1. If ABC DEF, then A? and BC?. D. EF 2. What is the distance between (3, 4) and ( 1, 5)? 17
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1 Warm Up 1. If ABC DEF, then A? and BC?. D EF 2. What is the distance between (3, 4) and ( 1, 5)? If 1 2, why is a b? Converse of Alternate Interior Angles Theorem 4. List methods used to prove two triangles congruent. SSS, SAS, ASA, AAS, HL
2 Unit 2C Day 5 Essential Question: How do you use CPCTC to prove parts of triangles are congruent?
3 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.
4 Remember! SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.
5 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.
6 Check It Out! Example 2 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.
7 If two triangles share a side you can use the reflexive property to prove the shared sides are congruent! If two triangles create a bowtie shape, you can use the Vertical Angles Theorem to prove the vertical angles are congruent!
8 Example 3: Proving Corresponding Parts Congruent Given: YW bisects XZ, XY YZ. Prove: XYW ZYW ZW Z WY
9 Prove: PQ PS Check It Out! Example 4 Given: PR bisects QPS and QRS. PR bisects QPS and QRS Given RP PR Reflex. Prop. PQR PSR ASA QRP SRP QPR SPR Def. of bisector PQ PS CPCTC
10 Helpful Hint Work backward when planning a proof. To show that ED GF, look for a pair of angles that are congruent. Then look for triangles that contain these angles. Don t forget You can use one of the Converse Theorems to prove two lines are parallel!
11 Example 5: Using CPCTC in a Proof Given: NO MP, N P Prove: MN OP
12 Example 5 Continued Statements Reasons 1. N P; NO MP 2. NOM PMO 3. MO MO 4. MNO OPM 5. NMO POM 6. MN OP 1. Given 2. Alt. Int. s Thm. 3. Reflex. Prop. of 4. AAS 5. CPCTC 6. Conv. Of Alt. Int. s Thm.
13 Check It Out! Example 6 Given: J is the midpoint of KM and NL. Prove: KL MN
14 Check It Out! Example 6 Continued Statements Reasons 1. J is the midpoint of KM and NL. 2. KJ MJ, NJ LJ 3. KJL MJN 4. KJL MJN 5. LKJ NMJ 6. KL MN 1. Given 2. Def. of mdpt. 3. Vert. s Thm. 4. SAS Steps 2, 3 5. CPCTC 6. Conv. Of Alt. Int. s Thm.
15 Example 7: Find the value of x.
16 Example 8: Find the value of x and m A.
17 Example 9: 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.
18 Step 2 Use the Distance Formula to find the lengths of the sides of each triangle.
19 So DE GH, EF HI, and DF GI. Therefore DEF GHI by SSS, and DEF GHI by CPCTC.
20 Check It Out! Example 10 Given: J( 1, 2), K(2, 1), L( 2, 0), R(2, 3), S(5, 2), T(1, 1) Prove: JKL RST Step 1 Plot the points on a coordinate plane.
21 Check It Out! Example 10 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.
22 Assignment: Page #3, 4, 7,17, 18, 24-28
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