Reflexive Property of Congruence If , then If and

U T V a Figure 3.5 SAS METHOD FOR PROVING TRIANGLES CONGRUENT A second way of establishing that two triangles are congruent involves showing that two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle. If two people each draw a triangle so that two of the sides measure 2 cm and 3 cm and their included angle measures 54°, then those triangles are congruent. See Figure 3.6. A C B b 54° a 2 3 Figure 3.6 54° b 2 3 Informally, the term include names the part of a triangle that is “between” two other named parts. EXAMPLE 4 In of Figure 3.5b: a Which angle is included by and ? b Which sides include ? c What is the included side for and ? d Which angles include ? Solution a because it is formed by and b and because these form c because it is the common side for and d and because is a side of each angle CB ∠B ∠C ∠B ∠A AB ∠B BC AB CB AC ∠C CB ∠B ∠A ∠B CB AC 䉭ABC If two sides and the included angle of one triangle are congruent to two sides and the in- cluded angle of a second triangle, then the triangles are congruent SAS. POSTULATE 13 The order of the letters SAS in Postulate 13 helps us to remember that the two sides that are named have the angle “between” them. That is, in each triangle, the two sides form the angle. In Example 5, which follows, the two triangles to be proved congruent share a com- mon side; the statement is justified by the Reflexive Property of Congruence, which is conveniently expressed as Identity. PN ⬵ PN In this context, Identity is the reason we cite when verifying that a line segment or an angle is congruent to itself; also known as the Reflexive Property of Congruence. DEFINITION 쮿 ASA METHOD FOR PROVING TRIANGLES CONGRUENT The next method for proving triangles congruent requires a combination of two angles and the included side. If two people each draw a triangle for which two of the angles measure 33° and 47° and the included side measures 5 centimeters, then those triangles are congruent. See the figure below. 1 2 N Q M P Figure 3.7 a 5 33° 47° Figure 3.8 b 5 33° 47° EXAMPLE 5 GIVEN: See Figure 3.7. PROVE: 䉭PNM ⬵ 䉭PNQ MN ⬵ NQ PN ⬜ MQ In Example 5, note the use of Identity and SAS as the final reasons. PROOF Statements Reasons

1. 2.

3. 4.

5. 䉭PNM ⬵ 䉭PNQ PN ⬵ PN MN ⬵ NQ ∠1 ⬵ ∠2 PN ⬜ MQ 1. Given 2. If two lines are , they meet to form adjacent 3. Given 4. Identity or Reflexive 5. SAS ∠s ⬵ ⬜ NOTE: In , step 3 and step 4 include ; similarly, and include in . Thus, SAS is used to verify that in reason 5. 䉭PNM ⬵ 䉭PNQ 䉭PNQ ∠2 PN NQ ∠1 PN MN 䉭PNM 쮿 If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent ASA. POSTULATE 14 Although this method is written compactly as ASA, you must be careful as you write these abbreviations For example, ASA refers to two angles and the included side, whereas SAS refers to two sides and the included angle. For us to apply any postulate, the specific conditions described in it must be satisfied. Exs. 3–6 SSS, SAS, and ASA are all valid methods of proving triangles congruent, but SSA is not a method and cannot be used. In Figure 3.9, the two triangles are marked to show SSA, yet the two triangles are not congruent. Another combination that cannot be used to prove triangles congruent is AAA. See Figure 3.10. Three congruent pairs of angles in two triangles do not guarantee congruent pairs of sides In Example 6, the triangles to be proved congruent overlap see Figure 3.11. To clarify relationships, the triangles have been redrawn separately in Figure 3.12. Note that the parts indicated as congruent are established as congruent in the proof. For state- ment 3, Identity or Reflexive is also used to justify that an angle is congruent to itself. a 20° 5 2 Figure 3.9 b 20° 5 2 Figure 3.10 A B D E F 2 1 C Figure 3.11 1 A C E Figure 3.12 2 C D B EXAMPLE 6 GIVEN: See Figure 3.11. PROVE: 䉭ACE ⬵ 䉭DCB ∠1 ⬵ ∠2 AC ⬵ DC Next we consider a theorem proved by the ASA postulate that is convenient as a reason in many proofs. AAS METHOD FOR PROVING TRIANGLES CONGRUENT PROOF Statements Reasons 1. See Figure 3.12. 2. 3. 4. 䉭ACE ⬵ 䉭DCB ∠C ⬵ ∠C ∠1 ⬵ ∠2 AC ⬵ DC 1. Given 2. Given 3. Identity 4. ASA 쮿 If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of a second triangle, then the triangles are congruent AAS. THEOREM 3.1.1 GIVEN: , , and See Figure 3.13 on page 134. PROVE: 䉭TSR ⬵ 䉭KJH SR ⬵ HJ ∠S ⬵ ∠J ∠T ⬵ ∠K Exs. 7–11 T S R Figure 3.13 K J H D B A C C A B 11 8 10 Exercises 2–4, 6 c D F E a b Warning Do not use AAA or SSA, because they are simply not valid for proving triangles congruent; with AAA the triangles have the same shape but are not necessarily congruent. PROOF Statements Reasons

1. 2.

