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 ABCLMK.

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: ACBECD

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.

Guided Practice

1. Fill in the blanks in the proof below.

Given: AB¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯, AC¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯

Prove: ABCDCB

[Figure 4]

 


2. Is the pair of triangles congruent? If so, write the congruence statement and why.

[Figure 5]

Answers:

1

2. The triangles are congruent because they have three pairs of sides congruent. DEFIGH.

Practice

Are the pairs of triangles congruent? If so, write the congruence statement and why.

1

[Figure 6]

2.

[Figure 7]

3.

[Figure 8]


4.

[Figure 9]

State the additional piece of information needed to show that each pair of triangles is congruent.

5. Use SSS

[Figure 10]

6. Use SSS

[Figure 11]

Fill in the blanks in the proofs below.

7. Given: B is the midpoint of DC¯¯¯¯¯¯¯¯, AD¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯. Prove: ABDABC

[Figure 12]

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 ABCXYZ.

[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 ABCRQP 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¯¯¯¯¯¯¯¯.

Watch this video for help with the examples above.

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.

Guided Practice

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

[Figure 18]

2. Fill in the blanks in the proof below.

Given:

SV¯¯¯¯¯¯¯WU¯¯¯¯¯¯¯¯¯

T is the midpoint of SV¯¯¯¯¯¯¯ and WU¯¯¯¯¯¯¯¯¯

Prove: WS¯¯¯¯¯¯¯¯¯UV¯¯¯¯¯¯¯¯

[Figure 19]

3. If two right triangles have congruent hypotenuses and one pair of non-right angles that are congruent, are the two right triangles definitively congruent?

Answers:

1. All we know is that two pairs of sides are congruent. Since we do not know if these are right triangles, we cannot use RHS Congruence theorem. We do not know if these triangles are congruent.

2

Note that even though these were right triangles, we did not use the RHS congruence theorem because we were not originally given that the two hypotenuses were congruent. The SAS congruence shortcut was quicker in this case.

3. Yes, by the AAS Congruence shortcut. One pair of congruent angles is the right angles, and another pair is given. The congruent pair of sides are the hypotenuses that are congruent. Note that just like in #2, even though the triangles are right triangles, it is possible to use a congruence shortcut other than RHS Congruence theorem to prove the triangles are congruent.

Practice

Using the RHS Congruence Theorem, what information do you need to prove the two triangles are congruent?

1. 

[Figure 20]

2. 

[Figure 21]

3. 

[Figure 22]

The triangles are formed by two parallel lines cut by a perpendicular transversal. C is the midpoint of AD¯¯¯¯¯¯¯¯. Complete the proof to show the two triangles are congruent.

[Figure 23]


Based on the following details, are the two right triangles definitively congruent?

4. The hypotenuses of two right triangles are congruent.

5. Both sets of legs in the two right triangles are congruent.

6. One set of legs are congruent in the two right triangles.

7. The hypotenuses and one pair of legs are congruent in the two right triangles.

8. One of the non right angles of the two right triangles is congruent.

9. All of the angles of the two right triangles are congruent.

10. All of the sides of the two right triangles are congruent.

11. Both triangles have one leg that is twice the length of the other.