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Chapter
5
Chapter Review
5
Chapter Review
Vocabulary Review
Resources
distance from a point to a line (p. 266)
equivalent statements (p. 281)
incenter of a triangle (p. 273)
indirect proof (p. 281)
indirect reasoning (p. 281)
inscribed in (p. 273)
altitude of a triangle (p. 275)
centroid (p. 274)
circumcenter of a triangle (p. 273)
circumscribed about (p. 273)
concurrent (p. 273)
contrapositive (p. 280)
coordinate proof (p. 260)
inverse (p. 280)
median of a triangle (p. 274)
midsegment (p. 259)
negation (p. 280)
orthocenter of a triangle (p. 275)
point of concurrency (p. 273)
Student Edition
Extra Skills, Word Problems, Proof
Practice, Ch. 5, p. 724
English/Spanish Glossary, p. 779
Postulates and Theorems, p. 770
Table of Symbols, p. 763
Choose the correct vocabulary term to complete each sentence.
1. A (centroid, median of a triangle) is a segment whose endpoints are a vertex
and the midpoint of the side opposite the vertex. median of a k
2. distance from the point
to the line
Vocabulary and Study Skills
worksheet 5F
Spanish Vocabulary and Study
Skills worksheet 5F
Interactive Textbook Audio
Glossary
Online Vocabulary Quiz
2. The length of the perpendicular segment from a point to a line is the
(midsegment, distance from the point to the line).
3. If T is a point on the perpendicular bisector of FG, then TF = TG because of
the (Perpendicular Bisector Theorem, Angle Bisector Theorem). # Bis. Thm.
4. The (altitude, median) of a triangle is a perpendicular segment from a vertex to
the line containing the side opposite the vertex. altitude
5. The notation ~q S ~p is the (inverse, contrapositive) of p S q. contrapositive
6. To write a(n) (indirect proof, negation), you start by assuming that the opposite
of what you want to prove is true. indirect proof
7. In #ABC, AB + BC . AC because of the (Comparison Property of
Inequality, Triangle Inequality Theorem). k Ineq. Thm.
8. The (circumcenter, incenter) of a triangle is the point of concurrency of the
angle bisectors of the triangle. incenter
PHSchool.com
For: Vocabulary quiz
Web Code: auj-0551
9. The (Angle Bisector Theorem, Triangle Inequality Theorem) says that if a point
is on the bisector of an angle, then it is equidistant from the sides of the angle.
l Bis. Thm.
10. A point where three lines intersect is a (point of concurrency, incenter).
point of concurrency
Spanish Vocabulary/Study Skills
Vocabulary/Study Skills
Name
Skills and Concepts
5-1 and 5-2 Objectives
To use properties of
midsegments to solve
problems
Class
5D: Vocabulary
ELL
L3
Date
For use with Chapter Review
Study Skill: When you read, your eyes make small stops along a line of
words. Good readers make fewer stops when they read. The more stops you
make when you read, the harder it is for you to comprehend what you’ve
read. Try to concentrate and free yourself of distractions as you read.
A midsegment of a triangle is a segment that connects the midpoints of two sides.
A midsegment is parallel to the third side, and is half its length.
Complete the crossword puzzle.
2
1
3
4
5
6
7
8
In a coordinate proof, a figure is drawn on a coordinate plane and formulas are
used to prove properties of the figure.
9
10
11
12
The distance from a point to a line is the length of the perpendicular segment from
the point to the line. The Perpendicular Bisector Theorem together with its
converse states that a point is on the perpendicular bisector of a segment if and
only if it is equidistant from the endpoints of the segment. The Angle Bisector
Theorem together with its converse states that a point is on the bisector of an angle
if and only if it is equidistant from the sides of the angle.
Chapter 5 Chapter Review
297
13
15
14
16
17
18
7.
8.
10.
15.
18.
ACROSS
the result of a single trial
a number pattern
an equation involving two or more variables
an equation that describes a function
a graph that relates two groups of data
4.
5.
6.
9.
11.
12.
13.
DOWN
14.
1. a conclusion you reach by inductive reasoning
2. has two parts, a base and an exponent
3. type of reasoning where conclusions are based
on patterns you observe
20
Reading and Math Literacy Masters
16.
17.
19.
19
set of second coordinates in an ordered pair
a term that has no variable
rational numbers and irrational numbers
each item in a matrix
a comparison of two numbers by division
multiplicative inverse
a data value that is much higher or lower than any
other data values in the set
a relation that assigns exactly one value in the range
to each value in the domain
operations that undo each other
the set of first coordinates in an ordered pair
each number in a sequence
© Pearson Education, Inc. All rights reserved.
To use properties of
perpendicular bisectors
and angle bisectors
Algebra 1
297
x 2 Algebra Find the value of x.
11.
12.
15
11
x⫹5
30
3x ⫺ 1
x
Use the figure to find each segment length
or angle measure.
13. m&BEF 40
15. EC 14
A
23⬚
C
F
14. FE 7
B
50⬚
7
D
16. m&CEA 80
E
5-3 Objectives
To identify properties of
perpendicular bisectors
and angle bisectors
To identify properties of
medians and altitudes of
a triangle
When three or more lines intersect in one point, they are concurrent.
The median of a triangle is a segment whose endpoints are a vertex and the
midpoint of the opposite side. The altitude of a triangle is a perpendicular
segment from a vertex to the line containing the opposite side.
For any given triangle, special segments and lines are concurrent:
• the perpendicular bisectors of the sides at the circumcenter, the center of the
circle that can be circumscribed about the triangle
• the bisectors of the angles at the incenter, the center of the circle that can be
inscribed in the triangle
• the medians at the centroid
• the lines containing the altitudes at the orthocenter of the triangle.
