# Congruence in Right Triangles

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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. LMX LOX AMD CNB Yes; use the congruent hypotenuses and leg BC to prove ABC DCB Not possible LM LO AM CN or MD NB 4-6

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 , 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

Overlapping triangles share part or all of one or more sides.
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

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

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 Look again at ZWM and YXM.  ZMW YMX because vertical angles are congruent, MW MX, and by subtraction  ZWM YXM, so ZWM YXM by ASA. Label point M where ZX intersects WY, as shown in the diagram. ZW YX by CPCTC if ZWM YXM. Look at MWX. MW MX by the Converse of the Isosceles Triangle Theorem. You can prove these triangles congruent using ASA as follows: Quick Check 4-7

Using Corresponding Parts of Congruent Triangles
GEOMETRY LESSON 4-7 Using Two Pairs of Triangles Write a paragraph proof. Given: XW YZ, XWZ and YZW are right angles. Prove: XPW YPZ 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 XW YZ. Because XWZ and YZW are right angles, XWZ YZW. WZ ZW, by the Reflexive Property of Congruence. Therefore, XWZ YZW by SAS. WXZ ZYW by CPCTC, and XPW YPZ because vertical angles are congruent. Therefore, XPW YPZ by AAS. Quick Check 4-7

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 CBE CDA. This congruence holds by SAS if CB CD. Statements Reasons Proof: 1. BCE DCA 1. Reflexive Property of Congruence 2. CA CE, BA DE 2. Given 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 Postulate CE – DE = CD 6. CB = CD 6. Substitution 7. CB CD 7. Definition of congruence 8. CBE CDA 8. SAS Quick Check 9. CBE CDA 9. CPCTC 4-7

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 XY KSR MRS SAS GHI IJG ASA XD XC by CPCTC if DXA CXB. This congruence holds by AAS if BAD ABC. Show BAD ABC by SSS. 4-7 4-7