Warm-Up:EOC Review

What is the negation of x ≤ 10?

A) x ≤ 10B) –x ≤ 10C) –x > 10D) x > 10

What is the inverse of p q?

A) q pB) ~q ~pC) p qD) ~p ~q

Inequalities in Triangles5.5

1. Use inequalities involving angles of triangles.

2. To use inequalities involving sides of triangles.

Today’s GoalsBy the end of class today, YOU should be able to…

Comparison Property of Inequality

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

Proof of the Comparison Property of Inequality

Given: a = b + c, and c > 0

Prove: a > b

Statement 1: c > 0 Given

Statement 2: b + c > b + 0

Addition Property of Inequality

Statement 3: b + c > b

Simplify

Statement 5: a > b Substitute a for b + c in Statement 3

Statement 4: a = b + c Given

Corollary to the Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.m<1 > m<2 and m<1 > m<3

Theorem 5-10

If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.If XZ > XY, then m<Y > m<Z.

Theorem 5-11

If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If m<A > m<B, then BC > AC.

Ex.1: Using Theorem 5-11

In TUV, which side is shortest?

Ex.1: Solution

By the Triangle Angle-Sum Theorem, m<T = 60. The smallest angle in TUV is U. It follows, by Theorem 5-11, that the shortest side is TV.

You Try…

Which side is shortest?

Which side is longest?

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.XY + YZ > XZYZ + ZX > YXZX + XY > ZY

Ex.2: Using the Triangle Inequality Theorem

Can a triangle have sides with the given lengths?

3ft, 7ft, 8ft

Ex.2: Solution

3 + 7 > 88 + 7 > 33 + 8 > 7

Yes, the sum of any two lengths is greater than the third length

You Try…

Can a triangle have sides with the given lengths?

3ft, 6 ft, 10 ft

Ex.3: Using the Triangle Inequality Theorem

A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side.

Ex.3: Solution

Let x represent the length of the third side.

8 + 10 > xx < 18

x + 10 > 8

x > -2

x + 8 > 10

x > 2

The third side must be longer than 2 cm and shorter than 18 cm.

You Try…

A triangle has sides of lengths 7 in and 4 in. Describe the lengths possible for the third side.

Homework

Page 276 #s 1, 6, 7, 14, 17, 20, 24, 25

Page 277 # 34The assignment can also be found

at:•http://www.pearsonsuccessnet.com/sn

papp/iText/products/0-13-037878-X/Ch05/05-05/PH_Geom_ch05-05_Ex.pdf