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# Triangle. Inequalities

## 1. G.5

TriangleInequalities

Modified by Lisa Palen

1

## 2. Triangle Inequality Theorem:

2Triangle Inequality Theorem:

Can you

make a

triangle?

Yes!

## 3. Triangle Inequality Theorem:

3Triangle Inequality Theorem:

Can you

make a

triangle?

NO

because

4 + 5 < 12

## 4. Triangle Inequality Theorem:

4Triangle Inequality Theorem:

The sum of the lengths of any two

sides of a triangle is greater than

the length of the third side.

A

b

C

a+b>c

a+c>b

b+c>a

c

a

B

## 5. Finding the range of the third side:

5Finding the range of the third side:

Example Given a triangle with sides of length 3 and 7, find

the range of possible values for the third side.

Solution Let x be the length of the third side of the triangle.

The maximum value:

x < 3 + 7 = 10

The minimum value:

x>7–3=4

So 4

x < 10

x>4

< x < 10 (x is between 4 and 10.)

## 6. Finding the range of the third side:

6Finding the range of the third side:

Given The lengths of two sides of a triangle

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

given cannot be larger than the other side, we

find the minimum value by subtracting the

two sides.

Difference < Third Side < Sum

## 7. Finding the range of the third side:

7Finding the range of the third side:

Example Given a triangle with sides of length a and b, find

the range of possible values for the third side.

Solution Let x be the length of the third side of the triangle.

The maximum value:

x<a+b

The minimum value:

x > |a – b|

x<a+b

x > |a – b|

|a – b|< x < a + b

(x is between |a – b| and a + b.)

So

## 8.

8In a Triangle:

The smallest angle is opposite the smallest side.

The largest angle is opposite the largest side.

The smallest side is opposite the smallest angle.

The largest side is opposite the largest angle.

m CBA = 80 °

B

m BC = 4.1 cm

m AB = 6.3 cm

C m BCA = 64 °

m CA = 6.9 cm

A

m BAC = 36 °

## 9.

9Theorem

•If one angle of a triangle is larger than a

second angle, then the side opposite the first

angle is larger than the side opposite the

second angle.

larger

angle B

shorter side

C

longer side

smaller angleA

## 10.

10Theorem

•If one side of a triangle is larger than a

second side, then the angle opposite the first

side is larger than the angle opposite the

second side.

## 11. Corollary #1:

11Corollary #1:

The perpendicular segment from a

point to a line is the shortest

segment from the point to the line.

This side is longer

because it is

opposite the

largest angle!

This is the

shortest segment!

## 12. Corollary #2:

12Corollary #2:

The perpendicular segment from a

point to a plane is the shortest

segment from the point to the plane.

This side is longer

because it is

opposite the

largest angle!

This is the

shortest segment!