Chapter 4 Section 7 : Triangle Congruence: CPCTC
Use CPCTC to prove parts of triangles are congruent.
Objectives
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.
What is CPCTC?
SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.
Remember !!!
A and B are on the edges of a ravine. What is AB?
Example 1: Engineering Application
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.
A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK?
Example 2
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.
Given: YW bisects XZ, XY YZ. Prove: XYW ZYW
Example 3Proofs
Z
solution
Given: PR bisects QPS and QRS. Prove: PQ PS
Example 4
solutionsPR bisects QPS and QRS
QRP SRPQPR SPR
Given Def. of bisector
RP PR
Reflex. Prop. of
∆PQR ∆PSR
PQ PS
ASA
CPCTC
Do problems 2 and3 in your book page 270
Student guided practice
Example 5
Prove: MN || OP Given: NO || MP, N P
solution
5. CPCTC5. NMO POM
6. Conv. Of Alt. Int. s Thm.
4. AAS4. ∆MNO ∆OPM
3. Reflex. Prop. of
2. Alt. Int. s Thm.2. NOM PMO
1. Given
ReasonsStatements
3. MO MO
6. MN || OP
1. N P; NO || MP
Example 6
Prove: KL || MN Given: J is the midpoint of KM and NL.
solution
5. CPCTC5. LKJ NMJ6. Conv. Of Alt. Int. s Thm.
4. SAS Steps 2, 34. ∆KJL ∆MJN
3. Vert. s Thm.3. KJL MJN
2. Def. of mdpt.
1. GivenReasonsStatements
6. KL || MN
1. J is the midpoint of KM and NL. 2. KJ MJ, NJ LJ
Do problem 4 in your book page 271
Student guided practice
Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3)
Prove: DEF GHI
Example 7
Step 1 Plot the points on a coordinate plane.
solution
Step 2 Use the Distance Formula to find the lengths of the sides of each triangle.
solution
So DE GH, EF HI, and DF GI. Therefore ∆DEF ∆GHI by SSS, and DEF
GHI by CPCTC.
solution
Given: J(–1, –2), K(2, –1), L(–2, 0), R(2, 3), S(5, 2), T(1, 1)
Prove: JKL RST Solution: Step 1 Plot the points on a coordinate plane
Example 8
Step 2 Use the Distance Formula to find the lengths of the sides of each triangle.
RT = JL = √5, RS = JK = √10, and ST = KL = √17.
So ∆JKL ∆RST by SSS. JKL RST by CPCTC.
solution
Do problems 7-13 in your book page 271
Student guided practice
Today we saw CPTCP and how we can prove corresponding parts to corresponding triangles
Next class we are going to learn about Introduction to coordinate proofs
Closure