# Proving Triangles Congruent

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Proving Triangles Congruent
Part 2

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: ÐM A: ÐMJL S: JL DJML J B C D A: ÐD A: ÐG S: 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: XW L: WV Right Ð: ÐM & ÐQ H: RS L: 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: ÐJ A: ÐH S: JK H A C B B is the midpoint of AC J S: BC A: ÐCBD S: DB AAS K SAS DHJK 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 CD HL A: ÐC S: CE A: ÐDEC DCBD Right Angles: ÐABD & ÐCBD H: CD L: BD ASA DDEC

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