GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Warm UpClassify each angle as acute, obtuse, or right.

1. 2.

3.

4. If the perimeter is 47, find x and the lengths of the

three sides.

rightacute

x = 5; 8; 16; 23

obtuse

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Homework: Page 410-411

You should be done with 1-16

For Tonight work on Problems#: 16*, 22, 23

Include a graph for each problem, use a ruler & compass!

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

An auxiliary line is a line that is added to a figure to aid in a proof.

An auxiliary line used in the

Triangle Sum Theorem

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.

Interior

Exterior

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side.

Interior

Exterior

4 is an exterior angle.

3 is an interior angle.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.

Interior

Exterior

3 is an interior angle.

4 is an exterior angle.

The remote interior angles of 4 are 1 and 2.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Apply SSS and SAS to construct triangles and solve problems.

Prove triangles congruent by using SSS and SAS.

Objectives

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

triangle rigidityincluded angle

Vocabulary

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

The property of triangle rigidity states that if the side lengths of a triangle are given, the triangle can have only one shape.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

For example, you only need to know that two triangles have three pairs of congruent corresponding sides.

This can be expressed as the following postulate.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.

Remember!

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

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.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

TEACH! Example 1

Use SSS to explain why ∆ABC ∆CDA.

It is given that AB CD and BC DA.

By the Reflexive Property of Congruence, AC CA.

So ∆ABC ∆CDA by SSS.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

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

B is the included angle between sides AB and BC.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

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.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

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

Caution

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Example 2: 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.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

TEACH! Example 2

Use SAS to explain why ∆ABC ∆DBC.

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

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angles, you can construct one and only one triangle.

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Proving Triangles Congruent

Given: BC ║ AD, BC ADProve: ∆ABD ∆CDB

ReasonsStatements

5. SAS Steps 3, 2, 45. ∆ABD ∆ CDB

4. Reflex. Prop. of

3. Given

2. Alt. Int. s Thm.2. CBD ADB

1. Given1. BC || AD

3. BC AD

4. BD BD

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

TEACH! Proving Triangles Congruent

Given: QP bisects RQS. QR QS

Prove: ∆RQP ∆SQP

ReasonsStatements

5. SAS Steps 1, 3, 45. ∆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

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Lesson Quiz: Part I

1. Show that ∆ABC ∆DBC, when x = 6.

ABC DBCBC BCAB DBSo ∆ABC ∆DBC by SAS

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

2. 3.none SSS

26°

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Lesson Quiz: Part II

4. 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