HONORS GEOMETRY
5.3. Inequalities in One Triangle
Do Now: Welcome Back!
Complete the Do Now given to you when you walked into
class today
Think < or > or =
Reminder:
• Inequalities involve either:
• A greater than sign ( > )
• OR
• A less than ( < ) sign
Definition of Inequality
• For any real numbers a and b, a>b if and only if there is a
positive number c such that a = b+c.
• Ex: If 5= 2 + 3 then 5>2 and 5>3
Recall:
• Interior Angles?
• Exterior Angles?
• Remote Interior angles?
• What does the exterior angle theorem claim?
So….
• Exterior angle theorem proves that
𝑚 < 1 + 𝑚 < 2 = 𝑚 < 4.
• Doesn’t this statement therefore imply that < 1 and
2 must be smaller than < 4?
<
Exterior Angle Inequality
• The measure of an exterior angle of a triangle is greater
than the measure of either of its corresponding remote
interior angles.
• Ex: 𝐴 > 𝐶 and 𝐴 > 𝐷
Example One:
• Use the exterior angle inequality theorem to list all the
angles that satisfy the stated condition.
• Measures less than 𝑚 < 7
• Measures greater than 𝑚 < 6
Example Two:
• Find the angles that….
• Measure less than 𝑚 < 1
• Measure greater than 𝑚 < 8
• Measures greater than 𝑚 < 4
You Try!
• Which of the following angles are
• Greater than 𝑚 < 2?
• Less than the 𝑚 < 5?
• Greater than the 𝑚 < 𝐴?
Triangle Relationship:
• In a triangle,
• the smallest angle is opposite the smallest side
• the largest angle is opposite the largest side
Example Three:
• List the angles of ∆𝑃𝑄𝑅 in order from smallest to largest.
Example Four:
• List the sides of ∆𝐹𝐺𝐻 in order from the shortest to the
longest.
You Try!
• List the angles of ∆𝐴𝐵𝐶 in order from smallest to largest.
• List the sides of ∆𝑀𝐴𝑇 from largest to smallest.
2 theorems:
• 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 smaller angle.
Example Five:
Example Six:
Given: 𝑚 < 𝐷𝐶𝐵 < 𝑚 < 𝐷𝐴𝐵
𝐴𝐷 ≅ 𝐷𝐶
Prove: 𝑚 < 𝐴𝐶𝐵 < 𝑚 < 𝐶𝐴𝐵
You Try!
• Given: 𝐶𝐵 ≅ 𝐶𝐴
Prove: 𝐶𝐷 > 𝐶𝐴
Practice Problems
• Try some on your own/in your table groups
• As always don’t hesitate to ask me and or your
tablemates questions if you are confused.
Exit Ticket: