# High School: Geometry

### Congruence HSG-CO.B.8

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow the definition of congruence in terms of rigid motions.

Basically, this standard requires students to fall in love with triangles. They should be so head-over-heels that triangles become tri-angels. (That's an angel that's three times as heavenly.)

Of course, falling in love with triangles isn't easy. If your students who think geometry is a four-letter word, they may not want to enter in any spelling bees in the near future. But more than that, they can use three-letter words to save themselves from drowning in the oddly symmetrical sea of geometric congruence.

Specifically, we're talking about the three-letter words SSS, SAS, and ASA. They're actually acronyms for "Side Side Side," "Side Angle Side," and "Angle Side Angle." But we don't judge.

These 3 acronyms represent different combinations of sides and angles that must correspond in order for two triangles to be congruent: all corresponding sides, two corresponding sides and their included angles, or two corresponding angles and any corresponding side. If two triangles meet just one of these three criteria, they're congruent.

Students should know not only what these three rules are and how they work, but also when to use which one based on the situation. Sometimes more than one is applicable!

We also recommend showing why these rules work. A good way to do this is to find all angles and side lengths of two triangles while only given enough to satisfy the SSS, SAS, or ASA rules. Students should understand that when even one of these rules applies, the triangles are congruent not because the rule "says so," but because it's a quick way of realizing that all side lengths and angles are congruent.

Basically, they should understand that the concept of congruence hasn't changed. You've just given them shortcuts. All the more reason to fall in love with triangles, we think. In fact, don't be surprised if your students have already picked a date for the wedding.

Need a recap? Here is a video to help you out on ASA, SAS, SAS....

...and SSS.

#### Drills

1. Which of the three rules of triangle congruence does not rely on side to angle correspondence?

SSS

Only SSS does not rely on one element corresponding to another. The SAS rule states in part that the mutual sides between equal corresponding sides must also be equal. In ASA, the triangles must share two equal corresponding angles and also any pair of corresponding sides. There is no such thing as the AAA rule (too bad for the car insurance company), so our only remaining option is the SSS rule.

2. Which of the following statements is not correct?

The SAS postulate states that if 2 angles of a triangle and the side between them are congruent to the corresponding parts in another, the triangles are congruent

Answers (A) and (B) are the true definitions of the SSS and SAS postulates, so they are incorrect. The ASA postulate actually says that if 2 angles and any side in one triangle are congruent to those same parts in another, then the triangles are congruent. The corresponding congruent sides in the triangles don't have to be "in between" the 2 corresponding angles. So the ASA definition provided is incorrect and therefore the right answer.

3. Which postulate would prove that ΔABC is congruent to ΔDEF?

None of the above

No postulate can prove these triangles are congruent. Why? Because they aren't. Notice the measurement of the hypotenuse in each case: 3 for ΔABC and 2.5 for ΔDEF. Although the triangles look congruent, looks can be deceiving. All it takes is for one pair of corresponding angles or sides to be different, and it's all over. The beautiful image of congruence is forever shattered, and no amount of three-letter rules can repair it. How depressing.

4. Can we prove ΔLMN is congruent to ΔXYZ? If so, which postulate can we use to do so?

No, we cannot prove they are congruent because none of the three postulates can be used

It is true that two corresponding sides are congruent and two corresponding angles are congruent. But the angle that is equal is not between the sides that are equal, so the SAS postulate cannot be used. Neither can SSS or ASA, as we don't have sufficient information to use either one. That makes answers (B) and (C) incorrect. Therefore, (D) is correct.

5. Two triangles have three pairs of corresponding angles. Which of the following is true?

We cannot prove they are congruent because we do not know the lengths of the sides

Though we know the two triangles have angles of equal measure, we would also need to know the sides are all of equal lengths. But the side lengths are not given here. Just because two triangles are equiangular does not automatically mean they are congruent. Answer (C) is incorrect because we do know the angle measurements. We just need the measurements of the corresponding sides.

6. Why is SSA not one of the postulates we can use to prove triangle congruence?

The side opposite the known angle can be flipped to create two different triangles

Right off the bat, we know that (B) is not true because of SAS and SSS. We also know that (C) is not true for any obtuse or right triangle, since the remaining angles will have to add up to 90° or less. If we draw an angle with one known side and two adjacent sides, we can create two triangles by flipping the last side back and forth. Since these two unique triangles can be made, we can't know for sure which one is which without more information.

7. If ABDE, ACDF, and ∠B ≅ ∠E, are ΔABC and ΔDEF congruent?

No, because the given information would mean SSA would be needed to prove congruence and there is no SSA postulate

Although two sides and one angle are congruent, the known angle is not in between the two sides. Because of this, we can't use SAS, but there is no such thing as SSA, either. Knowing two sides doesn't automatically mean we know the third, and there is no indication that ∠B and ∠E are right angles. Sight also isn't reliable to use when deciding on triangle congruence, so the only answer we can give is (D).

8. Even if we know that all three corresponding angles are congruent (AAA), we cannot prove congruence between two triangles. Why not?

The triangles could have the same shape but not the same size.

You can picture it. The three angles say nothing at all about any side (except for the fact that the sides go in the same direction). Yes, the two triangles are the same shape, but we can dilate and compress either triangle as much as we want. They won't be congruent then, so there's no way to prove it. The triangles are the same shape, but they could have very different sizes.

9. If a is congruent to d and b is congruent to e, are the two triangles congruent?

Yes, by the SSS postulate

Since a is congruent to d and b is congruent to e, we have two pairs of corresponding sides. That already eliminates (C) from the equation. To prove congruence, we just need the angles in between the two pairs of sides to correspond or a third pair of matching sides. The only other side we have is c, and we certainly know that it's congruent to itself. All three corresponding sides are congruent, which means we can use SSS to prove that both triangles are congruent.

10. Why are pairs of triangles that satisfy the SSS, SAS, and ASA postulates congruent?

Because all sides and angles will be congruent for triangles that satisfy these postulates