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5.5 – Use Inequalities in a Triangle
P
95º
5cm
2.6cm
55º
M
30º
5.9cm
N
Measure each side of the triangle in centimeters
and each angle in degrees. Write these
measurements on your triangle.
Theorem: If one ______
side of the triangle is longer
than the other side, then the angle
opposite
larger
____________
the __________
longer side is _________
opposite
shorter
than the angle ____________
the ____________
side
angle
Theorem: If one _________
of a triangle is
larger
__________
than another angle, then the side
larger
opposite the _____________
angle is
longer
___________
than the side opposite the smaller
angle.
1. List the sides and angles in order from smallest to
largest.
AB, BC , AC
Sides: _________________
C, A, B
Angles: ________________
1. List the sides and angles in order from smallest to
largest.
ST , RT , SR
Sides: _________________
R, S, T
Angles: ________________
1. List the sides and angles in order from smallest to
largest.
KJ , KH , JH
Sides: _________________
H, J, K
Angles: ________________
55°
1. List the sides and angles in order from smallest to
largest.
DF , EF , DE
Sides: _________________
33°
E, D, F
Angles: ________________
Construct a triangle with sides of 2, 2, and 6 cm.
2cm
6cm
Doesn’t work!
Why?
2cm
2cm
6cm
Too short!
What if I tried 3, 3, and 6cm?
3cm
3cm
6cm
Still doesn’t work!
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a
triangle is greater than the length of the
third side.
a+b>c
a+c>b
b+c>a
c
a
b
2. Is it possible to construct a triangle with the given side lengths?
If not, explain why not.
a.
3, 4, 5
3+4>5
3+5>4
4+5>3
Yes
2. Is it possible to construct a triangle with the given side lengths?
If not, explain why not.
b.
1, 4, 6
1+4>6
5>6
No
2. Is it possible to construct a triangle with the given side lengths?
If not, explain why not.
c.
17, 17, 33
17 + 17 > 33
17 + 33 > 17
17 + 33 > 17
Yes
2. Is it possible to construct a triangle with the given side lengths?
If not, explain why not.
d.
2.2, 2.6, 4.8
2.2 + 2.6 > 4.8
4.8 > 4.8
No
2. Is it possible to construct a triangle with the given side lengths?
If not, explain why not.
e.
6, 6 3, 12
6 + 10.4 > 12
6 + 12 > 10.4
10.4 + 12 > 6
Yes
6 3  10.4
2. Is it possible to construct a triangle with the given side lengths?
If not, explain why not.
f.
7, 52, 45
7 + 52 > 45
7 + 45 > 52
52 > 52
No
3. Describe the possible lengths of the third side of
the triangle given the lengths of the other two sides.
a.
x + 9 > 21
x > 12
21 + 9 > x
30 > x
or
x < 30
12 < x < 30
x + 21 > 9
x > -12
Trick!
Subtract the two sides to find the lowest and
add to find the highest.
3. Describe the possible lengths of the third side of the triangle
given the lengths of the other two sides.
b.
12in, 6in
12 – 6 < x < 12 + 6
6in < x < 18in
3. Describe the possible lengths of the third side of the triangle
given the lengths of the other two sides.
c.
3ft, 8ft
8–3< x< 8+3
5ft < x < 11ft
3. Describe the possible lengths of the third side of the triangle
given the lengths of the other two sides.
d.
12cm, 17cm
17 – 12 < x < 17 + 12
5cm < x < 29cm
3. Describe the possible lengths of the third side of the triangle
given the lengths of the other two sides.
e.
7yd, 13yd
13 – 7 < x < 13 + 7
6yd < x < 20yd
3. Describe the possible lengths of the third side of the triangle
given the lengths of the other two sides.
f.
9m, 9m
9–9< x< 9+9
0m < x < 18m
4. Solve the inequality AB + AC > BC for x.
5 + 7 > 2x – 2
12 > 2x – 2
14 > 2x
7>x
x<7
4. Solve the inequality AB + AC > BC for x.
3x + 1 + 4x – 2 > 5x + 7
7x – 1 > 5x + 7
2x – 1 > 7
2x > 8
x>4