Transcript Geometry Unit 3 - Notes Sections 5-2 and 5.4

Geometry ~
Chapter 5.2 and 5.4
Inequalities and Triangles
Inequalities
What are they?
 Angle measures can be compared using
inequalities:


m<a=m<b

m<a< m<b

m<a>m<b
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Exterior Angle Inequality Theorem:
If an angle is an exterior angle of a
triangle, then its measure is greater than
the measure of either of the remote interior
angles.
 Example:

 <1
is an exterior angle
 m < 1 > m <PQR
m
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< 1 > m<RPQ
1
The positions of the longest and shortest
sides of a triangle are related to the positions
of the largest and smallest angles.
If one side of a triangle is longer
than another side, then the angle
opposite the longer side has a
greater measure than the angle
opposite the shorter side.
If one angle of a triangle has a
greater measure than another
angle, then the side opposite the
greater angle is longer than the
side opposite the lesser angle.
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Ex. 1 - Write the angles in order from
smallest to largest.
The shortest side is GH, so the
smallest angle is opposite
GH…. F
The longest side is FH, so the
largest angle is G
The angles from smallest to largest are
F, H and G.
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Ex. 2 - Write the sides in order from shortest
to longest.
mR = 180° – (60° + 72°) = 48°
The smallest angle is R, so the
shortest side is PQ.
The largest angle is Q, so the
longest side is PR.
The sides from shortest to
longest are PQ, QR, and PR.
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Example 2A ~ If m  A = 9x – 7, m  B = 7x – 9
and m  C = 28 – 2x, list the sides of ABC in order
from shortest to longest.
Draw and label triangle ABC!!!
A 101°
(9x – 7) + (7x – 9) + (28 – 2x) = 180
(9x – 7)°
B
14x + 12 = 180
(7x – 9)° (28 – 2x)°
C
4°
75°
14x = 168
x = 12
The sides from shortest to longest are AB, AC, BC
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A triangle is formed by three segments, but not
every set of three segments can form a triangle.
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If you take 3 straws of lengths 8 inches, 5
inches and 1 inch and try to make a triangle
with them, you will find that it is not possible.
This illustrates the Triangle Inequality
Theorem.
A certain relationship must exist among the
lengths of three segments in order for them to
form a triangle.
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Ex. 3 – Applying the Triangle Inequality Thm.
Tell whether a triangle can have sides with the
given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
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Ex. 4 - Tell whether a triangle can have sides
with the given lengths. Explain.
2.3, 3.1, 4.6



Yes—the sum of each pair of lengths is
greater than the third length.
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Ex. 5 - Tell whether a triangle can have sides
with the given lengths. Explain.
8, 13, 21
No—by the Triangle Inequality Theorem,
a triangle cannot have these side lengths.
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Finding the RANGE of side lengths
The lengths of two sides of a triangle are 8 inches
and 13 inches. Find the range of possible lengths for
the third side.
Let x represent the length of the third side. Then apply the
Triangle Inequality Theorem.
x + 8 > 13
x>5
x + 13 > 8
x > –5
8 + 13 > x
21 > x
Combine the inequalities. So 5 < x < 21. The length
of the third side is greater than 5 inches and less than
21 inches.
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Ex. 6 - The lengths of two sides of a triangle
are 23 inches and 17 inches. Find the range of
possible lengths for the third side.
Let x represent the length of the third side. Then apply
the Triangle Inequality Theorem.
x + 23 > 17
x > –6
x + 17 > 23
x>6
23 + 17 > x
40 > x
Combine the inequalities. So 6 < x < 40. The
length of the third side is greater than 6 inches
and less than 40 inches.
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Lesson Wrap Up
1. Write the angles in order from smallest to
largest.
C, B, A
2. Write the sides in order from shortest to
longest.
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Lesson Wrap Up
3. The lengths of two sides of a triangle are 17 cm
and 11 cm. Find the range of possible lengths for
the third side.
6 cm < x < 28 cm
4. Tell whether a triangle can have sides with
lengths 2.7, 3.5, and 9.8. Explain.
No; 2.7 + 3.5 is not greater than 9.8.
5. Ray wants to place a chair so it is
10 ft from his television set. Can
the other two distances
shown be 8 ft and 6 ft? Explain.
Yes; the sum of any two lengths is
greater than the third length.
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