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Lesson 5-4 The Triangle Inequality
• 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
• 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 the plane.
Determine whether the measures
can be lengths of the sides of a triangle.
and
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
b. 14, 16, 27
Answer: no
Answer: yes
Multiple-Choice Test Item
In
and
A 7
B9
Which measure cannot be PR?
C 11
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
Given:
Prove:
line
through point J
Point K lies on t.
KJ < KH
Proof:
Statements
Reasons
1.
2.
1. Given
2. Perpendicular lines form right
angles.
are right angles.
3.
4.
5.
6.
7.
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
1.
is an altitude
2. in
3.
4.
are right angles.
Reasons
1. Given
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.
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