Graph kABC with vertices A(2, 3), B(–4, –3), and C(2, –3). Find the coordinates
of each point of concurrency.
17. circumcenter (–1, 0)
18. centroid (0, –1)
19. orthocenter (2, –3)
Determine whether AB is a perpendicular bisector, an angle bisector, a median, an
altitude, or none of these. Explain.
20.
A
21.
B
22.
B
B
l bisector, because it
bisects an l
5-4 Objectives
To write the negation
of a statement and
the inverse and
contrapositive of a
conditional statement
To use indirect reasoning
298
298
A altitude, because
it is # to a side
A
median, because it goes
through a midpoint
The negation of a statement has the opposite truth value. The inverse of a
conditional statement is the negation of both the hypothesis and the conclusion.
The contrapositive of a conditional statement switches the hypothesis and the
conclusion and negates both. Statements that always have the same truth value are
equivalent statements.
To use indirect reasoning, consider all possibilities and then prove all but one false.
The remaining possibility must be true.
Chapter 5 Chapter Review
24. Inverse: If an angle is
not obtuse, then its
measure is not greater
than 90 and less than
180. Contrapositive: If
an angle’s measure is
not greater than 90 and
less than 180, then it is
not obtuse.
25. Inverse: If a figure is not
a square, then its sides
are not congruent.
Contrapositive: If a
figure’s sides are not
congruent, then it is
not a square.
Alternative Assessment
Name
The three steps of an indirect proof are:
To use inequalities
involving angles of
triangles
To use inequalities
involving sides of
triangles
L4
Date
Form C
Chapter 5
Step 1 State as an assumption the opposite (negation) of what you want to prove.
Materials: ruler, compass
TASK 1
Step 2 Show that this assumption leads to a contradiction.
Harold wants to create a triangular garden plot. He wants to plant 6 ft of
irises along one side, 13 ft of daffodils along the second side, and 6 ft of tulips
along the third side. Write a convincing argument using indirect reasoning to
show that Harold’s garden plan is impossible.
Step 3 Conclude that the assumption must be false and that what you want to
prove must be true.
Write the inverse and the contrapositive of each statement.
TASK 2
23. If it is snowing, then it is cold outside. 23–25. See left.
24. If an angle is obtuse, then its measure is greater than 90 and less than 180.
25. If a figure is a square, then its sides are congruent.
26. If you are in Australia, then you are south of the equator.
See margin.
A locus is the set of points that meet a stated condition. How does the locus
of points in a plane equidistant from a single point differ from the locus of
points in a plane equidistant from the endpoints of a segment? Use sketches
to illustrate your answer.
Write a convincing argument that uses indirect reasoning. 27–30. See margin.
27. The product of two numbers is even. Show that at least one of the two numbers
must be even.
28. Show that a right angle cannot be formed by the intersection of
nonperpendicular lines.
29. Show that a triangle can have at most one obtuse angle.
30. Show that an equilateral triangle cannot have an obtuse angle.
5-5 Objectives
Class
Alternative Assessment
© Pearson Education, Inc. All rights reserved.
23. Inverse: If it is not
snowing, then it is not
cold outside.
Contrapositive: If it is
not cold outside, then
it is not snowing.
If two sides of a triangle are not congruent, then the larger angle lies opposite the
longer side. The converse is also true. If two angles are not congruent, then the
longer side lies opposite the larger angle.
The measure of an exterior angle of a triangle is greater than the measure of each
of its remote interior angles. The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
List the angles and sides in order from smallest to largest.
31.
S
32. O
80⬚
8
4
70⬚
R
T
lT, lR, lS; RS , TS , TR
F
G
5
lG, lO, lF; OF , FG , OG
Is it possible for a triangle to have sides with the given lengths? Explain.
34. Yes; each pair S 3rd.
33. 5 in., 8 in., 15 in. No; 5 ± 8 w 15.
34. 10 cm, 12 cm, 20 cm
35. Yes; each pair S 3rd.
35. 20 m, 22 m, 24 m
36. 3 ft, 6 ft, 8 ft Yes; each pair S 3rd.
37. 1 yd, 1 yd, 3 yd No; 1 ± 1 w 3.
38. 5 km, 6 km, 7 km
Yes; each pair S 3rd.
Two side lengths of a triangle are given. Write an inequality to show the range of
values, x, for the length of the third side.
39. 4 in., 7 in. 3 R x R 11
40. 8 m, 15 m 7 R x R 23
41. 2 cm, 8 cm 6 R x R 10
42. 12 ft, 13 ft 1 R x R 25
Chapter 5 Chapter Review
299
Geometry Chapter 5
Form C Test
25
26. Inverse: If you are not in
Australia, then you are
not south of the equator.
Contrapositive: If you
are not south of the
equator, then you are
not in Australia.
27. Assume that both
numbers are odd. The
product of 2 odd
numbers is always odd,
which contradicts that
the product is even.
Therefore, at least one
number must be even.
28. Assume a right l can be
formed by non-perp.
lines. Then by the def. of
# , the lines are # .
Therefore, the
assumption is false.
29. Assume that a k has 2
obtuse '. Then these '
by def. are greater than
90, which makes their
sum greater than 180.
But the sum of the
measures of the '
of a k ≠ 180, so the
assumption must
be false.
30. Assume an l is obtuse,
and therefore has
measure greater than
90. Since the k is
equilateral, it is
equiangular, and each l
measures 60.
299