5'5 Inequalities in One Triangle

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5.5 Inequalities in One Triangle

• Geometry
• Mrs. Spitz
• Fall, 2004

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Objectives

• Use triangle measurements to decide which side is
  longest or which angle is largest.
• Use the Triangle Inequality

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Assignment

• pp. 298-300 1-25, 34

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Objective 1 Comparing Measurements of a Triangle

• In activity 5.5, you may have discovered a
  relationship between the positions of the longest
  and shortest sides of a triangle and the position
  of its angles.

The diagrams illustrate Thms. 5.10 and 5.11.
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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.

m?A gt m?C
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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.

60
40
EF gt DF
You can write the measurements of a triangle in
order from least to greatest.
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Ex. 1 Writing Measurements in Order from Least
to Greatest

• Write the measurements of the triangles from
  least to greatest.
• m ?G lt m?H lt m ?J
• JH lt JG lt GH

100
45
35
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Ex. 1 Writing Measurements in Order from Least
to Greatest

• Write the measurements of the triangles from
  least to greatest.
• QP lt PR lt QR
• m ?R lt m?Q lt m ?P

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5
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Paragraph Proof Theorem 5.10

• Given?AC gt AB
• Prove ?m?ABC gt m?C
• Use the Ruler Postulate to locate a point D on AC
  such that DA BA. Then draw the segment BD. In
  the isosceles triangle ?ABD, ?1 ? ?2. Because
  m?ABC m?1m?3, it follows that m?ABC gt m?1.
  Substituting m?2 for m?1 produces m?ABC gt m?2.
  Because m?2 m?3 m?C, m?2 gt m?C. Finally
  because m?ABC gt m?2 and m?2 gt m?C, you can
  conclude that m?ABC gt m?C.

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NOTE

• The proof of 5.10 in the slide previous uses the
  fact that ?2 is an exterior angle for ?BDC, so
  its measure is the sum of the measures of the two
  nonadjacent interior angles. Then m?2 must be
  greater than the measure of either nonadjacent
  interior angle. This result is stated in Theorem
  5.12

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Theorem 5.12-Exterior Angle Inequality

• The measure of an exterior angle of a triangle is
  greater than the measure of either of the two non
  adjacent interior angles.
• m?1 gt m?A and m?1 gt m?B

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Ex. 2 Using Theorem 5.10

• DIRECTORS CHAIR. In the directors chair shown,
  AB ? AC and BC gt AB. What can you conclude about
  the angles in ?ABC?

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Ex. 2 Using Theorem 5.10Solution

• Because AB ? AC, ?ABC is isosceles, so ?B ? ?C.
  Therefore, m?B m?C. Because BCgtAB, m?A gt m?C by
  Theorem 5.10. By substitution, m?A gt m?B. In
  addition, you can conclude that m?A gt60, m?Blt
  60, and m?C lt 60.

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Objective 2 Using the Triangle Inequality

• Not every group of three segments can be used to
  form a triangle. The lengths of the segments
  must fit a certain relationship.

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Ex. 3 Constructing a Triangle

• 2 cm, 2 cm, 5 cm
• 3 cm, 2 cm, 5 cm
• 4 cm, 2 cm, 5 cm
• Solution Try drawing triangles with the given
  side lengths. Only group (c) is possible. The
  sum of the first and second lengths must be
  greater than the third length.

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Ex. 3 Constructing a Triangle

• 2 cm, 2 cm, 5 cm
• 3 cm, 2 cm, 5 cm
• 4 cm, 2 cm, 5 cm

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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 gt AC
• AC BC gt AB
• AB AC gt BC

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Ex. 4 Finding Possible Side Lengths

• A triangle has one side of 10 cm and another of
  14 cm. Describe the possible lengths of the
  third side
• SOLUTION Let x represent the length of the
  third side. Using the Triangle Inequality, you
  can write and solve inequalities.

• x 10 gt 14
• x gt 4
• 10 14 gt x
• 24 gt x
• ?So, the length of the third side must be greater
  than 4 cm and less than 24 cm.

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24 - homework

• Solve the inequality
• AB AC gt BC.
• (x 2) (x 3) gt 3x 2
• 2x 5 gt 3x 2
• 5 gt x 2
• 7 gt x

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5. Geography

• AB BC gt AC
• MC CG gt MG
• 99 165 gt x
• 264 gt x
• x 99 lt 165
• x lt 66
• 66 lt x lt 264