ΔABC Δ ΔBAC Δ ΔCAB Δ ΔACB Δ ΔBCA Δ ΔCBA Δ
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1 Geometry & Statistics Name: Guided Notes: Triangle Congruence Congruence Reminder: Two geometric figures are if they have exactly the same size and shape. Ex 1: ΔABC ΔPQR. List the corresponding congruent angles and corresponding congruent sides. Corresponding congruent angles Corresponding congruent sides Naming triangles: There are six different ways to write a congruence statement for a pair of triangles, but be sure that the corresponding angles are in the same order. ΔABC Δ ΔBAC Δ ΔCAB Δ ΔACB Δ ΔBCA Δ ΔCBA Δ
2 Interactive Activity: Triangle Congruences Go to You will need to activate Flash for this. 1.) Side-Side-Side Click on sides AB, BC, and AC. Construct a triangle Construct another one. Can you construct a triangle that isn t congruent to your first one? Page 2 2.) Side-Angle-Side Click on sides AB and BC and angle B. Construct a triangle. Construct another one. Can you construct a triangle that isn t congruent to your first one? 3) Angle-Side-Angle Click on side AB and angles A, B, and C. Construct a triangle. Construct another one. Can you construct a triangle that isn t congruent to your first one?
3 Page 3 4) Side-Side-Angle Click on sides AB and BC and angle C. Construct a triangle. Construct another one. Can you construct a triangle that isn t congruent to your first one? 5.) Angle-Angle-Side Click on side AB and angles B and C. Construct a triangle. Construct another one. Can you construct a triangle that isn t congruent to your first one? 6.) Angle-Angle-Angle Click on angles A, B, and C. Construct a triangle. Construct another one. Can you construct a triangle that isn t congruent to your first one?
4 Page 4 Now we have some shortcuts! Example 1: Does the diagram give enough information to show that the triangles are congruent? Write YES or NO. If YES, write a congruence statement. Congruence Statement: Congruence Statement:
5 Page 5 Remember that the included angle is the angle CREATED BY the two sides. SKETCH: INCLUDED ANGLE NOT AN INCLUDED ANGLE Answers: Example 2: Does the diagram give enough information to use SAS? Write YES or NO. If YES, write a congruence statement. Congruence statement Congruence Statement:
6 Page
7 Page 7 Homework:
8 Page
9 Page 9 ASA and AAS Congruence Words: If two and the side of one triangle are congruent to two angles and the of a second triangle, then the two triangles are congruent. Picture: Symbols: Words: If two and a side of one triangle are congruent to two angles and the side of a second triangle, then the two triangles are congruent. Picture: Symbols: BE CAREFUL! Order matters here. For example, Side-Side-Angle is not a valid congruence reason.
10 Page 10 a. b. c
11 Page 11
12 Page 12 Homework:
13 Page 13 Proving Triangles are Congruent with HL What are the four three-letter congruence postulates for triangles? What are the two that DO NOT prove triangle congruence? We are going to look at one of these NON-congruence ones in more depth Why doesn t SSA work? In the three diagrams below m A, AB, and BC don t change. In the first diagram C is an angle. BC is then swung out so it hits the horizontal side at a different point. In the third diagram C is an angle. SSA does NOT prove congruence because the other angle could be either acute or obtuse. But what if we could make sure the angle is always acute. Angles in a Right Triangle: In a triangle, no angle is greater than. Remember parts of a right triangle:
14 ulates 192 riangle The hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. Checkpoint: hypotenuse hypotenuse Tell whether the segment is a leg or the hypotenuse of the leg leg right triangle. 1. AC &* A hypotenuse hypotenuse 4. KL &* K Page 14 L Help TIP at the f a right led the elp LES s at leg IStudent Help ICLASSZONE.COM MORE EXAMPLES COM More examples at classzone.com 2. BC &* 3. AB &* B THEOREM Example 1: Is it possible to show that TJGH cthkj using the HL Congruence Theorem? Explain your reasoning. C 5. KJ &* 6. JL&* Hypotenuse-Leg Congruence Theorem (HL) Hypotenuse-Leg Congruence Theorem (HL) Words If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then Words If the the two hypotenuse triangles are and congruent. a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then Symbolsthe If TABC two triangles and TDEFare are congruent. A D right triangles, and Symbols If TABC and TDEF are A D H AC&* c DF &*, and right triangles, and L BC &* c EF&*, H AC&* c DF &*, and then TABC ctdef. B C E F L BC &* c EF&*, Solution then TABC ctdef. EXAMPLE 1 Determine When To Use HL In the diagram, you are given that TJGH and THKJ are right triangles. By the Reflexive Property, you know JH&* c JH&* (hypotenuse) and you are given EXAMPLE that JG&* c HK &** 2 (leg). Use the You HL can Congruence use the HL Congruence Theorem Theorem to EXAMPLE show Use that the TJGH 2diagram Use cthkj. to the prove HL that Congruence TPRQ ctprs. Theorem Example 2: Use the diagram to prove that TPRQ ctprs. S Solution Hypotenuse-Leg Congruence Theorem: HL Given PR &* SQ&* P R Solution PQ&* c PS &* S Given Prove PR &* TPRQ ctprs SQ&* P P R Prove PQ&* Statements c PS &* Reasons TPRQ ctprs 1. PR &* SQ&* 1. Given P 2. aprq and aprs are right A. 2. lines form right angles. J G B C E K H J F
15 Page 15 Checkpoint: Are the triangles congruent by HL? a) b) c) d) e) f) a) b) c) d) e) f) FIVE CONGRUENCE POSTULATES: Checkpoint: Determine whether you are given enough information to show that the triangles are congruent. Explain your answer. 7. D E 8. M P 9. U V S G F W N P R T
16 HL Congruence Theorem Determine whether you can use the HL Congruence Theorem to show that the triangles are congruent. Explain your reasoning. Page A D E 11. J 12. P M T P B C F K L R S Practice: You be the Judge Decide whether enough information is given to show that the triangles are congruent. If so, state the theorem or postulate you would use. Explain your reasoning. 15. A 16. J 17. L P P C B D K Y L X F E Z M N R 18. F 19. S T V 20. A B C G H J U E D 21. A J K D D B C L M C F E
17 Page 17 Homework:
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