Geometry Unit 3 - Notes Sections 5-2 and 5.4
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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.
mR = 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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