4-7 Warm Up Lesson Presentation Lesson Quiz Triangle Congruence: CPCTC Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry
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.
SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Remember!
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.
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.
Example 2: Proving Corresponding Parts Congruent Given: YW bisects XZ, XY YZ. Prove: XYW ZYW Z
Example 2 Continued WY ZW
Given: PR bisects QPS and QRS. Check It Out! Example 2 Prove: PQ PS Given: PR bisects QPS and QRS.
Check It Out! Example 2 Continued PR bisects QPS and QRS QRP SRP QPR SPR Given Def. of bisector RP PR Reflex. Prop. of ∆PQR ∆PSR PQ PS ASA CPCTC
Given: J is the midpoint of KM and NL. Check It Out! Example 3 Prove: KL || MN Given: J is the midpoint of KM and NL.
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.
Step 2 Use the Distance Formula to find the lengths of the sides of each triangle.
So DE GH, EF HI, and DF GI. Therefore ∆DEF ∆GHI by SSS, and DEF GHI by CPCTC.