3. 4. 䉭TSR ⬵ 䉭KJH SR ⬵ HJ ∠R ⬵ ∠H ∠S ⬵ ∠J ∠T ⬵ ∠K

1. Given 2. If two

of one are to two of another , then the third are also congruent 3. Given 4. ASA ∠s 䉭 ∠s ⬵ 䉭 ∠s General Rule: Methods of proof possible final reasons available in Section 3.1 are SSS, SAS, ASA, and AAS. Illustration: See Exercises 9–12 of this section. STRATEGY FOR PROOF 왘 Proving That Two Triangles Are Congruent Exercises 3.1 In Exercises 1 to 8, use the drawings provided to answer each question.

1. Name a common angle and a common side for

and . If , can you conclude that and are congruent? Can SSA be used as a reason for proving triangles congruent? 䉭ABD 䉭ABC BC ⬵ BD 䉭ABD 䉭ABC

4. With corresponding angles indicated, find if

and . m ∠C = 85° m ∠A = 57° m ∠E For Exercises 2 and 3, see the figure in the second column.

2. With corresponding angles indicated, the triangles are

congruent. Find values for a, b, and c.

3. With corresponding angles indicated, find if

. m ∠F = 72° m ∠A 5. In a right triangle, the sides that form the right angle are the legs; the longest side opposite the right angle is the hypotenuse. Some textbooks say that when two right triangles have congruent pairs of legs, the right triangles are congruent by the reason LL. In our work, LL is just a special case of one of the postulates in this section. Which postulate is that?

6. In the figure for Exercise 2, write a statement that the

triangles are congruent, paying due attention to the order of corresponding vertices. Exs. 12–14 A B D F E C

7. In , the midpoints of the sides are joined. What does

intuition tell you about the relationship between and ? We will prove this relationship later. 䉭FDE 䉭AED 䉭ABC

20. 䉭MNP ⬵ 䉭MQP

13. , ,

and

14. , ,

and B is the midpoint of

15. , ,

and bisects

16. , ,

and

17. , ,

and by Identity

18. and are

right , , and In Exercises 19 and 20, the triangles to be proved congruent have been redrawn separately. Congruent parts are marked. a Name an additional pair of parts that are congruent by Identity. b Considering the congruent parts, state the reason why the triangles must be congruent.

19. 䉭ABC ⬵ 䉭AED

∠A ⬵ ∠D AB ⬵ BD ∠s ∠2 ∠1 CB ⬵ CB AB ⬵ BD AC ⬵ CD AB ⬵ BD AC ⬵ CD ∠A ⬵ ∠D ∠ACD CB AC ⬵ CD ∠A ⬵ ∠D AD AC ⬵ CD ∠A ⬵ ∠D ∠1 ⬵ ∠2 AB ⬵ BD ∠A ⬵ ∠D In Exercises 13 to 18, use only the given information to state the reason why . Redraw the figure and use marks like those used in Exercises 9 to 12. 䉭ABC ⬵ 䉭DBC

12. 11.

10. In Exercises 9 to 12, congruent parts are indicated by like dashes sides or arcs angles. State which method SSS, SAS, ASA, or AAS would be used to prove the two triangles congruent.

9. 8. Suppose that you wish to prove that

. Using the reason Identity, name one pair of corresponding parts that are congruent. 䉭RST ⬵ 䉭SRV T V S R C A B F D E R S T X Y Z N P Q S R M G H J K L I D B 1 2 3 4 A C Exercises 13–18 N Q P M N Q P M P M D D E E C B B C A A A In Exercises 21 to 24, the triangles named can be proven congruent. Considering the congruent pairs marked, name the additional pair of parts that must be congruent for us to use the method named.

21. SAS

PROOF Statements Reasons 1. and 1. ? 2. ? 2. Identity 3. 3. ? 䉭ABC ⬵ 䉭CDA AD ⬵ CB AB ⬵ CD In Exercises 25 and 26, complete each proof. Use the figure at the top of the second column.

25. Given: and

Prove: 䉭ABC ⬵ 䉭CDA AD ⬵ CB AB ⬵ CD 22. ASA

23. SSS

24. AAS

A D E C B ⌬ABD ⬵ ⌬CBE ⌬WVY ⬵ ⌬ ZVX W X Y Z V ⌬MNO ⬵ ⌬OPM M P N O ⌬ EFG ⬵ ⌬ JHG E F G J H A B D C Exercises 25, 26

26. Given: and

Prove: PROOF Statements Reasons 1. 1. ? 2. 2. ? 3. ? 3. Given 4. ? 4. If two lines are cut by a transversal, alt. int. are 5. 5. ? 6. ? 6. ASA AC ⬵ AC ⬵ ∠s 7 ∠DCA ⬵ ∠BAC DC 7 AB 䉭ABC ⬵ 䉭CDA AD 7 BC DC 7 AB

27. Given: bisects

Prove:

28. Given: and

Prove:

29. Given: and

Prove: 䉭ABC ⬵ 䉭ABD BC ⬵ BD AB ⬜ BD AB ⬜ BC 䉭MQP ⬵ 䉭NQP ∠1 ⬵ ∠2 PQ ⬜ MN 䉭MQP ⬵ 䉭NQP MP ⬵ NP ∠MPN PQ In Exercises 27 to 32, use SSS, SAS, ASA, or AAS to prove that the triangles are congruent. M N P Q 1 2 Exercises 27, 28 C B D A