Geometry
§5.5-5.6
Name_________________________
Date______________ Pd_________
Using Inequalities in One and Two Triangles
Theorem –
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.
Conclusion:
B
8
5
A
Theorem –
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.
C
Conclusion:
B
50
30
A
C
Ex:
1. List the sides of RST in order from shortest to longest.
S
121
29
30
R
2. List the angles of ABC in order from least to greatest.
T
B
8
6
A
10
C
Triangle Inequality Theorem –
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
How to find the possible lengths of the missing side:
1. Find the sum of the two sides you are given
2. Find the difference of the two sides you are given
3. The possible lengths will be between those two values.
4. Write the sentence or inequality appropriately.
Muscarella, revised 2011
B
A
C
Ex:
3. A triangle has one side length of 8 and another of length 12. Describe the possible lengths of the
third side. Write your solution as a sentence and as an inequality if x represents the length of the
missing side.
Hinge Theorem –
If two sides of one triangle are congruent to two sides of
another triangle, and the included angle of the first is larger
than the included angle of the second, then the third side
of the first is longer than the third side of the second.
W
V
S
88
R
35
T
X
D
Converse of the Hinge Theorem –
If two side of one triangle are congruent to two sides of
another triangle, and the third side of the first is longer
than the third side of the second, then the included angle
of the first is larger than the included angle of the second.
A
9
12
C
B
F
E
Ex:
4. Use the diagram at the right. Explain how you know for each.
P
a. If PR = PS and mQPR  mQPS, which is longer, SQ or RQ ?
S
R
b. If PR = PS and RQ < SQ, which is larger, RPQ or SPQ ?
Q
5. Write and solve an inequality to describe the restriction on the value of x.
Do Now: Page 331 #6-26
Page 338 #3-9, 16, 17
Include all drawings!
Muscarella, revised 2011
Geometry
§5.5-5.6
Name_________________________
Date______________ Pd_________
Using Inequalities in One and Two Triangles
Theorem –
If one side of a triangle is
longer than another side,
then the angle opposite
the longer side is larger than
A
the angle opposite the shorter side.
B
8
Conclusion:
5
mC  mA
C
B
Theorem –
If one angle of a triangle is
larger than another angle,
50
then the side opposite the
A
larger angle is longer than the
side opposite the smaller angle.
Muscarella, revised 2011
Conclusion:
30
C
BC  AB
Ex:
1. List the sides of RST in order from shortest to longest.
S
The shortest side is across from the
smallest angle, and the longest side is
across from the largest angle…
121
29
30
R
ST  SR  RT
T
Common Mistake:
READING!!
Be sure you read the question to see the order
being asked for…least to greatest is different than
greatest to least.
ORDER MATTERS!!!
2. List the angles of ABC in order from least to greatest.
The smallest angle is across from the
shortest side, and the largest angle is
A
across from the longest side…
mA  mC  mB
Muscarella, revised 2011
B
8
6
10
C
Triangle Inequality Theorem –
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
B
A
C
Side 1 + Side 2 > Side 3
Ex:
3. A triangle has one side length of 8 and another of length 12.
Describe the possible lengths of the third side. Write your solution as a
sentence and as an inequality if x represents the length of the missing
side.
12 + 8 = 20
12 – 8 = 4
Which inequality would represent this if x were the length of the
missing side? Explain how you know.
a) 4 ≤ x ≤ 20
b) 4 < x < 20
Write your answer like this:
The lengths of the third side are between 4 and 20.
4 < x < 20
You cannot choose a) since the sum of any two sides of a triangle
must be greater than the length of the third side!!
4 + 8 = 12;
4 + 8 > 12 is false.
Muscarella, revised 2011
Hinge Theorem –
If two sides of one triangle are congruent to two sides of
another triangle, and the included angle of the first is larger
than the included angle of the second, then the third side
of the first is longer than the third side of the second.
W
V
S
88
R
35
T
X
D
Converse of the Hinge Theorem –
If two side of one triangle are congruent to two sides of
another triangle, and the third side of the first is longer
than the third side of the second, then the included angle
of the first is larger than the included angle of the second.
A
9
12
C
B
F
E
The Hinge Theorem and the Converse of the Hinge Theorem…
Think about it like this…
When you open a door, you can have it open only a little bit, or you
can open it up a lot. But, there are two things that won’t change…do
you know what they are? Look at the pictures below…
Muscarella, revised 2011
If you open the door a little bit, that angle will have to be less than if
you open it up a lot.
So, the side opposite the smaller opening will have a shorter distance
compared to the side opposite the bigger opening.
Smaller opening
Shorter distance
Larger opening
Longer distance
Ex:
4. Use the diagram at the right. Explain how you know for each.
a. If PR = PS and mQPR  mQPS, which is longer, SQ or RQ ?
P
Since QPR is the larger angle,
that means RQ is the longer side.
S
R
Q
Muscarella, revised 2011
b. If PR = PS and RQ < SQ, which is larger, RPQ or SPQ ?
P
Since SQ is the longer side,
that means SPQ is the larger angle.
S
R
Q
5. Write and solve an inequality to describe the
restriction on the value of x.
We have 2 different pairs of sides
that are the same, so we can apply
the Converse of the Hinge Theorem.
Since 115º is more than 45º, we
write the following inequality to
solve:
3x + 1 > x + 3 (1) Inequality
2x + 1 > 3
(2)
2x > 2
(3)
x>1
(4) Restriction on x
Muscarella, revised 2011
Include a drawing to help illustrate your point
as you summarize each of the following
in your own words.
Be ready to discuss in 3 minutes or less.
 If you’re given 2 angles in a triangle, describe how
to list all three sides from smallest to largest.
 Triangle Inequality Theorem
 Hinge Theorem
 Converse of the Hinge Theorem
Do Now: Page 331 #6-26
Page 338 #3-9, 16, 17
Include all drawings!
Muscarella, revised 2011