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```7/16/15 Triangle Congruence ì CPCTC OBJECTIVE: Use CPCTC to prove parts of triangles are congruent. 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 verCcal angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. CPCTC is an abbreviaCon for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a jusCﬁcaCon in a proof aHer you have proven two triangles congruent. Once we know the two triangles are congruent, we can say any part of them we want is congruent. ? ? Example 2:
Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY ≅ YZ. Prove: ∠XYW ≅ ∠ZYW Z By CPCTC, the third side pair is congruent, so AB = 18 mi. 1 7/16/15 Example 2 Continued
Example 3: Using CPCTC in a Proof
Z Statements Reasons 1. YW Bisects XZ 1. Given 2. XW ≅ WZ 2. DeﬁniCon of ∠ bisector 3. XY ≅ YZ 3. Given 4. YW ≅ YW 4. Reﬂex. Prop. of ≅ 5. ∆MNO ≅ ∆OPM 5. SSS 6. ∠NMO ≅ ∠POM 6. CPCTC Given: NO || MP, ∠N ≅ ∠P Prove: MN || OP Example 3 Continued
Helpful Hint Work backward when planning a proof. To show that NM || OP, look for a pair of angles that are congruent. Then look for triangles that contain these angles. Statements Reasons 1. ∠N ≅ ∠P; NO || MP 1. Given 2. ∠NOM ≅ ∠PMO 2. Alt. Int. ∠s Thm. 3. MO ≅ MO 3. Reﬂex. Prop. of ≅ 4. ∆MNO ≅ ∆OPM 4. AAS 5. ∠NMO ≅ ∠POM 5. CPCTC 6. MN || OP 6. Conv. Of Alt. Int. ∠s Thm. 2 7/16/15 Example 4: Using CPCTC In the Coordinate Plane
Step 2 Use the Distance Formula to ﬁnd the lengths of the sides of each triangle. 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. End of Lesson ì Did you meet these goals So DE ≅ GH, EF ≅ HI, and DF ≅ GI. Therefore ∆DEF ≅ ∆GHI by SSS and ∠DEF ≅ ∠GHI by CPCTC. •
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I know what CPCTC stands for I know when to use CPCTC I can write proofs where this principle is needed I remember alternate interior angles I remember the distance formula 3 ```
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