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Proving Triangles are Congruent: SSS, SAS Section 5.2 Objectives: • Show triangles are congruent using SSS and SAS. Key Vocabulary • Included angle • proof Postulates • 12 SSS Congruence Postulate • 13 SAS Congruence Postulate Review • Congruent Triangles ▫ Triangles that are the same shape and size are congruent. ▫ Each triangle has three sides and three angles. ▫ If all 6 parts (3 corresponding sides and 3 corresponding angles) are congruent….then the triangles are congruent. Review • CPCTC – Corresponding Parts of Congruent Triangles are Congruent • Be sure to label Δs with proper markings (i.e. if D L, V P, W M, DV LP, VW PM, and WD ML then we must write ΔDVW ΔLPM) So, to prove Δs we must prove ALL sides & ALL s are ? Fortunately, NO! There are some shortcuts… Two of which follow!!! Postulate 12: Side-Side-Side Congruence Postulate SSS Congruence - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If MN QR, NP RS , PM SQ then, MNP QRS Remember! Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. ∆PMN is adjacent to ∆PON, PN≅PN by the Reflexive Property. Example 1: Using SSS to Prove Triangle Congruence Use SSS to explain why ∆ABC ∆DBC. It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS. Your Turn Use SSS to explain why ∆ABC ∆CDA. It is given that AB CD and BC DA. By the Reflexive Property of Congruence, AC CA. So ∆ABC ∆CDA by SSS. Example #2 – SSS – Coordinate Geometry Use the SSS Postulate to show the two triangles are congruent. Find the length of each side. AC = 5 BC = 7 2 2 AB = 5 7 74 MO = 5 NO = 7 MN = 52 72 74 VABC VMNO Example 3 Does the diagram give enough information to show that the triangles are congruent? Explain. SOLUTION From the diagram you know that HJ LJ and HKLK. By the Reflexive Property, you know that JK JK. ANSWER Yes, enough information is given. Because corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that ∆HJK ∆LJK. Definition: Included Angle The angle between two sides in a figure. ∠B is included between: BA & BC INCLUDED ANGLE Y C A B X Z Angle in between two consecutive sides Use the diagram. Name the included angle between the pair of sides given. ∠MTR ∠RTQ ∠MRT ∠Q The 2nd Congruence Short Cut • It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. • SAS Postulate 13: Side-Angle-Side Congruence Postulate SAS Postulate – If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If PQ WX , QS XY , Q X then, PQS WXY Caution The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. S S S A S A Example 4: Engineering Application The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ ∆VWZ. It is given that XZ VZ and that YZ WZ. By the Vertical s Theorem. XZY VZW. Therefore ∆XYZ ∆VWZ by SAS. Example 5 Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. a. b. SOLUTION a. From the diagram you know that AB CB and DB DB. The angle included between AB and DB is ABD. The angle included between CB and DB is CBD. Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that ∆ABD ∆CBD. Example 5 b. You know that GF GH and GE GE. However, the congruent angles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate. Your Turn Use SAS to explain why ∆ABC ∆DBC. It is given that BA BD and ABC DBC. By the Reflexive Property of , BC BC. So ∆ABC ∆DBC by SAS. Example 6A: Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. ∆MNO ∆PQR, when x = 5. PQ = x + 2 =5+2=7 QR = x = 5 PQ MN, QR NO, PR MO ∆MNO ∆PQR by SSS. PR = 3x – 9 = 3(5) – 9 = 6 Example 6B: Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. ∆STU ∆VWX, when y = 4. ST = 2y + 3 = 2(4) + 3 = 11 TU = y + 3 =4+3=7 mT = 20y + 12 ST VW, TU WX, and T W. ∆STU ∆VWX by SAS. = 20(4)+12 = 92° Your Turn Show that ∆ADB ∆CDB, t = 4. DA = 3t + 1 = 3(4) + 1 = 13 DC = 4t – 3 = 4(4) – 3 = 13 mD = 2t2 = 2(16)= 32° ADB CDB Def. of . DB DB Reflexive Prop. of . ∆ADB ∆CDB by SAS. #1 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R S T ΔRST ΔYZX by SSS #2 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. P R S Q ΔPQS ΔPRS by SAS #3 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R T S Not congruent. Not enough Information to Tell #4 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. P S U Q R T ΔPQR ΔSTU by SSS #5 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. M P R N Q Not congruent. Not enough Information to Tell Proof • A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. • An important part of writing a proof is giving justifications to show that every step is valid. Two-Column Proof • Two-Column Proof – A proof format used in geometry in which an argument is presented with two columns, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof. • Two- Column Proof is a formal proof because it has a specific format. • Two-Column Proof ▫ Left column – statements in a logical progression. ▫ Right column – Reason for each statement (definition, postulate, theorem or property). Two-Column Proof Given: Prove: Proof: Statements Reasons How to Write A Proof 1. 