4.4 Prove Triangles Congruent by SSS

• You will use the side lengths to prove triangles are congruent.

• Essential Question:

How can you use side lengths to prove triangles congruent?

You will see how toanswer this question by learningthe SSS Congruence Postulate.

predict

Warm-Up Exercises

ANSWER ∆MNO ∆PRQ

1. Write a congruence statement.

M

N O

P

R Q

Warm-Up Exercises

ANSWER Third s Thm.

2. How do you know that N R?

M

N O

P

R Q

Warm-Up Exercises

ANSWER 30

3. Find x.

(7x – 50)º(2x + 10)º

(3x)º

Warm-Up ExercisesEXAMPLE 1 Use the SSS Congruence Postulate

Write a flow chart proof.

GIVEN KL NL, KM NM

LM LN.

KLM NLM

PROVE KLM NLM

given

KL = NL

KM = NM

Reflexive Property

givenSSS

Warm-Up Exercises

Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.

GUIDED PRACTICE for Example 1

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

Yes. The statement is true.

1. DFG HJK

Side DG HK, Side DF JH,and Side FG JK.

So by the SSS Congruence postulate, DFG HJK.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

2. ACB CAD

BC ADGIVEN :

PROVE : ACB CAD

PROOF: It is given that BC AD By Reflexive propertyAC AC, But AB is not congruent CD.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

QT TR , PQ SR, PT TSGIVEN :

PROVE : QPT RST

PROOF: It is given that QT TR, PQ SR, PT TS. So bySSS congruence postulate, QPT RST. Yes the statement is true.

QPT RST 3.

Warm-Up ExercisesEXAMPLE 2 Standardized Test Practice

SOLUTION

By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR.

d = y2 – y1( )2x2 – x1( )2 +

Warm-Up ExercisesEXAMPLE 2 Standardized Test Practice

= 25 5=

By the SSS Congruence Postulate, any triangle with side lengths 3, 4, and 5 will be congruent to PQR.

= + 1 – 4( )2– 1 (– 5( ) )2–PR

= 42 + (– 3 ) 2

= 42 + (– 3 2 5=(–4) – (–1)( )25 – 1)( 2 + = 25 )

The correct answer is A.ANSWER

The distance from (–1, 1) to (–1, 5) is 4. The distance from (–1, 5) to (–4, 5) is 3. The distance from (– 1, 1) to (–4, 5) is

Warm-Up ExercisesGUIDED PRACTICE for Example 2

JKL

has vertices J(–3, –2), K(0, –2), and L(–3, –8).

RST has vertices R(10, 0), S(10, – 3), and T(4, 0).

Graph the triangles in the same coordinate plane and show that they are congruent.

4.

KJ = SR = 3.ANSWER

JL = RT = 6.

LK = TS = 3 5.

Warm-Up ExercisesEXAMPLE 3 Solve a real-world problem

Structural Support

Explain why the bench with the diagonal support is stable, while the one without the support can collapse.

Warm-Up ExercisesEXAMPLE 3 Solve a real-world problem

The bench with a diagonal support forms triangles with fixed side lengths. By the SSS Congruence Postulate, these triangles cannot change shape, so the bench is stable. The bench without a diagonal support is not stable because there are many possible quadrilaterals with the given side lengths.

SOLUTION

Warm-Up ExercisesGUIDED PRACTICE for Example 3

Determine whether the figure is stable. Explain your reasoning.

SOLUTION

The figure is without a diagonal support is not stable Because there are many possible quadrilaterals with the given side lengths.

5.

Warm-Up ExercisesGUIDED PRACTICE for Example 3

Determine whether the figure is stable. Explain your reasoning.

SOLUTION

The diagonal support forms triangle with fixed side length by SSS congruence postulate, these triangles can not change shape. The figure is stable.

6.

Warm-Up ExercisesGUIDED PRACTICE for Example 3

Determine whether the figure is stable. Explain your reasoning.

SOLUTION

The diagonal support is not stable because the lower half of figure dies not have diagonal support.

7.

Warm-Up ExercisesDaily Homework Quiz

1. The vertices GHI and RST are G(–2, 5), H(2, 5), I(–2, 2), R(–9, 8), S(–5, 8), and T(–9, 5). Is GHI RST? Explain.

Yes. GH = RS = 4, HI = ST = 5, and IG = TR = 3. By the SSS post ., it follows that GHI RST.

ANSWER

Warm-Up ExercisesDaily Homework Quiz

Is ABC XYZ? Explain.2.

Yes. By the seg. Add. Post., AC XZ. Also , AB XY and BC YZ. So ABC XYZ by the SSS post.

ANSWER

• You will use the side lengths to prove triangles are congruent.

• Essential Question:

How can you use side lengths to prove triangles congruent?

• Two triangles with the same side lengths are congruent by the SSSCongruence Postulate.

Show that the sides can be matched so that all three pairs of corresponding sides are congruent.