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3-11-13 Unit 2 Triangles

Triangle Inequalities and Isosceles Triangles

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Triangle Inequality The shortest side is across from the smallest angle.

The largest angle is across from the longest side.

AB

= 4

.3 c

m

BC = 3.2 cm

AC = 5.3 cm

is thesmallest angle, is thesmallest side. A BC

is the largest angle, is the largest side. B AC

54°

37°

89°

B

C

A

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Triangle Inequality – examples…For the triangle, list the angles in order from least to greatest measure.

CA

B

4 cm

6 cm

5 cm

, ,

.

arg arg .

AB is the smallest side C smallest angle

BC is thel est side Ais

Angles in order from least to grea

the

tes

l est angle

t C B A

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Triangle Inequality – examples…For the triangle, list the sides in order from shortest to longest measure.

8x-10

7x+67x+8

CA

B(7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180°

22 x + 4 = 180 °

22x = 176

X = 8 m<C = 7x + 8 = 64 °

m<A = 7x + 6 = 62 °

m<B = 8x – 10 = 54 ° 64 °62 °

54 °

, ,

.

arg .

B is the smallest angle AC shortest side

C is thel est angle ABi

Sides in order from smallest to

s the

long

longest s

est AC BC AB

ide

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Triangle Inequality Rule:

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

c

b

a

B

C

A

a + b > c

a + c > b

b + c > a

Example: Determine if it is possible to draw a triangle with side measures 12, 11, and 17. 12 + 11 > 17 Yes

11 + 17 > 12 Yes

12 + 17 > 11 Yes

Therefore a triangle can be drawn.

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Finding the range of the third side:Since the third side cannot be larger than the other two added

together, we find the maximum value by adding the two sides.

Since the third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides.

Example: Given a triangle with sides of length 3 and 8, find the range of possible values for the third side.

The maximum value (if x is the largest side of the triangle) 3 + 8 > x

11 > x

The minimum value (if x is not that largest side of the ∆) 8 – 3 > x

5> x

Range of the third side is 5 < x < 11.

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Parts of an Isosceles Triangle

An isosceles triangle is a triangle with two congruent sides.

The congruent sides are called legs and the third side is called the base.

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Leg Leg

Base

21

Ð1 and Ð2 are base anglesÐ3 is the vertex angle

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Isosceles Triangle Rule

By the Isosceles Triangle Theorem,the third angle must also be x.Therefore, x + x + 50 = 180

2x + 50 = 1802x = 130x = 65

Example:

x°

50°

Find the value of x.

A

B C

, .If AB AC then B C

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

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Isosceles Triangle RuleIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Example: Find the value of x. Since two angles are congruent, the sides opposite these angles must be congruent.

3x – 7 = x + 152x = 22X = 11

A

B C

50° 50°

3x - 7x+15

A

B C

, .If B C then AB AC