Warmup In triangle ABC, answer the following questions: What side is opposite angle A? What angle is opposite side AB? What angle is included between sides AC and BC? What side is included between angles A and C? What is another way to write angle C? B A C

How can we use AAS and HL to prove triangles congruent? Agenda: Review SSS, SAS, ASA AAS and HL notes/practice Quiz Tomorrow

Practice (from yesterday) Textbook p. 245 #9 – 26 Textbook p. 254 #10 – 14, 16, 17, 19, 21

Determine what is missing in order to use the indicated reason

AAS Theorem If two angles and one of the non-included sides in one triangle are congruent to two angles and one of the non-included sides in another triangle, then the triangles are congruent.

AAS Looks Like… A: ÐK @ ÐM A: ÐKJL @ ÐMJL S: JL @ JL DJKL @ DJML G F A: ÐK @ ÐM A: ÐKJL @ ÐMJL S: JL @ JL DJKL @ DJML J B C D A: ÐA @ ÐD A: ÐB @ ÐG S: AC @ DF ACB  DFG M K L

AAS vs. ASA AAS ASA

Parts of a Right Triangle hypotenuse legs

HL Theorem RIGHT TRIANGLES ONLY! If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.

HL Looks Like… Right Ð: ÐTVW & ÐXVW Right Ð: ÐM & ÐQ H: TW @ XW N T X V Right Ð: ÐTVW & ÐXVW H: TW @ XW L: WV @ WV Right Ð: ÐM & ÐQ H: PN @ RS L: MP @ QS P R NMP  RQS WTV  WXV Q S

There’s no such thing as AAA AAA Congruence: These two equiangular triangles have all the same angles… but they are not the same size!

Recap: There are 5 ways to prove that triangles are congruent: SSS SAS ASA AAS HL

AAS SAS DMLN @ DHJK DABD @ DCBD D A: ÐL @ ÐJ A: ÐM @ ÐH S: LN @ JK A C Examples D M N L A: ÐL @ ÐJ A: ÐM @ ÐH S: LN @ JK H A C B B is the midpoint of AC J S: AB @ BC A: ÐABD @ ÐCBD S: DB @ DB AAS K SAS DMLN @ DHJK DABD @ DCBD

HL DABD @ DCBD ASA DBEA @ DDEC B A C D A: ÐA @ ÐC S: AE @ CE Examples B C A C B E D D A DB ^ AC AD @ CD HL A: ÐA @ ÐC S: AE @ CE A: ÐBEA @ ÐDEC DABD @ DCBD Right Angles: ÐABD & ÐCBD H: AD @ CD L: BD @ BD ASA DBEA @ DDEC

We cannot conclude whether the triangle are congruent. Examples W Z B A C X V D A: ÐWXV @ ÐYXZ S: WV @ YZ Y B is the midpoint of AC SSS DDAB @ DDCB Not Enough! We cannot conclude whether the triangle are congruent. S: AB @ CB S: BD @ BD S: AD @ CD

Practice Textbook p. 254 #15, 18, 25 Textbook p. 260 # 1-12, 15-23

State if the two triangles are congruent and state the reason (SSS, SAS, ASA, AAS, HL) 1. 2. 4. 3.