Advanced Geometry
Topic 5 – Relationships within Triangles
NOTES – Inequalities in One Triangle (5-7)
Name: _____________________________________________________
Date: _________________________________ Period: ____________
Property – Comparison of Property Inequality
If 𝑎 = 𝑏 + 𝑐 and 𝑐 > 0, then 𝑎 > 𝑏.
Corollary – Corollary to the Triangle Exterior Angle Theorem
Corollary
If…
Then…
The measure of an exterior angle of a
∠1 is an exterior angle
𝑚∠1 > 𝑚∠2 and
𝑚∠1 > 𝑚∠3
Theorem
If…
Then…
If two sides of a triangle are not
𝑋𝑍 > 𝑋𝑌
𝑚∠𝑌 > 𝑚∠𝑍
Theorem
If…
Then…
If two angles of a triangle are not
𝑚∠𝐴 > 𝑚∠𝐵
𝐵𝐶 > 𝐴𝐶
triangle is _________________ _________________
the measure of each of its _________________
_________________ angles.
Theorem 5-10
_________________, the _________________
_________________ lies opposite the
_________________ _________________.
Theorem 5-11
_________________, the _________________
_________________ lies opposite the
_________________ _________________.
Theorem 5-12 – Triangle Inequality Theorem
The sum of the lengths of any _________________ _________________ of a triangle is
_________________ _________________ the length of the _________________ _________________.
𝑿𝒀 + 𝒀𝒁 > 𝑿𝒁
𝒀𝒁 + 𝑿𝒁 > 𝑿𝒀
𝑿𝒁 + 𝑿𝒀 > 𝒀𝒁
Applying the Corollary
Why is 𝑚∠5 > 𝑚∠𝐶?
Using Theorem 5-10
Suppose the landscape architect wants to place a drinking fountain at the
corner with the second largest angle. Which two streets form the corner
with the second largest angle?
Using Theorem 5-11
In the figure, 𝑚∠𝑆 = 24 and 𝑚∠𝑂 = 130. Which side of ∆SOX is the shortest
side? Explain your reasoning.
Using the Triangle Inequality Theorem
Finding Possible Side Lengths