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

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You will use theorems about isosceles and equilateral
triangles.

*

SOLUTION

*

If HG HK , then ? ? .

If KHJ KJH, then ? ? .

If KHJ KJH, then ? ? .

*

Can an equilateral triangle have an

angle of 61?

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)

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

ANSWER

STU is equilateral, then its is equiangular

Thus ST = 5

( Base angle theorem )

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

*

SOLUTION

m ∠ M.

Find the values of x and y in the diagram.

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

*

LN = LM

3 = x

*

QPS PQR?

Explain why PQT is isosceles.

Show that PTS QTR.

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,

QPS PQR.

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.

Solve a multi-step problem

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.

*

SOLUTION

Find the values of x and y in the diagram.

y° = 120°

x° = 60°

SOLUTION

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

a.

QT + TS = QS and PT + TR = PR

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

PTS QTR

Since PT QT from part (b) and

from part (c) then,

ANSWER

1.

ANSWER

8

2.

ANSWER

3

Daily Homework Quiz

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

3.

ANSWER

4.

ANSWER

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 sides

conversely.

• If a triangle is equilateral, then it is equiangular and conversely.

*

*

SOLUTION

*

If HG HK , then ? ? .

If KHJ KJH, then ? ? .

If KHJ KJH, then ? ? .

*

Can an equilateral triangle have an

angle of 61?

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)

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

ANSWER

STU is equilateral, then its is equiangular

Thus ST = 5

( Base angle theorem )

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

*

SOLUTION

m ∠ M.

Find the values of x and y in the diagram.

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

*

LN = LM

3 = x

*

QPS PQR?

Explain why PQT is isosceles.

Show that PTS QTR.

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,

QPS PQR.

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.

Solve a multi-step problem

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.

*

SOLUTION

Find the values of x and y in the diagram.

y° = 120°

x° = 60°

SOLUTION

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

a.

QT + TS = QS and PT + TR = PR

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

PTS QTR

Since PT QT from part (b) and

from part (c) then,

ANSWER

1.

ANSWER

8

2.

ANSWER

3

Daily Homework Quiz

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

3.

ANSWER

4.

ANSWER

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 sides

conversely.

• If a triangle is equilateral, then it is equiangular and conversely.

*

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