The Triangle Inequality

&

Inequalities in Multiple Triangles

Objectives Apply the Triangle Inequality Theorem

Recognize and apply properties of inequalities to the measures of angles in a triangle

Recognize and apply properties of inequalities to the relationships between angles and sides of triangles

Inequalities

An inequality simply shows a relationship between any real numbers a and b such that if a > b then there is a positive number c so a = b + c.

All of the algebraic properties for real numbers can be applied to inequalities and measures of angles and segments (i.e. multiplication, division, and transitive).

Determine which angle has the greatest measure.

Explore Compare the measure of 1 to the measures of 2, 3, 4, and 5.

Plan Use properties and theorems of real numbers to compare the angle measures.

Example 1:

Solve Compare m3 to m1.

By the Exterior Angle Theorem, m1 m3 m4. Since angle measures are positive numbers and from the definition of inequality, m1 > m3.

Compare m4 to m1.

By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4.

Compare m5 to m1.

Since all right angles are congruent, 4 5. By the definition of congruent angles, m4 m5. By substitution, m1 > m5.

Example 1:

By the Exterior Angle Theorem, m5 m2 m3. By the definition of inequality, m5 > m2. Since we know that m1 > m5, by the Transitive Property, m1 > m2.

Compare m2 to m5.

Examine The results on the previous slides show that m1 > m2, m1 > m3, m1 > m4, and m1 > m5. Therefore, 1 has the greatest measure.

Answer: 1 has the greatest measure.

Example 1:

Determine which angle has the greatest measure.

Answer: 5 has the greatest measure.

Your Turn:

Exterior Angle Inequality Theorem

If an is an exterior of a ∆, then its measure is greater than the measure of either of its remote interior s.

m 1 > m 1 > m 33

m m 1 > m 1 > m 44

Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14.

By the Exterior Angle Inequality Theorem, m14 > m4, m14 > m11, m14 > m2, and m14 > m4 + m3.

Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7.

Answer: Thus, the measures of 4, 11, 9, 3, 2, 6, and 7 are all less than m14 .

Example 2a:

Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5.

By the Exterior Angle Inequality Theorem, m10 > m5, and m16 > m10, so m16 > m5, m17 > m5 + m6, m15 > m12, and m12 > m5, so m15 > m5.

Answer: Thus, the measures of 10, 16, 12, 15 and

17 are all greater than m5.

Example 2b:

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

a. all angles whose measures are less than m4

b. all angles whose measures are greater than m8

Answer: 5, 2, 8, 7

Answer: 4, 9, 5

Your Turn:

Theorem 5.9

If one side of a ∆ is longer than another side, then the opposite the longer side has a greater measure then the opposite the shorter side (i.e. the longest side is opposite the largest .)

m m 1 > m 1 > m 2 > m 2 > m 331

2

3

Determine the relationship between the measures of RSU and SUR.

Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

Example 3a:

Determine the relationship between the measures of TSV and STV.

Answer: The side opposite TSV is shorter than the side opposite STV, so mTSV < mSTV.

Example 3b:

Determine the relationship between the measures of RSV and RUV.

Answer: mRSV > mRUV

mRSU > mSUR

mUSV > mSUV

mRSU + mUSV > mSUR + mSUV

mRSV > mRUV

Example 3c:

Determine the relationship between the measures of the given angles.

a. ABD, DAB

b. AED, EAD

c. EAB, EDB

Answer: ABD > DAB

Answer: AED > EAD

Answer: EAB < EDB

Your Turn:

Theorem 5.10

If one of a ∆ has a greater measure than another , then the side opposite the greater is longer than the side opposite the lesser .

A

B C

AC > BC > CAAC > BC > CA

HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

Example 4:

Theorem 5.10 states that if one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Since X is opposite the longest side it has the greatest measure.

Answer: So, Ebony should tie the ends marked Y and Z.

Example 4:

KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing?

Answer: A and D

Your Turn:

Theorem 5.11∆ Inequality Theorem

The sum of the lengths of any two sides of a ∆ is greater than the length of the 3rd side.

The ∆ Inequality Theorem can be used to determine whether 3 sides can form a triangle or not.

do

g

d + o > g

o + g > d

g + d > o

Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle.

Determine whether the measures and

can be lengths of the sides of a triangle.

Example 1a:

HINT: If the sum of the two smaller sides is greater than the longest side, then it can form a ∆.

Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle.

Check each inequality.

Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.

Example 1b:

Determine whether the given measures can be lengths of the sides of a triangle.

a. 6, 9, 16

b. 14, 16, 27

Answer: no

Answer: yes

Your Turn:

A 7 B 9 C 11 D 13

Multiple-Choice Test ItemIn and Which measure cannot be PR?

Example 2:

Read the Test Item

You need to determine which value is not valid.

Solve the Test Item

Solve each inequality to determine the range of values for PR.

Example 2:

Graph the inequalities on the same number line.

The range of values that fit all three inequalities is

Example 2:

Examine the answer choices. The only value that does not satisfy the compound inequality is 13 since 13 is greater than 12.4. Thus, the answer is choice D.

Answer: D

Example 2:

A 4 B 9 C 12 D 16

Answer: D

Multiple-Choice Test ItemWhich measure cannot

be XZ?

Your Turn: