Using Corresponding Parts of Congruent TrianglesUsing Corresponding Parts of Congruent TrianglesGEOMETRY LESSON 4-7GEOMETRY LESSON 4-7

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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.

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 TrianglesUsing Corresponding Parts of Congruent TrianglesGEOMETRY LESSON 4-7GEOMETRY LESSON 4-7

Identify the overlapping triangles.

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Quick Check

Identifying Common Parts

Write a Plan for Proof that does not use overlapping triangles.

Given: ZXW YWX, ZWX YXWProve: ZW YX

You can prove these triangles congruent using ASA as follows:

Using Corresponding Parts of Congruent TrianglesUsing Corresponding Parts of Congruent TrianglesGEOMETRY LESSON 4-7GEOMETRY 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.

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Quick Check

Planning a Proof

Write a paragraph proof.

Given: XW YZ, XWZ and YZW are right angles.Prove: XPW YPZ

Using Corresponding Parts of Congruent TrianglesUsing Corresponding Parts of Congruent TrianglesGEOMETRY LESSON 4-7GEOMETRY LESSON 4-7

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Using Two Pairs of Triangles

Given: CA CE, BA DE

Write a two-column proof to show that CBE CDA.

3. CA – BA = CE – DE 3. Subtraction Property of Equality 4. CA – BA = CB, 4. Segment Addition Postulate

CE – DE = CD5. CB = CD 5. Substitution

Using Corresponding Parts of Congruent TrianglesUsing Corresponding Parts of Congruent TrianglesGEOMETRY LESSON 4-7GEOMETRY LESSON 4-7

Plan: CBE CDA by CPCTC if CBE CDA. This congruence holds by SAS if CB CD.

Proof: Statements Reasons

1. BCE DCA 1. Reflexive Property of Congruence 2. CA CE, BA DE 2. Given

6. CBE CDA 6. SAS 7. CBE CDA 7. CPCTC

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Quick Check

Separating Overlapping Triangles

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Using Corresponding Parts of Congruent TrianglesUsing Corresponding Parts of Congruent TrianglesGEOMETRY LESSON 4-7GEOMETRY 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 BCProve: XD XC

XY

KSR MRSSAS

GHI IJGASA

XD XC by CPCTC if DXA CXB.This congruence holds by AAS if BAD ABC. Show BAD ABC by SSS.

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