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Ways to Prove Triangles Congruent (ASA, SAS and SSS) SM1—Section 9.2 Date ____________ Objectives: 1. Use the SSS Postulate, the SAS Postulate and the ASA Postulate to prove two triangles congruent. 2. Identify corresponding parts, included angle and included side. 3. Understand why AAA and SSA are not sufficient to prove triangle congruence. If you know that two triangles are congruent, you can conclude that the six parts of one triangle are equal to the six parts of the other triangle. If you are not sure whether two triangles are congruent, however, it is not necessary to compare all six pairs of parts. You need to check only three pairs of parts. Activity – Compare Triangles with Different Sides Cut straws to makes side lengths of 4 cm, 5 cm and 6 cm. Make a triangle with the three lengths. Trace the result here: Mix the straws up, and make another triangle. If possible, make it different from the first. Trace your new triangle onto patty paper. Compare this triangle to the first triangle you drew above and write what you notice. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ In this activity, you can see that there is only one way to form a triangle given three side lengths. In general, any two triangles with the same three side lengths must be congruent. Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. L A N M C B If AB LM and AC LN and BC MN , then Postulate (SSS). ABC LMN by the Side Side Side O M N P Only two pairs of sides are marked as equal in MOP and NOP . Since OP OP by the Reflexive Property, we can conclude that MOP NOP by the SSS Postulate. Included Sides and Included Angles EG is opposite F E EG is included between E and G FG is opposite E FG is included between F and G F G Mark the included side: I 1. 2. 3. G I H Z F J P E Mark the included angle: 4. 5. M R 6. Y Z T X N O S Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. S In this figure, you are given that ST TP and T STR PTR . You can also see that TR TR by the Reflexive Property. R Therefore, RST RPT by SAS. P Name the congruent triangles and give a reason: M 7. L A D 8. 9. Y N B G O C J M K L A K ________________ _________________ _______________ ________________ _________________ _______________ Activity -- Does ASA work to show triangle congruence? Suppose you cut two angles out of a piece of paper and place them at a fixed distance apart. Can you form more than one triangle with a two given angles and given length as shown below? 50 30 5 cm In the space below, using a protractor, draw a 30° angle. Measure and mark one side of that angle to be 5 cm long. At that endpoint, construct a 50 ° angle. Could you draw more than one triangle using this exact combination? Explain. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ From this activity, you can see that there is only one way to form a triangle given two angles and the included side. Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. A R B C S Q QS is included between Q and S . BC is included between B and C . QS BC , B Q and C S . Therefore, QSR BCA by Angle-Side-Angle (ASA). Does AAA work to show triangle congruence? W S B X V T R A F C As you can see in the first set of triangles, VWX and RST , all the corresponding angles are congruent, but corresponding sides are not; therefore, the triangles are not congruent. In the second picture, ABC and AGF again have corresponding angles congruent, but the sides are not; therefore the triangles are not congruent. So, Angle-Angle-Angle DOES NOT prove triangle congruence. Name the congruent triangles and give a reason, if possible: 10. T Q 11. S 12. U R P G A D B X C H I Y Z ________________ _________________ _______________ ________________ _________________ _______________ Activity -- Does SSA work to show congruence of triangles? Draw a horizontal line using a straightedge like the first drawing below. Connect two straws of unequal length with at the ends, then position the straws so that they form a triangle with the segment you just drew. Make the shorter straw on the right side. Anchor the longer straw with tape, and label the vertices of the triangle A, B, and C, like the first picture. Now swing the straw representing BC so that the end of the straw touches the segment again. Label this point D. B A B C A D C Is triangle ABC congruent to triangle ABD? Is SSA a valid test for determining triangle congruence? Explain your reasoning. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ In this activity, you can see that when the angle is not included between the two corresponding congruent sides, two distinctly different triangles can be formed; therefore, you cannot use SSA to prove two triangles congruent. State the reason the following triangles can be proven congruent, if possible: 13. 14. 15. 16.