Inequalities and Triangles

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Lesson 5-2

• Inequalities and Triangles

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Transparency 5-2
5-Minute Check on Lesson 5-1
In the figure, A is the circumcenter of ?LMN. 1.
Find y if LO 8y 9 and ON 12y 11. 2.
Find x if m?APM 7x 13. 3. Find r if AN 4r
8 and AM 3(2r 11). In ?RST, RU is an
altitude and SV is a median. 4. Find y if m?RUS
7y 27. 5. Find RV if RV 6a 3 and RT
10a 14.
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Transparency 5-2
5-Minute Check on Lesson 5-1
In the figure, A is the circumcenter of ?LMN. 1.
Find y if LO 8y 9 and ON 12y 11. 5 2.
Find x if m?APM 7x 13. 11 3. Find r if AN
4r 8 and AM 3(2r 11). 12.5 In ?RST, RU
is an altitude and SV is a median. 4. Find y if
m?RUS 7y 27. 9 5. Find RV if RV 6a 3
and RT 10a 14. 27
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Objectives

• Recognize and apply properties of inequalities to
  the measures of angles of a triangle
• Recognize and apply properties of inequalities to
  the relationships between angles and sides of a
  triangle

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Vocabulary

• No new vocabulary words or symbols

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Theorems

• Theorem 5.8, Exterior Angle Inequality Theorem
  If an angle is an exterior angle of a triangle,
  then its measure is greater that the measure of
  either of it corresponding remote interior
  angles.
• Theorem 5.9 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.
• Theorem 5.10 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 lesser angle.

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Key Concept

• Step 1 Arrange sides or angles from smallest to
  largest or largest to smallest based on given
  information
• Step 2 Write out identifiers (letters) for the
  sides or angles in the same order as step 1
• Step 3 Write out missing letter(s) to complete
  the relationship
• Step 4 Answer the question asked

19 gt 14 gt 7 WT gt AW gt AT ?A
gt ?T gt ?W
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Determine which angle has the greatest measure.
Explore Compare the measure of ?1 to the measures
of ?2, ?3, ?4, and ?5.
Plan Use properties and theorems of real numbers
to compare the angle measures.
Solve Compare m?3 to m?1.
By the Exterior Angle Theorem, m?1 m?3 m?4.
Since angle measures are positive numbers and
from the definition of inequality, m?1 gt m?3.
Compare m?4 to m?1.
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Compare m?5 to m?1.
Compare m?2 to m?5.
Examine The results on the previous slides show
that m?1 gt m?2, m?1 gt m?3, m?1 gt m?4, and m?1 gt
m?5. Therefore, ?1 has the greatest measure.
Answer ?1 has the greatest measure.
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EXAMPLE 2
Order the angles from greatest to least measure.
Answer ?5 has the greatest measure ?1 and ?2
have the same measure ?4, and ?3 has the least
measure.
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EXAMPLE 3
Use the Exterior Angle Inequality Theorem to list
all angles whose measures are less than m?14.
By the Exterior Angle Inequality Theorem, m?14 gt
m?4, m?14 gt m?11, m?14 gt m?2, and m?14 gt m?4
m?3. Since ?11 and ?9 are vertical angles, they
have equal measure, so m?14 gt m?9. m?9 gt m?6 and
m?9 gt m?7, so m?14 gt m?6 and m?14 gt m?7. Answer
Thus, the measures of ?4, ?11, ?9, ? 3, ? 2, ?6,
and ?7 are all less than m?14 .
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EXAMPLE 4
Use the Exterior Angle Inequality Theorem to list
all angles whose measures are greater than m?5.
By the Exterior Angle Inequality Theorem, m?10 gt
m?5, and m?16 gt m?10, so m?16 gt m?5, m?17 gt m?5
m?6, m?15 gt m?12, and m?12 gt m?5, so m?15 gt m?5.
Answer Thus, the measures of ?10, ?16, ?12, ?15
and ?17 are all greater than
m?5.
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EXAMPLE 5
Answer ?5, ?2, ?8, ?7
Answer ?4, ?9, ?5
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EXAMPLE 6
Determine the relationship between the measures
of ?RSU and ?SUR.
Answer The side opposite ?RSU is longer than
the side opposite ?SUR, so m?RSU gt m?SUR.
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EXAMPLE 7
Determine the relationship between the measures
of ?TSV and ?STV.
Answer The side opposite ?TSV is shorter than
the side opposite ?STV, so m?TSV lt m?STV.
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EXAMPLE 8
Determine the relationship between the measures
of ?RSV and ?RUV.
m?RSU gt m?SUR
m?USV gt m?SUV
m?RSU m?USV gt m?SUR m?SUV
m?RSV gt m?RUV
Answer m?RSV gt m?RUV
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EXAMPLE 9
Answer ?ABD gt ?DAB
Answer ?AED gt ?EAD
Answer ?EAB lt ?EDB
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Summary Homework

• Summary
• The largest angle in a triangle is opposite the
  longest side, and the smallest angle is opposite
  the shortest side
• The longest side in a triangle is opposite the
  largest angle, and the shortest side is opposite
  the smallest angle
• Homework
• pg 251 (17-34, 46-50)