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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 5-1 Bisectors, Medians, and Altitudes
Lesson 5-2 Inequalities and Triangles
Lesson 5-3 Indirect Proof
Lesson 5-4 The Triangle Inequality
Lesson 5-5 Inequalities Involving Two Triangles
Example 1 Use Angle Bisectors
Example 2 Segment Measures
Example 3 Use a System of Equations to Find a Point
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
3.
4.
5.
2. Angle Sum Theorem
3. Substitution
4. Subtraction Property
5. Definition of angle
bisector
6. Angle Sum Theorem
7. Substitution
8. Subtraction Property
6.
7.
8.
Given:
.
Prove:
Proof:
Statements.
Reasons
1.
1. Given
2.
3.
4.
5.
2. Angle Sum Theorem
3. Substitution
4. Subtraction Property
5. Definition of angle
bisector
6. Angle Sum Theorem
7. Substitution
8. Subtraction Property
6.
7.
8.
ALGEBRA Points U, V, and W are the midpoints of
respectively. Find a, b, and c.
Find a.
Segment Addition Postulate
Centroid Theorem
Substitution
Multiply each side by 3 and simplify.
Subtract 14.8 from each side.
Divide each side by 4.
Find b.
Segment Addition Postulate
Centroid Theorem
Substitution
Multiply each side by 3 and simplify.
Subtract 6b from each side.
Subtract 6 from each side.
Divide each side by 3.
Find c.
Segment Addition Postulate
Centroid Theorem
Substitution
Multiply each side by 3 and simplify.
Subtract 30.4 from each side.
Divide each side by 10.
Answer:
ALGEBRA Points T, H, and G are the midpoints of
respectively. Find w, x, and y.
Answer:
COORDINATE GEOMETRY The vertices of HIJ are
H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of
the orthocenter of HIJ.
Find an equation of the altitude from
The slope of
so the slope of an altitude is
Point-slope form
Distributive Property
Add 1 to each side.
Next, find an equation of the altitude from I to
slope of
The
so the slope of an altitude is –6.
Point-slope form
Distributive Property
Subtract 3 from each side.
Then, solve a system of equations to find the point
of intersection of the altitudes.
Equation of altitude from J
Substitution,
Multiply each side by 5.
Add 105 to each side.
Add 4x to each side.
Divide each side
by –26.
Replace x with
y-coordinate.
in one of the equations to find the
Rename as improper fractions.
Multiply and simplify.
Answer: The coordinates of the orthocenter of
COORDINATE GEOMETRY The vertices of ABC are
A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of
the orthocenter of ABC.
Answer: (0, 1)
Example 1 Compare Angle Measures
Example 2 Exterior Angles
Example 3 Side-Angle Relationships
Example 4 Angle-Side Relationships
Inequalities
• For any real numbers a and b, a>b if and
only if there is a positive number c such
that a = b + c.
– If 6 = 4 + 2, 6 > 4 and 6 >2.
• Theorem 5.8: Exterior Angle Inequality –
If an angle is an exterior angle of a
triangle, then its measure is greater than
the measure of either of its corresponding
remote interior angles.
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 > 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.
Determine which angle has the greatest measure.
Answer: 5 has the greatest 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
More Inequality Theorems
• 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.
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
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?
Theorem 5.10 states that 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. Since X is opposite the longest side it has the
greatest measure.
Answer: So, Ebony should tie the ends marked Y and Z.
KITE ASSEMBLY Tanya is following directions for
making a kite. She has two congruent triangular
pieces of fabric that need to be sewn together along
their longest side. The directions say to begin sewing
the two pieces of fabric together
at their smallest angles.
At which two angles
should she begin
sewing?
Answer: A and D
Example 1 Stating Conclusions
Example 2 Algebraic Proof
Example 3 Use Indirect Proof
Example 4 Geometry Proof
State the assumption you would make to start an
indirect proof for the statement
is not a
perpendicular bisector.
Answer:
is a perpendicular bisector.
State the assumption you would make to start an
indirect proof for the statement
Answer:
State the assumption you would make to start an
indirect proof for the statement m1 is less than or
equal to m2.
If m1
m2 is false, then m1 > m2.
Answer: m1 > m2
State the assumption you would make to start an
indirect proof for the statement If B is the midpoint of
and
then
is congruent to
The conclusion of the conditional statement is
is
congruent to
The negation of the conclusion is
is
not congruent to
Answer:
is not congruent to
State the assumption you would make to start an
indirect proof of each statement.
a.
is not an altitude.
Answer:
b.
