PreAP Geometry Notes Name___________________________ Section 5-3 and 5-5 Triangle Inequalities Date ___________

Objectives: Recognize and apply properties of inequalities to the measures of the angles of

a triangle. Recognize and apply properties of inequalities to the relationship between the

angles and the sides of a triangle. Use the Triangle Inequality Theorem to identify possible

triangles. Prove triangle relationships using the Triangle Inequality Theorem.

Sketchpad Activity #1:

Let’s use a Geometer’s Sketchpad sketch of ABC to gather some data about the angle and

side measurements. Then draw a circle around the longest side length of each triangle &

draw a circle around the greatest angle measure. Draw a square around the shortest side

length of each triangle & draw a square around the least angle measure.

BC AC AB m A m B m C

Triangle 1

Triangle 2

Triangle 3

Triangle 4

Triangle 5

Angle-Side Relationships in Triangles Theorem –

If two sides of a triangle are not congruent, then the larger angle is opposite the _____________

side. If two angles of a triangle are not congruent, then the longer side is opposite the

_____________ angle.

Example: The figures are NOT to scale.

(a) List the angle measures of the triangle in

order from least to greatest.

12.6

15.2

15.1

E

D

F

(b) List the side lengths of the triangle in order

from greatest to least.

15°

65°

G

H

I

(c) List the letters in order from least to

greatest.

e

dc

b

a90°

20°

60°

80°

(d) List the letters in order from greatest to

least.

35°

50°

60°85°

30°

100°

g

f

e

d

c

b

a

Sketchpad Activity #2:

Let’s use a Geometer’s Sketchpad sketch of ABC to gather some data about the side

lengths which will guarantee a triangle.

B

A C

AB CB AB CB AC Triangle?

Data Set 1

Data Set 2

Data Set 3

Data Set 4

Data Set 5

Data Set 6

Triangle Inequality Theorem –

Example: Determine whether or not a triangle can have sides with the given lengths.

(a) 8, 13, and 21 (b) 40 cm, 70 cm, and 1 m (c) 5n , 2n , and 2n when

3n

Example: Given lengths of two sides of a triangle, find the range of possible lengths for the

3rd side.

(a) 6 cm and 11 cm

(b) 22 in and 17 in (c) 3 ft and 4 in

SHORTCUT:

ABC exists when _________AB CB AC .

ABC does not exist when _________AB CB AC .