5-6 Inequalities in One Triangle

Previously we dealt with congruent segments and angles (having equal lengths or measure)

We used the Properties of Equality and Congruence

Now we will deal with segments of unequal lengths and angles with unequal measures

We will use the Properties of Inequality and apply them to segment lengths and angle measures

Properties of Inequalities

We will use these properties with the understanding that they apply only to positive numbers.◦ Think positive lengths and angle measurements

Most of the properties that we will use are based on one main principle: the angle/segment addition property

Two (or more) parts added together make the whole.

Properties of Inequalities

Given: AC > AB and AB > BC Conclusion: AC _____ BC

Assign values for each segment length if it helps AC > BC

Example 1

A

B

C

If a>b and b>c, then a>c

Example: If AC>AB and AB>BC, then AC>BC

Example: If A > B, and B > C,then A > C

Properties of Inequalities- Transitive Property

Given: Conclusion:

Again, plug in values if it helps

The total is greater than each of the parts

Example 2- Using the Angle Addition Postulate

A

B

C

D

If a = b + c, and c > 0, then a > b.

Example: If x = y + 1, then x>y and x>1(as long as x and y are greater than 0)

Example: If XY + YZ = XZ, then XY < XZ and YZ < XZ◦ Segment Addition Postulate◦ This tells us that if 2 parts add up to the whole,

then the whole must be larger than the two parts.

Properties of Inequalities- Comparison Property of Inequalities

The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles.

<BCD is the remote angle and <A and <B are the two remote interior angles

Since AB || CE, <B ≅<1 and <A ≅ <2 m<BCD = m<1 + m<2 or m<BCD = m<A + m<B

Properties of Equalities- reviewExterior Angle Theorem

A C

B

D

E

12

The measure of an exterior angle of a triangle will be greater than the measure of each of the two remote interior angles.

Since m<A + m<B = m<BCD, then

Again, this tells us that if 2 parts add up to the whole, then the whole must be larger than the two parts.

Corollary to the Triangle Exterior Angle Theorem

A C

B

D

If we have 2 segments: ◦ MO with midpoint N, and QS with midpoint R◦ MO > QS

Tell if we know the following are true: 1) MN > RS 2) 2MN = MO 3) 2QR > NO 4) QR + RS = QS

Properties of Inequalities

ONM SRQ

Side Lengths & Angle Lengths We have seen in isosceles triangles that we

have two congruent sides.◦ What do we know about their opposite angles?

Side Lengths & Angle Lengths In any triangle, however, the same principle

is at work.

If two congruent sides have congruent opposite angles, then two non-congruent sides will have two non-congruent opposite angles.

Theorem 5-10 If two sides of a triangle are not congruent,

then the larger angle lies opposite the longer side.

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

Theorem 5-11

Side Lengths & Angle Measures In a triangle, the angle opposite the largest

side is the largest angle and vice versa. Also, the angle opposite the smallest side is

the smallest angle and vice versa.

Triangle Inequality Theorem In a triangle, the three sides have specific

relationships and ratios that exist in every triangle.

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

Triangle Inequality Theorem Are the following sets of values possible

lengths for a triangle?

1) 2, 5, 6 2) 6, 11, 12 3) 15, 15, .00001 4) 7, 9, 21

Triangle Inequality Theorem Given the first two values, what is the

possible range of side for the third?

5, 8, ??? = How can we find this set of values?

Triangle Inequality Theorem Given the first two values, what is the

possible range of side for the third?

1) 9, 12, ___

2) 1, 5, ___

3) 18, 22, ___

5-6 worksheet

Classwork

5-6 worksheet (what is not completed in class)

Homework