Answer:
is an altitude.
State the assumption you would make to start an
indirect proof of each statement.
c. mABC is greater than or equal to mXYZ.
Answer: mABC < mXYZ
d. If
is an angle bisector of MLP, then MLH
is congruent to PLH.
Answer: MLH is not congruent to PLH.
Write an indirect proof.
Given:
Prove:
Indirect Proof:
Step 1 Assume that
.
Step 2 Substitute –2 for y in the equation
Substitution
Multiply.
Add.
This is a contradiction because the
denominator cannot be 0.
Step 3 The assumption leads to a contradiction.
Therefore, the assumption that
must be
false, which means that
must be true.
Write an indirect proof.
Given:
Prove:
Indirect Proof:
Step 1 Assume that
Step 2 Substitute –3 for a in the inequality
Substitution
Multiply.
Add.
This is a contradiction because the
denominator cannot be 0.
Step 3 The assumption leads to a contradiction.
Therefore, the assumption that
must be
false, which means that
must be true.
CLASSES Marta signed up for three classes at a
community college for a little under $156. There was
an administration fee of $15, but the class costs
varied. How can you show that at least one class cost
less than $47?
Answer:
Given: Marta spent less than $156.
Prove: At least one of the classes x cost less than $47.
That is,
Indirect Proof:
Step 1 Assume that none of the classes cost less than $47.
Step 2
then the minimum total amount Marta
spent is
However, this is a
contradiction since Marta spent less than $156.
Step 3 The assumption leads to a contradiction of a
known fact. Therefore, the assumption that
must be false. Thus, at least one of the classes
cost less than $47.
SHOPPING David bought four new sweaters for a little
under $135. The tax was $7, but the sweater costs
varied. How can you show that at least one of the
sweaters cost less than $32?
Answer:
Given: David spent less than $135.
Prove: At least one of the sweaters x cost less than $32.
That is,
Indirect Proof:
Step 1 Assume that none of the sweaters cost less
than $32.
Step 2
then the minimum total amount David
spent is
However, this is a
contradiction since David spent less than $135.
Step 3 The assumption leads to a contradiction of a
known fact. Therefore, the assumption that
must be false. Thus, at least one of the sweaters
cost less than $32.
Write an indirect proof.
Given: JKL with side lengths 5, 7, and 8 as shown.
Prove: mK < mL
Indirect Proof:
Step 1 Assume that
Step 2 By angle-side relationships,
By substitution,
. This inequality is a false statement.
Step 3 Since the assumption leads to a contradiction, the
assumption must be false. Therefore, mK < mL.
Write an indirect proof.
Given: ABC with side lengths 8, 10, and 12 as shown.
Prove: mC > mA
Indirect Proof:
Step 1 Assume that
Step 2 By angle-side relationships,
By substitution,
This inequality is a false statement.
Step 3 Since the assumption leads to a contradiction, the
assumption must be false. Therefore, mC > mA.
Example 1 Identify Sides of a Triangle
Example 2 Determine Possible Side Length
Example 3 Prove Theorem 5.12
Theorem 5.11: Triangle Inequality Theorem
• The sum of the lengths of any two sides of
a triangle is greater than the length of the
third side.
• This theorem will be used to determine
whether three segments can form a
triangle.
Determine whether the measures
and
can be lengths of the sides of a triangle.
Answer: Because the sum of two measures is not greater
than the length of the third side, the sides cannot
form a triangle.
Determine whether the measures 6.8, 7.2, and 5.1 can
be lengths of the sides of a triangle.
Check each inequality.
Answer: All of the inequalities are true, so 6.8, 7.2, and
5.1 can be the lengths of the sides of a triangle.
Determine whether the given measures can be
lengths of the sides of a triangle.
a. 6, 9, 16
Answer: no
b. 14, 16, 27
Answer: yes
Multiple-Choice Test Item
In
and
be PR?
A 7
B9
C 11
Which measure cannot
D 13
Read the Test Item
You need to determine which value is not valid.
Solve the Test Item
Solve each inequality to determine the range of values
for PR.
Graph the inequalities on the same number line.
The range of values that fit all three inequalities is
Examine the answer choices. The only value that does not
satisfy the compound inequality is 13 since 13 is greater
than 12.4. Thus, the answer is choice D.
Answer: D
Multiple-Choice Test Item
Which measure cannot
be XZ?
A 4
Answer: D
B9
C 12
D 16
Perpendicular Segments
• Theorem 5.12: The perpendicular
segment from a point to a line is the
shortest segment from the point to the line.
