Warm-Up 1Warm-Up 1

If the lengths of two sides of an isosceles triangle are 3 and 7, what is the perimeter of the triangle?

Warm-Up 2Warm-Up 2

Use your straightedge to draw an acute angle. How could you use your compass to turn this angle into an isosceles triangle?

Definition: ReviewDefinition: Review

A triangle is an isosceles triangleisosceles triangle if and only if it has at least two congruent sides.

4.7 Use Isosceles and Equilateral 4.7 Use Isosceles and Equilateral TrianglesTriangles

Objectives:

1. To complete and use the Base Angles Theorem and its converse

2. To deduce the Equilateral Triangle Theorem from the Base Angles Theorem

Base Angles Theorem:Base Angles Theorem:

If two sides of a triangle are congruent, then the angles opposite them are congruent.

Example 1Example 1

C

A

B

M

C

A

B

M

C

A

B

M

C

A

B

MedianMedian

1. AC = AB Given2. BM = CM Def. of Median3. AM = AM Reflex4. AMC = AMB SSS5. <B = <C CPCTC

Example 2Example 2

AltitudeAltitude

Try this one in your notebook

Example 3Example 3

Find the value of x.

2x+5

30

35

Example 4: SATExample 4: SAT

If AB = BC, what is y in terms of x?

3x - 2

2y

B

C

A

y=2x-4

Converse of the Base Angles Converse of the Base Angles Theorem:Theorem:

If two angles of a triangle are congruent, then the sides opposite them are congruent.

Investigation 3Investigation 3

In this Investigation, we will use some more compass and straightedge wizardry to construct a perfect equilateral triangle.

Investigation 3Investigation 3

1. Draw AB.

A B

Investigation 3Investigation 3

2. Using A as center and AB as the radius, draw a quarter-circle arc through B.

A BA B

Investigation 3Investigation 3

3. Using B as center and AB as the radius, draw a quarter-circle arc through A.

A BA B

Investigation 3Investigation 3

4. Label the point of intersection C.

A B

C

A B

Investigation 3Investigation 3

5. Draw AC and BC.

C

A B

C

A B

Investigation 3Investigation 3

Click button below to watch a construction video

Investigation 3Investigation 3

In ΔABC, measure each angle with the protractor. What do you notice about the angle measurements?

Use deductive reasoning to show that these measurements must be true based on the Base Angles Theorem.

Equilateral Triangle TheoremEquilateral Triangle Theorem

A triangle is equilateral if and only if it is equiangular.

Example 5Example 5

Find the values of x and y in the diagram.

Answer in your notebook

Example 6: SATExample 6: SAT

Find the value of x.x

24

18

60 60

30

Example 7: SATExample 7: SAT

In the figure shown, if O is the center of the circle, what is the value of x?

70

O

x

55

Example 8: SATExample 8: SAT

In the figure shown, if AB = AC and AC||BD, what is the value of x?

130

D

CB

A

x65