Congruent Triangles

1. Congruent Triangles • Can I have a volunteer read today’s objective? • SWBAT justify that two triangles are congruent using the congruence postulate theorems.

2. Quick Review: 6 minutes If you get stuck, try to think about these questions or hints. • What does opposite mean? • The smallest side is opposite the smallest angle. • The biggest side is opposite the biggest angle. • Triangles have the EXACT same measures for all angles and sides.

3. POINTS I MUST KNOW !!!

4. Remember… • Tests try to trip you up! Watch out so you can beat them! For Geometry, that means • Never assume an angle is right unless you see the square • Use what you know, not what it looks like. This is really important for size & measurements!

5. Congruent Triangles • Congruent triangles have three congruent sides and and three congruent angles. • However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles.

6. 30° 30° Example 1. Why aren’t these triangles congruent? 2. What do we call these triangles?

7. FOR ALL TRIANGLES SSS ASA AAS SAS FOR RIGHT TRIANGLES ONLY HL LL HA LA The Triangle Congruence Postulates &Theorems

8. 1. Write the theorem or postulate that matches with the picture.2. Write the congruency statement. Be sure to match up the appropriate parts.

9. So, how do we prove that two triangles really are congruent?

10. A C B D F E ASA (Angle, Side, Angle) • If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . . then the 2 triangles are CONGRUENT!

11. A C B D F E AAS (Angle, Angle, Side)Special case of ASA • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . . then the 2 triangles are CONGRUENT!

12. A C B D F E SAS (Side, Angle, Side) • If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . . then the 2 triangles are CONGRUENT!

13. A C B D F E SSS (Side, Side, Side) • In two triangles, if 3 sides of one are congruent to three sides of the other, . . . then the 2 triangles are CONGRUENT!

14. A C B D F E HL (Hypotenuse, Leg) • If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

15. A C D B F E HA (Hypotenuse, Angle) • If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

16. A C B D F E LA (Leg, Angle) • If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

17. A C B D F E LL (Leg, Leg) • If both pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT!

18. A C B Example 1 A S A D E F

19. A C B D E F Example 2 • S S A

20. A C B D Example 3 • S S S

21. D F E A C B Example 4 SAS

22. B C A D Example 5 S S S

23. Example 6 A SSS D B C

24. Q P T R S Example 7 mQSR = mPRS = 90° • Given: • Are the Triangles Congruent? Why? R H S QSR  PRS = 90°

25. Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS – Pairs of congruent angles and the side not contained between them. SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides

26. Summary ---for Right Triangles Only: HL – Pair of sides including the Hypotenuse and one Leg HA – Pair of hypotenuses and one acute angle LL – Both pair of legs LA – One pair of legs and one pair of acute angles