5.5/5.6 Proving Triangles Congruent

SSS and SAS What You'll Learn You will learn to use the SSS and SAS tests for congruency.

4) Construct a segment congruent to CB. 5) Label the intersection F. SSS and SAS 4) Construct a segment congruent to CB. 5) Label the intersection F. 2) Construct a segment congruent to AC. Label the endpoints of the segment D and E. 1) Draw an acute scalene triangle on a piece of paper. Label its vertices A, B, and C, on the interior of each angle. 6) Draw DF and EF. 3) Construct a segment congruent to AB. A C B D E F This activity suggests the following postulate.

Triangles are congruent. sides three corresponding SSS and SAS Postulate 5-1 SSS Postulate If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the two Triangles are congruent. sides three corresponding A B C R S T If AC  RT and AB  RS and BC  ST then ΔABC  ΔRST

In two triangles, ZY  FE, XY  DE, and XZ  DF. SSS and SAS In two triangles, ZY  FE, XY  DE, and XZ  DF. Write a congruence statement for the two triangles. X D Z Y F E Sample Answer: ΔZXY  ΔFDE

In a triangle, the angle formed by two given sides is called the SSS and SAS In a triangle, the angle formed by two given sides is called the ____________ of the sides. included angle C is the included angle of CA and CB A B C A is the included angle of AB and AC B is the included angle of BA and BC Using the SSS Postulate, you can show that two triangles are congruent if their corresponding sides are congruent. You can also show their congruence by using two sides and the ____________. included angle

If ________ and the ____________ of one triangle are SSS and SAS Postulate 5-2 SAS Postulate If ________ and the ____________ of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent. two sides included angle A B C R S T If AC  RT and A  R and AB  RS then ΔABC  ΔRST

NO! On a piece of paper, write your response to the following: SSS and SAS On a piece of paper, write your response to the following: Determine whether the triangles are congruent by SAS. If so, write a statement of congruence and tell why they are congruent. If not, explain your reasoning. P R Q F E D NO! D is not the included angle for DF and EF.

End of Section 5.5

ASA and AAS What You'll Learn You will learn to use the ASA and AAS tests for congruency.

It is the one side common to both angles. ASA and AAS The side of a triangle that falls between two given angles is called the ___________ of the angles. included side It is the one side common to both angles. A B C AC is the included side of A and C CB is the included side of C and B AB is the included side of A and B You can show that two triangles are congruent by using _________ and the ___________ of the triangles. two angles included side

If _________ and the ___________ of one triangle are ASA and AAS Postulate 5-3 ASA Postulate If _________ and the ___________ of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent. two angles included side A B C R S T If A  R and AC  RT and C  T then ΔABC  ΔRST

CA and CB are the nonincluded ASA and AAS CA and CB are the nonincluded sides of A and B A B C You can show that two triangles are congruent by using _________ and a ______________. two angles nonincluded side

If _________ and a ______________ of one triangle are ASA and AAS Theorem 5-4 AAS Theorem If _________ and a ______________ of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. two angles nonincluded side A B C R S T If A  R and C  T and CB  TS then ΔABC  ΔRST

ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to ASA and AAS ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent by AAS? If F and M are marked congruent, then FE and MN would be included sides. However, AAS requires the nonincluded sides. Therefore, D and L must be marked. D F E L M N

End of Section 5.6