GEOMETRY 8-1 Proving Triangles Congruent SSS & SAS Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

GEOMETRY

8-1 Proving Triangles Congruent SSS & SAS

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

GEOMETRY


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

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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!

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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

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

Interior

Exterior

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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.

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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.

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Apply SSS and SAS to construct triangles and solve problems.

Prove triangles congruent by using SSS and SAS.

Objectives

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triangle rigidityincluded angle

Vocabulary

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

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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.

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

Remember!

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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.

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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.

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

B is the included angle between sides AB and BC.

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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.

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

Caution

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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.

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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.

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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.

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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

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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

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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°

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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

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GEOMETRY 8-1 Proving Triangles Congruent SSS & SAS Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz