Ch. 4 – Triangle Congruence (Notes-Renes-Honors Geometry)
4-7 Triangle Congruence: CPCTC
Warm Up
1. If ∆ABC  ∆DEF, then A  ? and BC  ? .
3. If 1  2, why is a||b?
2. What is the distance between (3, 4) and (–1, 5)?
4. List methods used to prove two triangles
congruent.
Objectives
 Use CPCTC to prove parts of triangles are congruent.
***CPCTC is an abbreviation for the phrase “C______________ P_________ of C_______________
T_____________ are C________________.” It can be used as a justification in a proof
after you have
proven two triangles congruent.
Example 1: Engineering Application
A and B are on the edges of a ravine. What is AB?
A landscape architect sets up the triangles shown in the figure to find the distance JK
across a pond. What is JK?
Example 2: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY  YZ.
Prove: XYW  ZYW
Example 3: Using CPCTC in a Proof
Given: PR bisects QPS and QRS.
Prove: PQ  PS
***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.***
Given: NO || MP, N  P
Prove: MN || OP
___Statements
Given: J is the midpoint of KM and NL.
___Statements
Reasons____
Prove: KL || MN
Reasons____
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
4-7-ext Lines and Slopes
Objective :
Prove the slope criteria for parallel and perpendicular lines.
Slopes can be used to determine if two lines in a coordinate plane are parallel or perpendicular. In this lesson, you will prove the
Parallel Lines Theorem and the Perpendicular Lines Theorem. Suppose that L1 and L2 are two lines in the coordinate plane with
slopes m1 and m2. The proof of the Parallel Lines Theorem can be broken into three parts:
1.
2.
3.
Example 1: Proving the Parallel Lines Theorem
Are the lines parallel? Explain.
Complete the two-column proof, using the figure in Example 1
Statements
Given: m1 = m2
Prove: L1 || L2
Reasons
Example 2 : Proving the Perpendicular Lines
Statements
Are the lines perpendicular? Explain.
Reasons
4-8 Intro to Coordinate Proof
Objectives
-Position figures in the coordinate plane for use in coordinate proofs.
-Prove geometric concepts by using coordinate proof.
A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is
to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof
simpler.
Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance
Formula, or the Midpoint Formula to prove statements about the figure.
Write a coordinate proof.
Given: Rectangle ABCD with A(0, 0), B(4, 0), C(4, 10), and D(0, 10)
Prove: The diagonals bisect each other.
Use the information in Example 2 (p. 268) to write a coordinate proof showing that the area of ∆ADB is one half the
area of ∆ABC.
Proof: ∆ABC is a right triangle with height AB and base BC.
Position each figure in the coordinate plane and give the coordinates of each vertex.
-rectangle with width m and length twice the width
-right triangle with legs of lengths s and t
-a square with side length 4p in the coordinate plane and give the coordinates of each vertex.
4-9 Isosceles and Equilateral Triangles
Objectives


Prove theorems about isosceles and equilateral triangles.
Apply properties of isosceles and equilateral triangles.
Vocabulary
legs of an isosceles triangle
vertex angle
base
base angles
Recall that an isosceles triangle has at least _____ congruent sides. The congruent sides are called the ____. The
_____________ ____ is the angle formed by the legs. The side opposite the vertex angle is called the ________, and
the ________________ are the two angles that have the base as a side.
____is the vertex angle.
____and _____ are the base angles.
Example 1: Astronomy Application
The length of YX is 20 feet.
Explain why the length of YZ is the same.
Example 2A: Finding the Measure of an Angle
Find mF.
Find mG.
Find mN.
Find the value of x.
Find the value of y.
Example 4: Using Coordinate Proof
Prove that the segment joining the midpoints of two sides of an isosceles triangle is half the base.
Given: In isosceles ∆ABC, X is the mdpt. of AB, and Y is the mdpt. of AC.
Prove: XY = 1/2AC.