Unit 5 Triangle Congruence

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1 Unit 5 Triangle Congruence Day Classwork Homework Wednesday 10/25 Unit 4 Test D1 - Proving SAS through Rigid Motions Watch Video Thursday 10/26 Friday 10/27 Monday 10/30 Proving SAS through Rigid Motions Using SAS to Prove Triangles Congruent CPCTC Proving ASA and SSS through Rigid Motions D2 - Using SAS to Prove Triangles Congruent CPCTC Watch Video D3 - Proving ASA and SSS through Rigid Motions Watch Video D4 - AAS, HL Watch Video Tuesday 10/31 Wednesday 11/1 Thursday 11/2 Friday 11/3 Monday 11/6 Tuesday 11/7 Wednesday 11/8 AAS, HL Unit 5 Quiz 1 Adding and Subtracting Segments/Angles Overlapping Triangle proofs Unit 5 Quiz 2 Double Triangle Proofs Review Unit 5 Quiz 3 Review Unit 5 Test D5 - Adding and Subtracting Segments/Angles Watch Video D6 - Overlapping Triangle proofs Watch Video D7 - Double Triangle Proofs Watch Video Review Review Study 1

2 Day 1 Proving SAS through Rigid Motions Recall: We have agreed to use the word congruent to mean there exists a of basic rigid motions of the plane that maps one figure to the other. We are going to show that there are criteria that refer to a few parts of the two triangles and a correspondence between them that guarantee congruency (i.e., existence of rigid motion). We start with the Side-Angle-Side (SAS) criteria. In the figure below, 2 pairs of sides are congruent and the included angles are congruent. AB AC A There are 3 rigid motions that will map ABC to A' B ' C '. Describe each rigid motion in words and symbols B C A B''' 2

3 Examples: 1. Case Diagram Transformations Needed Shared Side B C A B''' Shared Vertex B A B" C C" 2. Given: Triangles with a pair of corresponding sides of equal length and a pair of included angles of equal measure. Sketch and label three phases of the sequence of rigid motions that prove the two triangles to be congruent. Transformation Sketch 3

4 Directions: Justify whether the triangles meet the SAS congruence criteria; explicitly state which pairs of sides or angles are congruent and why. If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one triangle onto the other. 1. Given: LNM LNO, MN ON. Do LMN and LON meet the SAS criteria? 2. Given: HGI JIG, HG JI. Do HGI and JIG meet the SAS criteria? 3. Given: AB CD, AB CD. Do ABD and CDB meet the SAS criteria? 4

5 4. Given: m R = 25, RT = 7", SU = 5", ST = 5". Do RSU and RST meet the SAS criteria? 5. Given: KM and JN bisect each other. Do JKL and NML meet the SAS criteria? 6. Given: AE bisects angle BCD, BC DC. Do CAB and CAD meet the SAS criteria? 5

6 Day 2 Using SAS to Prove Triangles Congruent Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria to prove the triangles are congruent. 1. Given: AB DC Prove: ABC DCB. 2. Given: AB RS AB RS Prove: ABC RST. Example: We already know that the base angles of an isosceles triangles are congruent. We are going to prove this fact in two ways: (1) by using transformations, and (2) by using SAS triangle congruence criteria. Prove Base Angles of an Isosceles are Congruent Using Transformations Given: Isosceles ABC, with AB AC. Prove: B C. 6

7 Prove Base Angles of an Isosceles are Congruent Using SAS Given: Isosceles ABC, with AB AC. Prove: B C. CPCTC Corresponding parts of congruent triangles are congruent. 1. Given: EA AD FD AD AC DB AE DF E F Prove: EC FB A B C D 7

8 2. Given: KM bisects LKJ LK JK K Prove: M is the midpoint of LJ L M J K 3. Given: KM LJ LM JM Prove: KLJ is isosceles L M J 8

9 4. Given: C is a midpoint of AE BC DC 1 2 BF DF A B C D E Prove: AF EF F T 5. Given: TR TS TM TN Prove: 1 2 M N 1 2 R S 9

Day 3 Proving ASA and SSS through Rigid Motions A second criteria that guarantees congruency (i.e., existence of rigid motion) is Angle-Side-Angle. 1.

