

[Figure 16]

We know the two triangles are right triangles. The have one pair of legs that is congruent and their hypotenuses are congruent. This means that △ABC≅△RQP by RHS Congruence theorem.

Example C

Determine the additional piece of information needed to show the two triangles are congruent by RHS Congruence theorem.

[Figure 17]

We already know one pair of legs is congruent and that they are right triangles. The additional piece of information we need is that the two hypotenuses are congruent, UT¯¯¯¯¯¯¯≅FG¯¯¯¯¯¯¯¯.

Vocabulary

Two figures are congruent if they have exactly the same size and shape. By definition, two triangles are congruent if the three corresponding angles and sides are congruent. The symbol ≅ means congruent. There are shortcuts for proving that triangles are congruent. The RHS Congruence Theorem states that if the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, then the two triangles are congruent. A right triangle has exactly one right (90∘) angle. The two sides adjacent to the right angle are called legs and the side opposite the right angle is called the hypotenuse.