Download Overlapping triangles

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
Label point M where ZX intersects WY, as shown in the
diagram. ZW YX by CPCTC if ZWM
YXM.
You can prove these triangles congruent using ASA as follows:
Look at MWX. MW
Theorem.
MX by the Converse of the Isosceles Triangle
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.
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
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
Plan: CBE CDA by CPCTC if CBE
congruence holds by SAS if CB CD.
Reasons
Proof: Statements
1. BCE DCA
2. CA CE, BA DE
CDA. This
1. Reflexive Property of Congruence
2. Given
3. CA – BA = CE – DE
4. CA – BA = CB,
CE – DE = CD
5. CB = CD
3. Subtraction Property of Equality
4. Segment Addition Postulate
6. CBE
7. CBE
6. SAS
7. CPCTC
CDA
CDA
CDA.
5. Substitution
4-7
Quick Check
Using Corresponding Parts of Congruent Triangles
GEOMETRY LESSON 4-7
XY
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.
KSR
SAS
4. Plan a proof.
Given: AC BD, AD
Prove: XD XC
BC
MRS
GHI
ASA
IJG
XD XC by CPCTC if DXA
CXB.
This congruence holds by AAS if
BAD
ABC. Show BAD
ABC
by SSS.
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