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Congruence in Right Triangles

Yes; use the congruent hypotenuses and leg BC to prove ABC DCB. LM LO. AM CN or MD NB. Congruence in Right Triangles. GEOMETRY LESSON 4-6. For Exercises 1 and 2, tell whether the HL Theorem can be used to prove

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Congruence in Right Triangles

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  1. Yes; use the congruent hypotenuses and leg BC to prove ABCDCB LM LO AM CN or MD NB Congruence in Right Triangles GEOMETRY LESSON 4-6 For Exercises 1 and 2, tell whether the HL Theorem can be used to prove the triangles congruent. If so, explain. If not, write not possible. 1.2. For Exercises 3 and 4, what additional information do you need to prove the triangles congruent by the HL Theorem? 3.LMXLOX4.AMD CNB Notpossible 4-6

  2. Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 (For help, go to Lessons 1-1 and 4-3.) 1. How many triangles will the next two figures in this pattern have? 2. Can you conclude that the triangles are congruent? Explain. a.  AZK and DRS b.  SDR and JTN c.  ZKA and NJT For every new right triangle, segments connect the midpoint of the hypotenuse with the midpoints of the legs of the right triangle, creating two new triangles for every previous new triangle. The first figure has 1 triangle. The second has 1 + 2, or 3 triangles. The third has 3 + 4, or 7 triangles. The fourth will have 7 + 8, or 15 triangles. The fifth will have 15 + 16, or 31 triangles. b. Two pairs of angles are congruent. One pair of sides is also congruent, and, since it is opposite a pair of corresponding congruent angles, the triangles are congruent by AAS. c.  Since AZK  DRS and SDR  JTN, by the Transitive Property of , ZKA NJT. a. Two pairs of sides are congruent. The included angles are congruent. Thus, the two triangles are congruent by SAS. Check Skills You’ll Need 4-7

  3. Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 Overlapping triangles share part or all of one or more sides. Some triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a common side or angle. You can simplify your work with overlapping triangles by separating and redrawing the triangles. 4-7

  4. Identify the overlapping triangles. Parts of sides DG and EG are shared by DFG and EHG. These parts are HG and FG, respectively. Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 Identifying Common Parts Name the parts of their sides that DFG and EHG share. Quick Check 4-7

  5. Label point M where ZX intersects WY, as shown in the diagram. ZWYX by CPCTC if ZWMYXM. Look at MWX. MWMX by the Converse of the Isosceles Triangle Theorem. Look again at ZWMand YXM.  ZMWYMX because vertical angles are congruent, MWMX, and by subtraction  ZWMYXM, soZWMYXM by ASA. Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 Planning a Proof Write a Plan for Proof that does not use overlapping triangles. Given: ZXWYWX, ZWXYXW Prove: ZW YX You can prove these triangles congruent using ASA as follows: Quick Check 4-7

  6. Plan:XPW YPZ by AAS if WXZZYW. These angles are congruent by CPCTC if XWZ YZW. These triangles are congruent by SAS. Proof: You are given XWYZ. Because XWZ and YZW are right angles,XWZYZW. WZZW, by the Reflexive Property of Congruence. Therefore, XWZYZW by SAS. WXZZYW by CPCTC, andXPWYPZ because vertical angles are congruent. Therefore, XPWYPZ by AAS. Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 Using Two Pairs of Triangles Write a paragraph proof. Given: XWYZ, XWZ and YZW are right angles. Prove: XPWYPZ Quick Check 4-7

  7. 1. BCE DCA 1. Reflexive Property of Congruence 2. CACE, BADE 2. Given 7. CBCD 7. Definition of congruence 8. CBECDA 8. SAS 9. CBE CDA 9. CPCTC Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 Separating Overlapping Triangles Given: CA CE, BA DE Write a two-column proof to show that CBE CDA. Plan: CBE CDA by CPCTC if CBECDA. This congruence holds by SAS if CBCD. Statements Reasons Proof: 3. CA = CE, BA = DE 3. Definition of congruent segments. 4. CA – BA = CE – DE 4. Subtraction Property of Equality 5. CA – BA = CB, 5. Segment Addition PostulateCE – DE = CD 6. CB = CD 6. Substitution Quick Check 4-7

  8. XY GHI IJG ASA KSR MRS SAS XD XC by CPCTC if DXA CXB.This congruence holds by AAS if BAD ABC. Show BAD ABCby SSS. Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 1. Identify any common sides and angles in AXY and BYX. For Exercises 2 and 3, name a pair of congruent overlapping triangles. State the theorem or postulate that proves them congruent. 2. 3. 4. Plan a proof. Given: AC BD, AD BC Prove: XD XC 4-7 4-7

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