Triangle Congruencies

1. Triangle Congruencies HOMEWORK: Proofs WS & Triangle Packet - evens Quiz Friday January11, 2012 Lessons 4.3 – 4.5 Triangle Sum, Triangle Inequalities, Side-Angle, Triangle Congruency

2. Congruent Triangles Two Triangles are congruent if they are the same shape and size. To PROVE two triangles congruent means to show that all corresponding parts are congruent. Each triangle has 6 parts: 3 sides & 3 angles

3. However, shortcuts have been discovered and they make our job easier. Possible Triangle Congruency Conjecture Combinations SSS ORDER MATTERS!! S – SIDE AAA The order of the corresponding parts in the triangles must mimic the order of the letters in the Congruency Statements SAS A – ANGLE SSA ASA SAA

4. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides Possible Triangle Congruency Combinations SSS AAA SAS SSA ASA SAA

5. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent sides YES! √ SSS AAA SAS SSA ASA SAA SSS

6. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles 25° 65° 90° √ SSS AAA SAS SSA ASA SAA

7. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Three pairs of congruent angles 65° 90° 25° NO! 65° 90° 25°

8. All Equilateral/Equiangular triangles have three 60° angles. 60° 60° 60° 60° 60° 60° 60° 60° 60° A Counterexample to an AAA congruence conjecture

9. Notes - Congruent Triangles Are these guarantees of triangle congruence? So far this is what has been determined from the list √ SSS Now let’s try combinations with 2 sides and an angle AAA SAS SSA ASA SAA

10. Notes - Congruent Triangles Is this a guarantee of triangle congruence? One pair of congruent angles, two pairs of congruent sides 25° 25° √ SSS AAA SAS SSA ASA SAA

11. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and the included angle √ SSS AAA √ SAS 25° 25° SSA YES! ASA SAS! SAA An included angle is between the two given sides.

12. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and the included angle ABC DEF D A √ SSS E AAA 92° √ SAS 92° SSA B C ASA SAA F

13. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle 65° √ SSS AAA √ SAS SSA ASA SAA

14. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle 65° √ SSS AAA √ SAS SSA ASA SAA

15. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle √ SSS AAA 65° √ SAS SSA ASA SAA

16. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle √ SSS AAA 65° √ SAS SSA ASA SAA

17. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent sides and a non-included angle A ABC DEF B C E √ SSS AAA √ SAS SSA ASA F D SAA NO!

18. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side 25° 90° √ SSS AAA √ SAS SSA ASA SAA

19. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side 25° 25° 90° 90° √ SSS AAA √ SAS SSA ASA YES! SAA ASA!

20. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the included side ABC B DEF 55° A 65° C E √ SSS AAA √ SAS 65° SSA F 55° √ ASA D SAA ASA!

21. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the non-included side 35° 90° √ SSS AAA √ SAS SSA √ ASA SAA

22. Notes - Congruent Triangles Is this a guarantee of triangle congruence? Two pairs of congruent angles and the non-included side 35° 35° 35° 90° YES! √ SSS NO AAA √ SAS SSA √ ASA √ NO SAA

23. Notes - Congruent Triangles These are the triangle congruencies. SSS √ SSS AAA SAS √ SAS SSA √ ASA ASA √ SAA SAA

24. One More Congruency If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. Using Properties of Right triangles In Combination with Triangle Congruencies Hypotenuse –Leg Congruence Theorem (HL)

25. HL (Hypotenuse - Leg) • is not like any of the previous congruence postulates... actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate.  • HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!

26. SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Methods of Proving Triangles Congruent SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

27. Congruent Triangles Name the congruence Is ? FRS PQD R P 120° Q F 120° 35° D 35° ASA S

28. Congruent Triangles Name the congruence Is QSR ? FRS R Q 42° 42° F Shared Side – Reflexive Prop S SSA WHY NOT? SSA is NOT a valid Triangle congruence

29. Congruent Triangles Name the congruence Is ? FRS QSR R 50° Q F Shared Side – Reflexive Prop 50° S SAS

30. Congruent Triangles Name the congruence Is ? MNR PTB B R N P SSS ? T M WHY NOT? Names of the triangles in the congruence statement are not in corresponding order.