Ch. 4 – Triangle Congruence (Notes-Renes-Honors Geometry)

4-7Triangle Congruence: CPCTCWarm Up

1. If ∆ABC ∆DEF, then A ? andBC ? .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.

***CPCTCis 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

___StatementsReasons____

Given:J is the midpoint of KM and NL. Prove:KL || MN

___StatementsReasons____

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 TheoremAre the lines parallel? Explain.

Complete the two-column proof, using the figure in Example 1 Given: m1 = m2Prove: L1 || L2

StatementsReasons

Example 2 : Proving the Perpendicular LinesAre the lines perpendicular? Explain.

Statements Reasons

4-8 Intro to Coordinate Proof

Objectives

-Position figures in the coordinate plane for use in coordinate proofs.

-Prove geometric concepts by usingcoordinate 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 trianglebase

vertex anglebase 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. ofAB, and Y is the mdpt. ofAC.

Prove:XY = 1/2AC.