Transcript Lesson 1 Contents

Lesson 5-2
Inequalities and Triangles
5-Minute Check on Lesson 5-1
Transparency 5-2
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.
6.
Standardized Test Practice:
circumcenter of WXY?
A
WX  XY
B
Which congruence statement is true if P is the
WP  XP
C
WP  WX
D
WY  XY
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.
5
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.
6.
Standardized Test Practice:
circumcenter of WXY?
A
WX  XY
B
27
Which congruence statement is true if P is the
WP  XP
C
WP  WX
D
WY  XY
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
Vocabulary
• No new vocabulary words or symbols
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.
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
A
19
>
14
>
7
7
14
T
19
W
WT >
AW > AT
A > T
> W
Determine which angle has the greatest
measure.
Explore Compare the measure of 1 to the measures
of 2, 3, 4, and 5.
Plan
Solve
Use properties and theorems of real numbers
to compare the angle measures.
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 > m3.
Compare m4 to m1.
By the Exterior Angle Theorem, m1 m3 m4.
By the definition of inequality, m1 > m4.
Compare m5 to m1.
Since all right angles are congruent, 4 5.
By the definition of congruent angles, m4 m5.
By substitution, m1 > m5.
Compare m2 to m5.
By the Exterior Angle Theorem, m5 m2 m3.
By the definition of inequality, m5 > m2.
Since we know that m1 > m5, by the
Transitive Property, m1 > m2.
Examine The results on the previous slides show that
m1 > m2, m1 > m3, m1 > m4, and
m1 > m5. Therefore, 1 has the greatest
measure.
Answer: 1 has the greatest measure.
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.
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 > m4,
m14 > m11, m14 > m2, and m14 > m4 + 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.
Answer: Thus, the measures of 4, 11, 9,  3,  2, 6,
and 7 are all less than m14 .
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 > m5,
and m16 > m10, so m16 > m5, m17 > m5 + m6,
m15 > m12, and m12 > m5, so m15 > m5.
Answer: Thus, the measures of 10, 16, 12, 15 and
17 are all greater than m5.
Use the Exterior Angle Inequality Theorem to list all of
the angles that satisfy the stated condition.
a. all angles whose measures are less than m4
Answer: 5, 2, 8, 7
b. all angles whose measures are greater than m8
Answer: 4, 9, 5
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 > mSUR.
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 < mSTV.
Determine the relationship between the measures of
RSV and RUV.
mRSU > mSUR
mUSV > mSUV
mRSU + mUSV > mSUR + mSUV
mRSV > mRUV
Answer: mRSV > mRUV
Determine the relationship between the measures of
the given angles.
a. ABD, DAB
Answer: ABD > DAB
b. AED, EAD
Answer: AED > EAD
c. EAB, EDB
Answer: EAB < EDB
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: 4-15