2. 3. 4. List the given information first Use information from the diagram Give a reason for every statement Use given information, definitions, postulates, and theorems as reasons 5. List statements in order. If a statement relies on another statement, list it later than the statement it relies on 6. End the proof with the statement you are trying to prove Geometric Proof • Since geometry also uses variables, numbers, and operations, many of the algebraic properties of equality are true in geometry. For example: Property Segments Angles Reflexive AB = AB m ∠1 = m ∠1 Symmetric If AB = CD, then CD = AB. If m ∠1 = m ∠2, then m ∠2 = m ∠1. Transitive If AB = CD and CD = EF, then AB = EF. If m ∠1 = m ∠2 and m ∠2 = m ∠3, then m ∠1 = m ∠3. • These properties can be used to write geometric proofs. Remember! Numbers are equal (=) and figures are congruent (). Example 7 Write a two-column proof that shows ∆JKL ∆NML. JL NL L is the midpoint of KM. ∆JKL ∆NML SOLUTION The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove. Example 7 Statements Reasons 1. JL NL 1. Given 2. L is the midpoint of KM. 2. Given 3. JKL NML 3. Vertical Angles Theorem These are the given statements. This information is from the diagram. Example 7 Statements Reasons 4. KL ML 4. Definition of midpoint 5. ∆JKL ∆NML 5. SAS Congruence Postulate Statement 4 follows from Statement 2. Statement 5 follows from the congruences of Statements 1, 3, and 4. Example 8 You are making a model of the window shown in the figure. You know that DR AG and RA RG. Write a proof to show that ∆DRA ∆DRG. D A R SOLUTION 1. Make a diagram and label it with the given information. G Example 8 2. Write the given information and the statement you need to prove. DR AG, RA RG ∆DRA ∆DRG 3. Write a two-column proof. List the given statements first. Example 8 Statements Reasons 1. RA RG 1. Given 2. DR AG 2. Given 3. DRA and DRG are right angles. 4. DRA DRG 3. lines form right angles. 5. DR DR 5. Reflexive Property of Congruence 6. ∆DRA ∆DRG 6. SAS Congruence Postulate 4. Right angles are congruent. Your Turn: 1. Fill in the missing statements and reasons. CB CE, AC DC ∆BCA ∆ECD Statements Reasons 1. CB CE 1. _____ ? ANSWER Given 2. _____ ? 2. Given ANSWER AC DC 3. BCA ECD 3. _____ ? ANSWER Vertical Angles Theorem 4. ∆BCA ∆ECD ANSWER SAS Congruence Postulate 4. _____ ? Example 9: Given: QR UT, RS TS, QS = 10, US = 10 Prove: ΔQRS ΔUTS U Q 10 R 10 S T Example 9: Given: QR UT, RS TS, QS = 10, US = 10 Prove: ΔQRS ΔUTS Q U 10 R 10 S Statements Reasons________ 1. QR UT, RS TS, 1. Given QS=10, US=10 2. QS = US 2. Substitution 3. QS US 3. Def of segs. 4. ΔQRS ΔUTS 4. SSS Postulate T Your turn: Proof by SSS Given: RP RT RX bisects PT Prove: PRX TRX Statement 1) RP RT (s) 2) RX bisects PT 3) X is midpoint of PT 4) PX XT (s) 5) RX RX (s) 6) PRX TRX R P 48 X T Reason 1) 2) 3) 4) 5) 6) Given Given Def. line bisector Def. midpoint Reflexive Postulate SSS Postulate Example 10: Proving Triangles Congruent Given: BC ║ AD, BC AD Prove: ∆ABD ∆CDB Statements Reasons 1. BC || AD 1. Given 2. CBD ABD 2. Alt. Int. s Thm. 3. BC AD 3. Given 4. BD BD 4. Reflex. Prop. of 5. ∆ABD ∆ CDB 5. SAS Your Turn Given: QP bisects RQS. QR QS Prove: ∆RQP ∆SQP Statements Reasons 1. QR QS 1. Given 2. QP bisects RQS 2. Given 3. RQP SQP 3. Def. of bisector 4. QP QP 4. Reflex. Prop. of 5. ∆RQP ∆SQP 5. SAS Summary – SSS & SAS Hints to Proofs! • If two triangles share a side, in the statement column state that the side is congruent to itself. The reason is the Reflexive Property. • If two triangles share a vertex, in the statement column state that the angles are congruent. The reason is the Vertical Angles Theorem. Hints to Proofs! • If a vocabulary word is in the given, you will need its definition as a reason somewhere in the proof, otherwise you wouldn’t need to know that piece of information. • If two lines are parallel, look for congruent angles pairs, such as Alternate Interior Angles, Corresponding Angles, or Alternate Exterior Angles. Assignment • Pg. 245 -249 #1 – 25 odd, 29 – 37 odd