5-5 INEQUALITIES IN
TRIANGLES
Objectives:
Students will be able to:
1) Use inequalities involving angles of triangles
2) Use inequalities involving sides of triangles
Think back….
What do we remember about the exterior angles of a
triangle?
What is the largest angle shown?
The Exterior Angle Theorem States: mÐ1= mÐ2 + mÐ3
Since mÐ1 is the sum of mÐ2 and mÐ3,
Exterior Ð1 must be greater than both
remote interior angles Ð2 and Ð3
Also remember that corollaries flow
directly from theorems…
Why is m<4 is greater than m<5?
Try this….
Lets use sketchpad!
• Lets see if there is a relationship between angles
and sides in triangles!
What did we see???
The largest angle was always opposite the largest side!
The smallest angle is always opposite the smallest side!
Help Mr. Tessalone!
• Mr. Tessalone wants to
build a bench in the
largest corner of his
triangular deck.
• Which angle should he
build it in and why?
• What if he wants to put
a plant in the smallest
angle? Where would
that go and why?
What if there is no diagram?
Think…
• Could we have a case where there is not one
specific angle that is the largest?
• Come up with an example
• Be sure to label your angles with possible
measurements correctly and use markings if
needed.
• What type or types of triangles did you come up
with?
What if we knew angle measures…
• Could we find the largest side?
• Could we find the smallest side?
• What do you think?
Converse to Theorem 5-10
List The Sides From Longest to Shortest!
• What is the measure
of <Y?
• m<y = 80
• What is the Largest
Angle? The Smallest?
• XZ, XY, YZ
List the sides in order from shortest to
longest.
Hungry?
• Lets have some pasta!
Your Challenge!
• Break your spaghetti into three pieces
that will not form a triangle.
Compare the lengths…
• Compare the two combined lengths of the
smaller pieces to the larger one.
• What do you see?
Triangle Inequality Theorem
Are Triangles Possible With these
Lengths?
Got that?
What if I only know 2 Sides?
• What are all the possible lengths of the third side
of a triangle if the other sides are 9 in and 17 in?
Lets say the third side is
the Smallest Side.
x + 9 > 17
x>8
Lets say the third side is
the Largest Side.
9 + 17 > x
26 > x
A Triangles sides are 3 in and 12 in.
What could the third side be?
• Say x is a smaller side.
• Say x is the Largest
Side