GeometryLesson 5 – 6

Inequalities in Two Triangles

Objective:Apply the Hinge Theorem or its converse to make comparisons in

two triangles.Prove triangle relationships using the Hinge Theorem or its converse.

Inequalities in Two Triangles

Hinge Theorem If two sides of a triangle are congruent to two

sides of another triangle, and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

Converse of Hinge TheoremIf two sides of a triangle are congruent to two sides of another triangle, and the third side in the first triangle is longer than the third side in the second triangle, then the include angle measure of the first triangle is greater than the included angle measure in the second.

Compare the given measures

WX and XY

BFCmandFCDm

WX < XY

BFCmFCDm

Compare the given measures

JK and MQ

VRTmandSRTm

JK > MQ

VRTmSRTm

AD and BD

Compare the given measures

BDCmandABDm

AD > BD

BDCmABDm

Real WorldTwo groups of snowmobilers leave from the same base camp. Group A goes 7.5 miles due west and then turns 35 degrees north of west and goes 5 miles. Group B goes 7.5 miles due east then turns 40 degrees north of east and goes 5 miles. At this point, which group is farther from the base camp? Explain.Draw a picture:

Group A is farther from campsince the included angle is larger than Group B.

Find the range of possible values for x.

6x + 15 > 65 6x + 15 > 0 6x + 15 < 180Angle has to be greater than 0, but less than 180.

6x > 50

3

18x

6x < 165

2

127x

2

127

3

18: xRange

Don’t have to solve since we already said has to be greater than 65.Double check each time!

Find the range of possible values for x.

9a + 15 < 141 9a + 15 > 0 9a + 15 < 180

9a < 126

a < 14

9a > -15

3

21a

Don’t have to solve since we already said has to be less than 141.Double check each time!

143

21: aRange

Find the range of possible values for x.

5x + 2 < 47 5x + 2 > 0 The length of a sidemust be positive.Do not need < 180 since 180 is for an anglenot a side, and side hasno limit on length.

5x < 45 5x > -2

5

2xx < 9

95

2: xRange

Homework

Pg. 371 1 – 8 all, 10 – 22 E, 38, 44 – 58 E