10 Day 3 Proving ASA and SSS through Rigid Motions A second criteria that guarantees congruency (i.e., existence of rigid motion) is Angle-Side-Angle. 1. In the diagram below, describe the rigid motion that maps ABC to ABC '''. Will the triangles be congruent by SAS? Why or why not? Since the triangles have two pairs of corresponding congruent angles and the included side congruent, the triangles are congruent by Angle-Side-Angle. 2. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the ASA triangle congruence criteria to prove the triangles are congruent. Given: ABC DCB Prove: ABC DCB. A Given: AD BC Prove: ABD ACD B D C 10

11 3. In the diagram below, describe the rigid motion that maps ABC to AB ' C. Will the triangles be congruent? Why or why not? Since the triangles have three pairs of corresponding congruent sides, the triangles are congruent by Side-Side-Side. 4. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SSS triangle congruence criteria to prove the triangles are congruent. Given: KM and JN bisect each other at L Prove: JKL NML. Given: R is the midpoint of KL Prove: JKR JLR 11

12 1. Given: M is the midpoint of HP, H P. Prove: GHM RPM 2. Given: Rectangle JKLM with diagonal KM. Prove: JKM LMK 12

13 3. Given: Circles with centers A and B intersect at C and D. Prove: CAB DAB. 4. Given: w x and y z. Prove: (1) ABE ACE. (2) AB AC and AD BC. 13

14 Day 4 AAS & HL Triangle Congruence 1. Consider a pair of triangles that meet the AAS criteria (see below). If you knew that two angles of one triangle corresponded to and were equal in measure to two angles of the other triangle, what conclusions can you draw about the third angles of each triangle? Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are congruent? Therefore, the AAS criterion is actually an extension of the criterion. triangle congruence Practice: Using only the information given, decide whether the triangles are congruent by ASA or AAS. 1. AB BC DC BC 2. BC AB AD DC

15 5. Consider the two right triangles below that have congruent hypotenuses and one pair of congruent legs. Are the triangles congruent? Why? This is our last criteria to prove triangles congruent: Hypotenuse-Leg (HL) **Note that only RIGHT triangles can be proved congruent using HL** Practice: Using only the information given, decide by which method the triangles could be proved congruent ED BA AC DF 3. HG HI 4. E is the midpoint of CD. A and B are right angles 15

16 Criteria that do not determine two triangles as congruent: Two sides and a non-included angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as well as the non-included angle of 23. Yet, the triangles are not congruent. Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show their possible locations. The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria might be congruent, but they might not be; therefore we cannot categorize SSA as congruence criterion. Practice: Decide if the triangles are congruent. If so, by what method? 1. Given: A X, BA YX, ZX CA 2. Given: CA ZX, BC YZ, A X 16

17 Three Congruent Angles (AAA): Observe the diagrams below. What is the measure of the missing angles? Notice that the triangles have 3 pairs of congruent angles. Yet, the triangles are not congruent The pattern of AAA cannot guarantee congruence criteria. In other words, two triangles under AAA criteria might be congruent, but they might not be; therefore we cannot categorize AAA as congruence criterion. B 1. Given: BD AC A C Prove: ABD CBD A D C 2. Given: PQ RS QO RO P O X 1 Q Prove: XO YO R 2 S Y 17

18 Day 5 - ADDING & SUBTRACTING SEGMENTS & ANGLES If a pair of congruent segments/angles are added to another pair of congruent segments/angles, then the resulting segments/angles are congruent. Similarly, if a pair of congruent segments/angles are subtracted from a pair of congruent segments/angles, then the resulting segments/angles are congruent. Statements Reasons 1. AB CD 1. Given 2. BC BC 2. Reflexive Property 3. AB BC CD BC 3. Addition Property 4. AC AB BC 4. A whole = s the sum of its BD CD BC parts 5. AC BD 5. Substitution Statements Reasons 1. AC BD 1. Given 2. BC BC 2. Reflexive Property 3. AC AB BC 3. A whole = s the sum of its BD CD BC parts 4. AB BC CD BC 4. Substitution 5. AC BD 5. Subtraction Property Examples 1. 18

21 8. Given: PS UR PQ UT QR TS Prove: Q T 9. Given: Prove:

22 Day 6 & 7 Overlapping and Double Triangle Proofs 1. Given: AB BC, BC DC DB bisects ABC, AC bisects DCB EB EC Prove: BEA CED 2. Given: BF AC, CE AB AE AF Prove: ACE ABF 22

23 3. Given: XJ YK, PX PY, ZXJ ZYK Prove: JY KX 4. Given: JK JL, JK XY Prove: XY XL 23

24 5. Given: 1 2, 3 4 Prove: AC BD 6. Given: 1 2, 3 4, AB AC Prove: (a) ABD ACD (b)

25 7. Given: AB AC, RB RC, Prove: SB SC 8. Given: JK JL, JX JY Prove: KX LY 25

26 9. Given: AD DR, AB BR, AD AB Prove: DCR BCR 26

DO NOW Geometry Regents Lomac Date. due. Complex Congruence Proofs. My reason will be: Complete each statement below:

DO NOW Geometry Regents Lomac 2014-2015 Date. due. Complex Congruence Proofs (DN) Write did #1 on your do now sheet and complete problem number 1 below. (#1 includes the rest of this page.) Name Per LO: