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4.8 Use Isosceles and Equilateral Triangles You will use theorems about isosceles and equilateral...

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4.8 Use Isosceles and Equilateral Triangles

• You will use theorems about isosceles and equilateral triangles.

• Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles?

You will learn how to answer this question by learning the Base Angles Theorem and its converse.

Warm-Up ExercisesEXAMPLE 1 Apply the Base Angles Theorem

SOLUTION

In DEF, DE DF . Name two congruent angles.

DE DF , so by the Base Angles Theorem, E F.

Warm-Up ExercisesGUIDED PRACTICE for Example 1

SOLUTION

Copy and complete the statement.

1. If HG HK , then ? ? .

HGK HKG

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Copy and complete the statement.

If KHJ KJH, then ? ? .If KHJ KJH, then ? ? .2. 2.

SOLUTION

If KHJ KJH, then , KH KJ

Warm-Up ExercisesEXAMPLE 2 Find measures in a triangle

Find the measures of P, Q, and R.

The diagram shows that PQR is equilateral. Therefore, by the Corollary to the Base Angles Theorem, PQR is equiangular. So, m P = m Q = m R.

3(m P) = 180o

Triangle Sum Theorem

m P = 60o

Divide each side by 3.

The measures of P, Q, and R are all 60° .

ANSWER Can an equilateral triangle have anangle of 61?

Warm-Up ExercisesGUIDED PRACTICE for Example 2

3. Find ST in the triangle at the right.

SOLUTION

STU is equilateral, then its is equiangular

Thus ST = 5 ( Base angle theorem )

ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 2

4. Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain.

SOLUTION

No; it is not possible for an equilateral triangle to have angle measure other then 60°. Because the triangle sum theorem and the fact that the triangle is equilateral guarantees the angle measure 60° because all pairs of angles could be considered base of an isosceles triangle

Warm-Up ExercisesEXAMPLE 3 Use isosceles and equilateral triangles

ALGEBRA

Find the values of x and y in the diagram.

SOLUTION

STEP 2 Find the value of x. Because LNM LMN, LN LM and LMN is isosceles. You also know that LN = 4 because KLN is equilateral.

STEP 1 Find the value of y. Because KLN is equiangular, it is also equilateral and KN KL . Therefore, y = 4.

Explain how you could findm ∠ M.

Warm-Up ExercisesEXAMPLE 3 Use isosceles and equilateral triangles

LN = LM Definition of congruent segments

4 = x + 1 Substitute 4 for LN and x + 1 for LM.

3 = x Subtract 1 from each side.

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

Lifeguard Tower

In the lifeguard tower, PS QR and QPS PQR.

QPS PQR?

a. What congruence postulate can you use to prove that

b. Explain why PQT is isosceles.

c. Show that PTS QTR.

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

SOLUTION

Draw and label QPS and PQR so that they do not overlap. You can see that PQ QP , PS QR , and QPS PQR. So, by the SAS Congruence Postulate,

a.

QPS PQR.

b. From part (a), you know that 1 2 because corresp. parts of are . By the Converse of the Base Angles Theorem, PT QT , and

PQT is isosceles.

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

c. You know that PS QR , and 3 4 because corresp. parts of are . Also, PTS QTR by the Vertical Angles Congruence Theorem. So, PTS QTR by the AAS Congruence Theorem.

Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4

5. Find the values of x and y in the diagram.

SOLUTION

y° = 120°

x° = 60°

Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4

SOLUTION

QPS PQR. Can be shown by segment addition postulate i.e

a. QT + TS = QS and PT + TR = PR

6. Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that PTS QTR

Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4

Since PT QT from part (b) and

TS TR from part (c) then,

QS PR

PQ PQ Reflexive Property and

PS QR Given

Therefore QPS PQR . By SSS Congruence Postulate

ANSWER

Warm-Up ExercisesDaily Homework Quiz

Find the value of x.

1.

ANSWER 8

Warm-Up ExercisesDaily Homework Quiz

Find the value of x.

2.

ANSWER 3

Warm-Up ExercisesDaily Homework Quiz

If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles?

3.

ANSWER 34°, 34°

Warm-Up ExercisesDaily Homework Quiz

Find the perimeter of triangle.4.

ANSWER 66 cm

• You will use theorems about isosceles and equilateral triangles.

• Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles?

• Angles opposite congruent sidesof a triangle are congruent andconversely.• If a triangle is equilateral, then it is equiangular and conversely.

If two sides of a triangle are congruent, then the angles opposite them are congruent. The converse is also true.

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