Transcript Document
Congruent Figures GEOMETRY LESSON 4-1 In Exercises 1 and 2, quadrilateral WASH quadrilateral NOTE. 1. List the congruent corresponding parts. WA NO, AS OT, SH TE, WH NE; E W N, A O, S T, H 2. mO = mT = 90 and mH = 36. Find mN. 144 3. Write a statement of triangle congruence. Sample: DFH ZPR 4. Write a statement of triangle congruence. Sample: ABD CDB 5. Explain your reasoning in Exercise 4 above. Sample: Two pairs of corresponding sides and two pairs of corresponding angles are given. C A because all right angles are congruent. BD BD by the Reflexive Property of . ABD CDB by the definition of congruent triangles. 4-1 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 (For help, go to Lesson 2-5.) What can you conclude from each diagram? 1. 1. According to the tick marks on the sides, AB DE. According to the tick marks on the angles, C F. 2. 3. 2. The two triangles share a side, so PR PR. According to the tick marks on the angles, QPR SRP and Q S. Check Skills You’ll Need 4-2 3. According to the tick marks on the sides, TO NV. The tick marks on the angles show that M S. Since MO || VS, by the Alternate Interior Angles Theorem MON SVT. Since OV OV by the Reflexive Property, you can use the Segment Addition Property to show TV NO. Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 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. 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC. 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 Caution The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. 4-2 SSA: Side-Side-Angle THIS DOES NOT WORK! It is a very common trap and students fall for it easily. Example: are these triangles congruent? S S A S NO! A S SSA Why doesn’t it work? Why doesn’t it work? These triangles are not congruent even though two sides and an angle are congruent. Decide if the following triangles are congruent using SSS or SAS. Use SSS or SAS. by SSS X Y S S W S S S Z Not (uses SSA) E M A S S A S A T by SAS L S K A S A S J H by SAS A S S S A KA S S D T Not (uses SSA) E L S S N A S V A S O by SSS S S S S S Summary To prove two triangles are congruent use or Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 Using SSS Given: M is the midpoint of XY, AX Prove: AMX AMY AY Write a paragraph proof. Copy the diagram. Mark the congruent sides. You are given that M is the midpoint of XY, and AX AY. Midpoint M implies MX MY. AM AM by the Reflexive Property of Congruence, so AMX AMY by the SSS Postulate. Quick Check 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 Using SAS AD BC. What other information do you need to prove by SAS? ADC BCD It is given that AD BC. Also, DC CD by the Reflexive Property of Congruence. You now have two pairs of corresponding congruent sides. Therefore if you know ADC BCD, you can prove ADC BCD by SAS. Quick Check 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 Are the Triangles Congruent? Quick Check Given: RSG RSH, SG SH From the information given, can you prove RSG RSH? Explain. Copy the diagram. Mark what is given on the diagram. It is given that RSG RSH and SG SH. RS RS by the Reflexive Property of Congruence. Two pairs of corresponding sides and their included angles are congruent, so RSG RSH by the SAS Postulate. 4-2 Triangle Congruence by SSS and SAS GEOMETRY LESSON 4-2 1. In VGB, which sides include B? BG and BV 2. In STN, which angle is included between NS and TN? N 3. Which triangles can you prove congruent? Tell whether you would use the SSS or SAS Postulate. APB XPY; SAS 4. What other information do you need to prove DWO DWG? If you know DO DG, the triangles are by SSS; if you know DWO DWG, they are by SAS. 5. Can you prove SED BUT from the information given? Explain. No; corresponding angles are not between corresponding sides. 4-2