Do Now

1. ∆QRS ∆LMN. Name all pairs of congruent corresponding parts.

2. Find the equation of the line through the points (3, 7) and (5, 1)

QR LM, RS MN, QS LN, Q L, R M, S N

y - 1 = -3(x - 5) or y - 7 = -3(x - 3)

or y = -3x + 16

TARGET: SWBAT Prove triangles congruent by using SSS and SAS.

4.2 Triangle Congruence

Example 1: Using SSS to Prove Triangle Congruence

Use SSS to explain why ∆ABC ∆DBC.

It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS.

Example 2 – More with SSS

Write a Congruence Statement for the following triangles.

It is given that AB CD and BC DA.

By the Reflexive Property of Congruence, AC CA.

So ∆ABC ∆CDA by SSS.

An included angle is an angle formed by two adjacent sides of a polygon.

B is the included angle between sides AB and BC.

It can also be shown that only two pairs of congruent

corresponding sides are needed to prove the

congruence of two triangles if the included angles are also

congruent.

This is Postulate 4-2 (SAS)

The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

Caution

Example 3: Engineering Application

The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ ∆VWZ.

It is given that XZ VZ and that YZ WZ. By the Vertical s Theorem. XZY VZW. Therefore ∆XYZ ∆VWZ by SAS.

Example 4 – More with SAS

Write a Congruence Statement for the following triangles.

It is given that BA BD and ABC DBC. By the Reflexive Property of congruence, BC BC. So ∆ABC ∆DBC by SAS.

Example 5: Proving Triangles Congruent

Given: BC ║ AD, BC AD

Prove: ∆ADB ∆CBD

ReasonsStatements

5. Side-Angle-Side5. ∆ADB ∆ CBD

2. Reflex. Prop. of 1. Given

4. Alt. Int. s Thm.4. CBD ADB

3. Given3. BC || AD

1. BC AD

2. BD BD

Example 6 – More with SAS

Given: QP bisects RQS. QR QS

Prove: ∆RQP ∆SQP

ReasonsStatements

5. Side-Angle-Side5. ∆RQP ∆SQP

4. Reflex. Prop. of

1. Given

3. Def. of bisector3. RQP SQP

2. Given2. QP bisects RQS

1. QR QS

4. QP QP

Assignment #33 - Pages 230-233

Foundation: 1-3, 8, 9, 11-14

Core: 16, 19, 24-26

Lesson Quiz: Part I

Which postulate, if any, can be used to prove the triangles congruent?

1. 2.none SSS

Lesson Quiz: Part II

3. Given: PN bisects MO, PN MO

Prove: ∆MNP ∆ONP

1. Given2. Def. of bisect3. Reflex. Prop. of 4. Given5. Def. of 6. Rt. Thm.7. SAS Steps 2, 6, 3

1. PN bisects MO2. MN ON3. PN PN4. PN MO 5. PNM and PNO are rt. s6. PNM PNO

7. ∆MNP ∆ONP

Reasons Statements