Chapter 5 Section 5.5 Inequalities in a triangle.

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Presentation transcript:

Chapter 5 Section 5.5 Inequalities in a triangle

-3 x < > -x -2 < x Don’t forget to change it! -2x 3 < 2x < 2x 22 6 < x6 < x

Compare Sides to Order Angles Theorem Theorem 5.10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. A B C CB > CA > AB m  A > m  B > m  C

Compare Angles to Order Sides Theorem Theorem 5.11 If one Angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Q P R QR > PR > PQ m  P > m  Q > m  R

m  M > m  K > m  L KL > LM > KM

m  M > m  O > m  N ON > MN > OM

m  B > m  A > m  C AC > BC> AB

m  F > m  E > m  G EG > GF > EF

m  J > m  I > m  H HI > HJ > IJ

m  K > m  L > m  M LM > KM > KL

In  ABC CB > AB > AC In  CBD BD > CD > CB Thus BD > CD > CB > AB > AC

In  LKM LM > KL > KM In  LMN MN > LN > LM Thus MN > LN > LM > LN > LM

Theorem Theorem 5.12 Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of the two nonadjacent interior angles. m  TQR > m  R  TQR is an exterior angle of  QRP Inequalities in a triangle Q P R T And m  TQR > m  P

Theorem Theorem 5.13 Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AB + AC > BC BC + AC > AB Inequalities in a triangle

The sum of any two sides must be greater than the third side! XZ + YZ > XY XZ + XY > YZ YZ + XY > XZ > XY 5 > XY 2 + XY > 3 XY > XY > 2 XY > -1 5 > XY > 1

The sum of any two sides must be greater than the third side! XZ + YZ > XY XZ + XY > YZ YZ + XY > XZ > XY 18 > XY 8 + XY > 10 XY > XY > 8 XY > > XY > 2

CC AF DS x > x 60 + x > > x x > -20 x + 40 > 60 x > > x > 20 No, the distance must be less than 100 miles