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4-7
TriangleCongruence:
Congruence: CPCTC
CPCTC
4-7 Triangle
Warm Up
Lesson Presentation
Lesson Quiz
Holt
HoltGeometry
McDougal Geometry
4-7 Triangle Congruence: CPCTC
CPCTC is an abbreviation for the phrase
“Corresponding Parts of Congruent
Triangles are Congruent.” It can be used
as a justification in a proof after you have
proven two triangles congruent.
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Remember!
SSS, SAS, ASA, AAS, and HL use
corresponding parts to prove triangles
congruent. CPCTC uses congruent
triangles to prove corresponding parts
congruent.
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Example 1: Engineering Application
A and B are on the edges
of a ravine. What is AB?
One angle pair is congruent,
because they are vertical
angles. Two pairs of sides
are congruent, because their
lengths are equal.
Therefore the two triangles are congruent by
SAS. By CPCTC, the third side pair is congruent,
so AB = 18 mi.
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Check It Out! Example 1
A landscape architect sets
up the triangles shown in
the figure to find the
distance JK across a pond.
What is JK?
One angle pair is congruent,
because they are vertical
angles.
Two pairs of sides are congruent, because their
lengths are equal. Therefore the two triangles are
congruent by SAS. By CPCTC, the third side pair is
congruent, so JK = 41 ft.
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Example 2: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY  YZ.
Prove: XYW  ZYW
Z
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Example 2 Continued
ZW
WY
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Check It Out! Example 2
Given: PR bisects QPS and QRS.
Prove: PQ  PS
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Check It Out! Example 2 Continued
QRP  SRP
PR bisects QPS
and QRS
Given
RP  PR
QPR  SPR
Reflex. Prop. of 
Def. of  bisector
∆PQR  ∆PSR
ASA
PQ  PS
CPCTC
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Check It Out! Example 3
Given: J is the midpoint of KM and NL.
Prove: KL || MN
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Example 4: Using CPCTC In the Coordinate Plane
Given: D(–5, –5), E(–3, –1), F(–2, –3),
G(–2, 1), H(0, 5), and I(1, 3)
Prove: DEF  GHI
Step 1 Plot the
points on a
coordinate plane.
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
Step 2 Use the Distance Formula to find the lengths
of the sides of each triangle.
Holt McDougal Geometry
4-7 Triangle Congruence: CPCTC
So DE  GH, EF  HI, and DF  GI.
Therefore ∆DEF  ∆GHI by SSS, and DEF  GHI
by CPCTC.
Holt McDougal Geometry