Exercise 33 Page 276

a Combinations:

SSS	SAS	ASA	AAS
1. TU≅ XY, UV≅ YZ, and TV≅ XZ	2. TU≅ XY, ∠ T ≅ ∠ X, and TV≅ XZ	5. ∠ T ≅ ∠ X, TU≅ XY, and ∠ U ≅ ∠ Y	8. ∠ T ≅ ∠ X, ∠ U ≅ ∠ Y, and UV≅ YZ
	2. TU≅ XY, ∠ T ≅ ∠ X, and TV≅ XZ	5. ∠ T ≅ ∠ X, TU≅ XY, and ∠ U ≅ ∠ Y	9. ∠ T ≅ ∠ X, ∠ U ≅ ∠ Y, and TV≅ XZ
	3. TU≅ XY, ∠ U ≅ ∠ Y, and UV≅ YZ	6. ∠ U ≅ ∠ Y, UV≅ YZ, and ∠ V ≅ ∠ Z	10. ∠ U ≅ ∠ Y, ∠ V ≅ ∠ Z, and TV≅ XZ
	3. TU≅ XY, ∠ U ≅ ∠ Y, and UV≅ YZ	6. ∠ U ≅ ∠ Y, UV≅ YZ, and ∠ V ≅ ∠ Z	11. ∠ U ≅ ∠ Y, ∠ V ≅ ∠ Z, and TU≅ XY
	4. UV≅ YZ, ∠ V ≅ ∠ Z, and TV≅ XZ	7. ∠ V ≅ ∠ Z, TV≅ XZ, and ∠ T ≅ ∠ X	12. ∠ V ≅ ∠ Z, ∠ T ≅ ∠ X, and TU≅ XY
	4. UV≅ YZ, ∠ V ≅ ∠ Z, and TV≅ XZ	7. ∠ V ≅ ∠ Z, TV≅ XZ, and ∠ T ≅ ∠ X	13. ∠ V ≅ ∠ Z, ∠ T ≅ ∠ X, and UV≅ YZ

Practice makes perfect

a Let's first list the statements that can be used to claim congruence between two non-right triangles:

&SSS Congruence Theorem &SAS Congruence Theorem &ASA Congruence Theorem &AAS Congruence Theorem Let's go through these statements one at a time for the given triangles. Note that there is no particular order you have to pick the statements. As long as they fulfill the criteria of the congruence statements, it's all good.

SSS Congruence Theorem

To prove congruence using the SSS Congruence Theorem, we need to know all three sides. Therefore, there is only one combination of the given statements that allows us to use SSS: 1. TU≅ XY, UV≅ YZ, and TV≅ XZ

SAS Congruence Theorem

The SAS Congruence Theorem requires us to know two sides and the included angle. There will be three combinations: &1. TU≅ XY, ∠ T ≅ ∠ X, and TV≅ XZ &2. TU≅ XY, ∠ U ≅ ∠ Y, and UV≅ YZ &3. UV≅ YZ, ∠ V ≅ ∠ Z, and TV≅ XZ

ASA Congruence Theorem

The ASA Congruence Theorem requires us to know two angles and the included side. Since there are three sides, there will be three combinations. &1. ∠ T ≅ ∠ X, TU≅ XY, and ∠ U ≅ ∠ Y &2. ∠ U ≅ ∠ Y, UV≅ YZ, and ∠ V ≅ ∠ Z &3. ∠ V ≅ ∠ Z, TV≅ XZ, and ∠ T ≅ ∠ X

AAS Congruence Theorem

The AAS Congruence Theorem requires us to know two angles and a non-included side. Here we will have 6 combinations as the non-included side could be in two places: &1. ∠ T ≅ ∠ X, ∠ U ≅ ∠ Y and UV≅ YZ. &2. ∠ T ≅ ∠ X, ∠ U ≅ ∠ Y and TV≅ XZ. &3. ∠ U ≅ ∠ Y, ∠ V ≅ ∠ Z and TV≅ XZ. &4. ∠ U ≅ ∠ Y, ∠ V ≅ ∠ Z and TU≅ XY. &5. ∠ V ≅ ∠ Z, ∠ T ≅ ∠ X and TU≅ XY. &6. ∠ V ≅ ∠ Z, ∠ T ≅ ∠ X and UV≅ YZ.

Number of statements

In total there are 13 combinations of the given statements that will provide enough information to prove that ∠ TUV is congruent to ∠ XYZ.

b To calculate probability, we divide the number of favorable outcomes by the number of possible outcomes.

P=Number of favorable outcomes/Number of possible outcomes

Number of favorable outcomes

From A, we found that there are 13 favorable combinations. However, due to the fact that order did not matter, some of the total possible outcomes are repeats of those 13. Here, we take those repeats into account. But how many repeats are there?

Handling the repeated outcomes

Consider only the case of SSS where we draw only statements regarding sides each time. The first pick could be any of the three statements about sides. The second pick must be one of the remaining two statements about sides. Finally, we are only left with one statement about sides.

SSS Permutations
1.TU ≅ XY and UV ≅ YZ and TV ≅ XZ
2.TU ≅ XY and TV ≅ XZ and UV ≅ YZ
3.UV ≅ YZ and TU ≅ XY and TV ≅ XZ
4.UV ≅ YZ and TV ≅ XZ and TU ≅ XY
5.TV ≅ XZ and TU ≅ XY and UV ≅ YZ
6.TV ≅ XZ and UV ≅ YZ and TU ≅ XY

Thus, there are 6 ways to pick the three statements about sides. As it happens, there are 6 ways to get each of the 13 statements found in part A. This means that we have 6* 13 = 78 possible outcomes that are favorable.

Number of possible outcomes

Having made the first pick, we have 5 remaining statements from which to choose for the second pick. Similarly, when we make the third pick, we have 4 remaining statements from which to choose. That means for each of the 6 possible first picks, we have the following.

That leaves us with 6*5*4=120 total different possible outcomes.

Probability

All-in-all, there are 120 different permutations. Since 78 of them result in enough information to prove congruence, the probability of picking such a combination is P=78/120=13/20=65 %.

Subchapter links

Communicate Your Answer p.269

Monitoring Progress p.271-273