Download Geometry 5-3 ppt inequalities in triangles

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Splash Screen
Over Lesson 5–2
Which of the following points is the center of
gravity of a triangle?
A. centroid
B. circumcenter
C. incenter
D. orthocenter
5-Minute Check 5
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You found the relationship between the angle
measures of a triangle.
• Recognize and apply properties of
inequalities to the measures of the angles of
a triangle.
• Recognize and apply properties of
inequalities to the relationships between the
angles and sides of a triangle.
Then/Now
• 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: 5=2+3, then 5>2 and 5>3
Concept
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• In every triangle, the angle
opposite the largest side is the
largest angle
• The angle opposite the smallest
side is the smallest angle
Concept
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Order Triangle Angle Measures
List the angles of ΔABC in order from smallest to
largest.
The sides from the shortest to longest are AB, BC, and
AC. The angles opposite these sides are C, A, and
B, respectively. So, according to the Angle-Side
Relationship, the angles from smallest to largest are
C, A, B.
Answer: C, A, B
Example 2
Order Triangle Side Lengths
List the sides of ΔABC in order from
shortest to longest.
Answer: AC, AB, BC
Example 3
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List the angles of ΔTVX in
order from smallest to largest.
A. X, T, V
B. X, V, T
C. V, T, X
D. T, V, X
Example 2
Angle-Side Relationships
HAIR ACCESSORIES Ebony is following directions
for folding a handkerchief to make a bandana for
her hair. After she folds the handkerchief in half, the
directions tell her to tie the two smaller angles of
the triangle under her hair. If she folds the
handkerchief with the dimensions shown, which
two ends should she tie?
Example 4
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List the sides of ΔRST in order from
shortest to longest.
A. RS, RT, ST
B. RT, RS, ST
C. ST, RS, RT
D. RS, ST, RT
Example 3
Concept
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Concept
Use the Exterior Angle Inequality Theorem
Example 1
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Use the Exterior Angle Inequality Theorem
By the Exterior Angle Inequality Theorem,
m14 > m4 and m14 > m11. In addition,
m14 > m2 and m14 > m4 + m3, so
m14 > m4 and m14 > m3.
Since 11 and 9 are vertical angles, they
have equal measure, so m14 > m9.
m9 > m6 and m9 > m7, so m14 > m6
and m14 > m7.
Example 1
Use the Exterior Angle Inequality Theorem
Example 1
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Use the Exterior Angle Inequality Theorem
By the Exterior Angle Inequality Theorem,
m10 > m5 and m16 > m10, so m16 >
m5. Since 10 and 12 are vertical angles,
m12 > m5. m15 > m12, so m15 > m5.
In addition, m17 > m5 + m6, so m17 >
m5.
Example 1
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B.
C.
D.
Example 1
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A.
B.
C.
D.
Example 1
Homework: Pg. 348 #’s 8-18 EVENS
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