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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 For Exercises 3 and 4, what additional information do you need to prove the triangles congruent by the HL Theorem? 3. LMX LOX4. AMD CNB Not possible Congruence in Right Triangles GEOMETRY LESSON 4-6 Yes; use the congruent hypotenuses and leg BC to prove ABC DCB LM LOAM CN or MD NB 4-6

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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 Using Corresponding Parts of Congruent Triangles 4-7 Check Skills Youll Need 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. a. Two pairs of sides are congruent. The included angles are congruent. Thus, the two triangles are congruent by SAS. 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.

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Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 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. Overlapping triangles share part or all of one or more sides.

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

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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: Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 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. 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. 4-7 Quick Check Planning a Proof

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Write a paragraph proof. Given: XW YZ, XWZ and YZW are right angles. Prove: XPW YPZ Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 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. 4-7 Quick Check Using Two Pairs of Triangles

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Given: CA CE, BA DE Write a two-column proof to show that CBE CDA. 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 = CD6. Substitution Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 4-7 Plan: CBE CDA by CPCTC if CBE CDA. This congruence holds by SAS if CB CD. Proof: StatementsReasons 1. BCE DCA 1. Reflexive Property of Congruence 2. CA CE, BA DE 2. Given 7. CB CD7. Definition of congruence 8. CBE CDA8. SAS 9. CBE CDA9. CPCTC 4-7 Quick Check Separating Overlapping Triangles

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4-7 Using Corresponding Parts of Congruent Triangles GEOMETRY LESSON 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 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

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