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4.7 – Isosceles Triangles. Geometry Ms. Rinaldi. Remember that a triangle is isosceles if it has at least two congruent sides. When an isosceles triangle has exactly two congruent sides, these two sides are the legs . The angle formed by the legs is the vertex angle . - PowerPoint PPT Presentation

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4.7 – Isosceles Triangles Geometry Ms. Rinaldi

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4.7 – Isosceles Triangles

GeometryMs. Rinaldi

Isosceles Triangles

• Remember that a triangle is isosceles if it has at least two congruent sides.

• When an isosceles triangle has exactly two congruent sides, these two sides are the legs.

• The angle formed by the legs is the vertex angle.

• The third side is the base of the isosceles triangle.

• The two angles adjacent to the base are called base angles.

Base Angles Theorem

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

If , thenACAB CB

Converse of Base Angles Theorem

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

If , thenCB ACAB

EXAMPLE 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.

EXAMPLE 2 Apply the Base Angles Theorem

In . Name two congruent angles.QRPQPQR ,

P

RQ

EXAMPLE 3 Apply the Base Angles Theorem

Copy and complete the statement.

1. If HG HK , then ? ? .

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

EXAMPLE 4 Apply the Base Angles Theorem

P

R

Q

(30)°

Find the measures of the angles.

SOLUTION

Since a triangle has 180°, 180 – 30 = 150° for the other two angles.

Since the opposite sides are congruent, angles Q and P must be congruent.

150/2 = 75° each.

EXAMPLE 5 Apply the Base Angles Theorem

P

R

Q

(48)°

Find the measures of the angles.

EXAMPLE 6 Apply the Base Angles Theorem

P

R

Q(62)°

Find the measures of the angles.

EXAMPLE 7 Apply the Base Angles Theorem

Find the value of x. Then find the measure of each angle.

P

RQ(20x-4)°

(12x+20)°

SOLUTION

Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12x + 20 = 20x – 4

20 = 8x – 4

24 = 8x

3 = xPlugging back in,

And since there must be 180 degrees in the triangle,

564)3(20

5620)3(12

Rm

Pm

685656180Qm

EXAMPLE 8 Apply the Base Angles Theorem

Find the value of x. Then find the measure of each angle.

P

R

Q(11x+8)° (5x+50)°

EXAMPLE 9 Apply the Base Angles Theorem

Find the value of x. Then find the length of the labeled sides.

P

R

Q(80)° (80)°

SOLUTION

Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40

4x = 40

x = 107x 3x+40

Plugging back in,

QR = 7(10)= 70PR = 3(10) + 40 = 70

EXAMPLE 10 Apply the Base Angles Theorem

Find the value of x. Then find the length of the labeled sides.

P

RQ

(50)°

(50)°

10x – 2

5x+3

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