5-4: Inequalities for Sides and Angles of a Triangle

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5-4 Inequalities for Sides and Angles of a
Triangle

• Expectation
• G1.2.2 Construct and justify arguments and solve
  multi-step problems involving angle measure, side
  length, perimeter and area of all types of
  triangles.

2

• The highest and lowest temperatures recorded in
  New York one year were 38 degrees Celsius and -21
  degrees Celsius. The next year the highest and
  lowest temperatures were 36C and -25ºC.
• What was the difference in the lowest and highest
  temperatures over the two years?

3
Scalene Triangles
a. Draw a scalene triangle.
b. Determine the measures of all of the sides and
all of the angles.
c. List the angles in terms of their measures
from smallest to largest.
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Scalene Triangles
d. List the sides in terms of their measures from
smallest to biggest.
e. Conjecture a relationship between the measures
of the angles and the lengths of the sides of the
triangle.
5
Unequal Sides Theorem

• If one side of a triangle is longer than another,
  then the angle opposite the longer side has
  ___________ measure than the angle opposite the
  shorter side.

The larger angle is opposite the longer side.
6
Unequal Angles Theorem

• If one angle of a triangle has greater measure
  than another angle, then the side opposite the
  larger angle is _____________ than the side
  opposite the smaller angle.

The longer side is opposite the larger angle.
7
Order the sides from longest to shortest.
C
37
105
A
B
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9
Order the angles from largest to smallest.
Y
27
38
X
42
Z
10
Triangle ABC has vertices A(0,0), B(4,7) and
C(-2,-3). List the angles in terms of size from
smallest to largest.
11
List the sides of ?ABC in order from shortest to
greatest

• m?A 9x - 4, m?B 4x - 16, m?C 68 - 2x

12
Distance Between a Point and a Line Theorem

• The shortest distance from a line to a point not
  on the line is the length of a perpendicular
  segment from the point to the line.

13
Prove the Distance Between a Point and a Line
Theorem by using an indirect proof.

• Given TR ? m, R and L are distinct points
• Prove TL gt TR

T
R
L
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Distance Between a Point and a Plane Corollary

• The shortest distance from a plane to a point not
  on the plane is the length of a perpendicular
  segment from the point to the plane.

16
Name the longest segment in ?CED.
E
A
55
30
50
D
100
40
B
C
17
Name the longest segment in the figure.
E
A
55
30
50
D
100
40
B
C
18
Given AD gt CD, m?A gt m?ADB and m?ABD gt m?DBC,
which of the following statements must be true?

1. AB gt BD
2. m?BCD lt m?ADB
3. ?A ? ?C
4. m?A lt m?C
5. BD gt CD

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Assignment

• pages 263-264,
• 15 26 (all), 29 and 32