Lesson 4-4

Proving Congruence:SSS and SAS

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Refer to the figure.1. Identify the congruent triangles.

2. Name the corresponding congruentangles for the congruent triangles.

3. Name the corresponding congruent sides for the congruent triangles.

Refer to the figure.4. Find x.

5. Find mA.

6. Find mP if OPQ WXY and mW = 80, mX = 70, mY = 30.Standardized Test Practice:

A CB D30 70 80 100

5-Minute Check on Lesson 4-35-Minute Check on Lesson 4-35-Minute Check on Lesson 4-35-Minute Check on Lesson 4-3 Transparency 4-4

Refer to the figure.1. Identify the congruent triangles.

LMN RTS2. Name the corresponding congruent

angles for the congruent triangles.L R, N S, M T

3. Name the corresponding congruent sides for the congruent triangles.LM RT, LN RS, NM ST

Refer to the figure.4. Find x.3

5. Find mA. 63

6. Find mP if OPQ WXY and mW = 80, mX = 70, mY = 30.Standardized Test Practice:

A CB D30 70 80 100

Objectives

• Use the SSS Postulate to test for triangle congruence

• Use the SAS Postulate to test for triangle congruence

Vocabulary

• Included angle – the angle formed by two sides sharing a common end point (or vertex)

Postulates

• Side-Side-Side (SSS) Postulate: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

• Side-Angle-Side (SAS) Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.

Side – Angle – Side (SAS)

Statements Reasons

Vertical Angles Theorem

SAS Postulate

ACB DCE (included angle)

AC = CD Given in problem

BC = CE Given

Given: AC = CD BC = CE

Prove: ABC = DEC

ABC DEC

Given: EI FH; FE HI; G is the midpoint of both EI and FH.

ENTOMOLOGY The wings of a moth form two triangles. Write a two-column proof to prove that FEG HIG if EI FH, FE HI, and G is the midpoint of both EI and FH.

Prove: FEG HIG

1. Given1.

Proof: ReasonsStatements

3. SSS3. FEG HIG

2. Midpoint Theorem2.

3. SSS

1. Given2. Reflexive

Proof: ReasonsStatements

1. 2.3. ABC GBC

Write a two-column proof to prove that ABC GBC if

Use the Distance Formula to show that the corresponding sides are congruent.

COORDINATE GEOMETRY Determine whether WDV MLP for D(–5, –1), V(–1, –2), W(–7, –4), L(1, –5), P(2, –1), and M(4, –7). Explain.

Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore, WDV MLP by SSS.

Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore, ABC DEF by SSS.

Determine whether ABC DEF for A(5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1). Explain.

Write a flow proof.

Given:

Prove: QRT STR

Answer:

Write a flow proof.

Given:

Prove: ABC ADC

Proof:

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Answer: SAS

Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. The triangles are congruent by SAS.

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Answer: SSS

Each pair of corresponding sides are congruent. Two are given and the third is congruent by Reflexive Property. So the triangles are congruent by SSS.

Answer: SAS

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

a.

Answer: not possible

b.

Summary & Homework

• Summary:– If all of the corresponding sides of two triangles

are congruent, then the triangles are congruent (SSS).

– If two corresponding sides of two triangles and the included angle are congruent, then the triangles are congruent (SAS).

• Homework: – Pg 266-8: 4, 16-19, 24, 25