• Corollary 5.1: The perpendicular segment
from a point to a plane is the shortest
segment from the point to a plane.
Given:
Prove:
line
through point J
Point K lies on t.
KJ < KH
Proof:
Statements
1.
2.
are right angles.
3.
4.
5.
6.
7.
Reasons
1. Given
2. Perpendicular lines form right
angles.
3. All right angles are congruent.
4. Definition of congruent angles
5. Exterior Angle Inequality Theorem
6. Substitution
7. If an angle of a triangle is greater
than a second angle, then the side
opposite the greater angle is
longer than the side opposite the
lesser angle.
Given:
is an altitude in ABC.
Prove: AB > AD
Proof:
Statements
is an altitude
1.
Reasons
1. Given
in
2.
3.
are right angles.
4.
2. Definition of altitude
3. Perpendicular lines form
right angles.
4. All right angles are congruent.
Proof:
Statements
5.
6.
7.
8.
Reasons
5. Definition of congruent angles
6. Exterior Angle Inequality Theorem
7. Substitution
8. If an angle of a triangle is greater
than a second angle, then the side
opposite the greater angle is
longer than the side opposite the
lesser angle.
Example 1 Use SAS Inequality in a Proof
Example 2 Prove Triangle Relationships
Example 3 Relationships Between Two Triangles
Example 4 Use Triangle Inequalities
Write a two-column proof.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
3.
2. Alternate interior angles
are congruent.
3. Substitution
4.
5.
6.
7.
4. Subtraction Property
5. Given
6. Reflexive Property
7. SAS Inequality
Write a two-column proof.
Given: m1 < m3
E is the midpoint of
Prove: AD < AB
Proof:
Statements
Reasons
1. E is the midpoint
of
1. Given
2.
3.
4.
5.
6.
7.
2. Definition of midpoint
3. Reflexive Property
4. Given
5. Definition of vertical angles
6. Substitution
7. SAS Inequality
Given:
Prove:
Proof:
Statements
Reasons
1.
2.
3.
1. Given
2. Reflexive Property
3. Given
4.
4. Given
5.
6.
5. Substitution
6. SSS Inequality
Given: X is the midpoint of
MCX is isosceles.
CB > CM
Prove:
Proof:
Statements
1. X is the midpoint of
2.
3. MCX is isosceles.
4.
5.
6.
7.
Reasons
1. Given
2. Definition of midpoint
3. Given
4. Definition of isosceles
triangle
5. Given
6. Substitution
7. SSS Inequality
SAS Inequality/Hinge Theorem
• If two sides of a triangle are congruent to
two sides of another triangle and the
included angle in one triangle has a
greater measure than the included angle
in the other, then the third side of the first
triangle is longer than the third side of the
second triangle.
SSS Inequality
• If two sides of a triangle are congruent to
two sides of another triangle and the third
side in one triangle is longer than the third
side in the other, then the angle between
the pair of congruent sides in the first
triangle is greater than the corresponding
angle in the second triangle.
Write an inequality comparing mLDM and mMDN
using the information in the figure.
The SSS Inequality allows us to conclude that
Answer:
Write an inequality finding the range of values
containing a using the information in the figure.
By the SSS Inequality,
SSS Inequality
Substitution
Subtract 15 from each side.
Divide each side by 9.
Also, recall that the measure of any angle is always
greater than 0.
Subtract 15 from each side.
Divide each side by 9.
The two inequalities can be written as the compound
inequality
Answer:
Write an inequality using the information in the figure.
a.
Answer:
b. Find the range of values
containing n.
Answer: 6 < n < 25
HEALTH Doctors use a straight-leg-raising test to
determine the amount of pain felt in a person’s back.
The patient lies flat on the examining table, and the
doctor raises each leg until the patient experiences
pain in the back area. Nitan can tolerate the doctor
raising his right leg 35° and his left leg 65° from the
table. Which foot can Nitan raise higher above the
table?
Assume both of Nitan’s legs have the same
measurement, the SAS Inequality tells us that the height
of the left foot opposite the 65° angle is higher than the
height of his right foot opposite the 35° angle. This means
that his left foot is raised higher.
Answer: his left foot
HEALTH Doctors use a straight-leg-raising test to
determine the amount of pain felt in a person’s back.
The patient lies flat on the examining table, and the
doctor raises each leg until the patient experiences
pain in the back area. Megan can lift her right foot 18
inches from the table and her left foot 13 inches from
the table. Which leg makes the greater angle with the
table?
Answer: her right leg
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