MM1G3b

-Understand and use the triangle inequality, the side-angle inequality, and the exterior angle inequality.

Important Triangle Facts A triangle has 6 parts:

3 sides 3 angles

The sum of the angles in a triangle is always 180°.

A triangle is named by its vertices.

A

BC

R

TS

Name the triangle and its parts.

∆ ABC

ACBCAB

<ACB

<CBA

<BAC

∆ RST

RTSTRS

<RST

<STR

<TRS

Classify triangles by the side lengths.EquilateralIsoscelesScalene

Classify triangles by angle measures.RightAcuteObtuseEquiangular

Do Parts I and II of the Triangle Notes Handout.

Classifying Triangles

– all sides are equal

– no sides are equal– at least two sides are equal

– has one 90 angle⁰– all angles are less than 90⁰

– has one angle greater than 90⁰– all angles are equal

Triangle Side Angle Inequalities Smaller angles are opposite shorter sides.

Larger angles are opposite longer sides.

Example.

98⁰47⁰

35⁰

1. In ∆ABC , list the sides in order from smallest to largest.

2. In ∆JKL , list the angles in order from smallest to largest.

11 in 15 in

24 in

AB, BC, AC <JLK, <KJL, <JKL

Handout straight edges and compasses Construct triangles:

3”, 4”, 5”2”, 3”, 6”

Triangle Inequality Theorem The sum of any two lengths of any two sides of a

triangle is greater than the length of the third side.Could say that the sum of the two shorter sides must be

greater than the longest side.If the 3rd side is equal to or less than the sum of the 2 other

sides, then it can not form a triangle.

Examples –

Can these three sides form a triangle?

A. 5, 8, 16

B. 6, 11, 14

C. 8, 13, 5

5 + 8 16< NO

6 + 11 14> YES

5 + 8 13= NO

Triangle Inequality Theorem If two sides of a triangle are given, describe the possible

lengths of the third side.

A. 2 yd, 6 yd What possible values would work? Compare the sum of 2 shorter

sides to longest side.

So the third side has to be bigger than 4 and less than 8 or 4 < x < 8.

If 3rd side is 1: 1 + 2 > 6

If 3rd side is 2: 2 + 2 > 6

If 3rd side is 3: 3 + 2 > 6

If 3rd side is 4: 4 + 2 > 6

If 3rd side is 5: 5 + 2 > 6

If 3rd side is 6: 6 + 2 > 6

If 3rd side is 7: 6 + 2 > 7

If 3rd side is 8: 6 + 2 > 8

If 3rd side is 9: 6 + 2 > 9

If 3rd side is 10: 6 + 2 > 10

No

No

No

No

Yes

Yes

Yes

No

No

No

Triangle Inequality Theorem If two sides of a triangle are given, describe the possible

lengths of the third side.

A. 2 yd, 6 yd So the third side has to be bigger than 4 and less than 8 or 4 < x < 8.

In other words,

2 + x > 6 or 2 + 6 > x

– 2 – 2

x > 4 or

Therefore, the range is going to be x has to greater than the difference or less than the sum of the two given sides.

8 > x

x < 8

Triangle Inequality Theorem If two sides of a triangle are given, describe the possible

lengths of the third side.

B. 4 in, 12 in 4 + x > 12 and 4 + 12 > x

– 4 – 4

x > 8 and

16 > x

x < 16

21 > x

x < 21x < 15

C. 3 ft, 18 ft3 + x > 18 and 3 + 18 > x – 3 – 3

8 < x < 16 15 < x < 21

Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater

than the measure of either of the nonadjacent (remote) interior angles.

The measure of an exterior angle is the sum of the remote interior angles.

Example

1

2

3 4

53

65°

3 (3x – 5)°6

5What do you know about x?

What relationships do we know about the angles listed?

<5 > <2

<4 > <1

<6 > <1

3x – 5 > 53<5 = <2 + <3

<4 = <1 + <2

<6 = <1 + <3

<5 > <3

<4 > <2

<6 > <3

3x – 5 > 65

3x – 5 = 53 + 653x – 5 = 118

+ 5 = +53x = 123

x = 41

Classwork/Homework

Textbook p287 (4-9,13-24 all)