2.5
1 PLAN
PACING
Basic: 2 days
Average: 2 days
Advanced: 2 days
Block Schedule: 0.5 block with 2.4
0.5 block with 2.6
What you should learn
GOAL 1 Justify statements
about congruent segments.
GOAL 2 Write reasons for
steps in a proof.
Why you should learn it
FE
Properties of congruence
allow you to justify segment
relationships in real life, such
as the segments in the trestle
bridge shown and in
Exs. 3–5.
AL LI
RE
LESSON OPENER
APPLICATION
An alternative way to approach
Lesson 2.5 is to use the Application
Lesson Opener:
•Blackline Master (Chapter 2
Resource Book, p. 69)
•
Transparency (p. 12)
Proving Statements
about Segments
MEETING INDIVIDUAL NEEDS
• Chapter 2 Resource Book
Prerequisite Skills Review (p. 5)
Practice Level A (p. 73)
Practice Level B (p. 74)
Practice Level C (p. 75)
Reteaching with Practice (p. 76)
Absent Student Catch-Up (p. 78)
Challenge (p. 81)
• Resources in Spanish
•
Personal Student Tutor
GOAL 1
PROPERTIES OF CONGRUENT SEGMENTS
A true statement that follows as a result of other true statements is called a
theorem. All theorems must be proved. You can prove a theorem using a
two-column proof. A two-column proof has numbered statements and reasons
that show the logical order of an argument.
THEOREM
THEOREM 2.1
Properties of Segment Congruence
Segment congruence is reflexive, symmetric, and transitive.
Here are some examples:
Æ
Æ
Æ
Æ
Æ
Æ
Æ
SYMMETRIC
If AB £ CD , then CD £ AB .
TRANSITIVE
If AB £ CD , and CD £ EF , then AB £ EF .
Æ
Æ
You can prove the Symmetric Property
of Segment Congruence as follows.
Æ
Æ
Æ
Æ
Statements
Æ
X
WARM-UP EXERCISES
of equality
STUDENT HELP
Study Tip
When writing a reason
for a step in a proof,
you must use one of
the following: given
information, a definition,
a property, a postulate,
or a previously proven
theorem.
3. If EF = GH and GH = IJ then
EF = IJ. Transitive prop. of
equality
4. If EF = 8 and EF = GH, then
GH = 8. Substitution prop. of
equality
102
P
Y
102
Æ
q
Reasons
1. PQ ⬵ XY
1. Given
2. PQ = XY
2. Definition of congruent segments
3. XY = PQ
3. Symmetric property of equality
4. XY ⬵ PQ
4. Definition of congruent segments
Æ
equality
Æ
GIVEN PQ ⬵ XY
NEW-TEACHER SUPPORT
See the Tips for New Teachers on
pp. 1–2 of the Chapter 2 Resource
Book for additional notes about
Lesson 2.5.
2. m2 = m2 Reflexive prop.
Æ
Symmetric Property of Segment Congruence
EXAMPLE 1
PROVE XY ⬵ PQ
Transparency Available
Give the property that justifies
each statement.
1. If m1 = m2, then m2 =
m1. Symmetric prop. of
Æ
For any segment AB, AB £ AB .
REFLEXIVE
Æ
You are asked to complete proofs for the Reflexive and Transitive Properties
of Segment Congruence in Exercises 6 and 7.
..........
A proof can be written in paragraph form, called paragraph proof. Here is
a paragraph proof for the Symmetric Property of Segment Congruence.
Æ
Æ
Paragraph Proof You are given that PQ £ XY. By the definition of congruent
segments, PQ = XY. By the symmetric property of equality, XY = PQ. Therefore,
Æ
Æ
by the definition of congruent segments, it follows that XY £ PQ.
Chapter 2 Reasoning and Proof
T H E O R E M 2 . 1 P R O P E RT I E S O F S E G M E N T C O N G R U E N C E
GOAL 2
USING CONGRUENCE OF SEGMENTS
MOTIVATING THE LESSON
Ask students if they have ever tried
to measure a room by pacing it off
with their feet. How does this give
them an approximate room measure? The logical sequence of steps
they use to justify the measurement
is an application of using segment
congruence to prove a statement.
Using Congruence
EXAMPLE 2
Proof
2 TEACH
Use the diagram and the given information to complete
the missing steps and reasons in the proof.
Æ
K
J
Æ
GIVEN 䉴 LK = 5, JK = 5, JK ⬵ JL
Æ
Æ
PROVE 䉴 LK ⬵ JL
L
Statements
Reasons
a.㛭㛭
1. 㛭㛭㛭㛭㛭
b.㛭㛭
2. 㛭㛭㛭㛭㛭
2. Given
3. LK = JK
3. Transitive property of equality
4. LK ⬵ JK
5. JK ⬵ JL
c.㛭㛭
4. 㛭㛭㛭㛭㛭
5. Given
d.㛭㛭
6. 㛭㛭㛭㛭㛭
6. Transitive Property of Congruence
1. Given
Æ
Æ
Æ
Æ
EXTRA EXAMPLE 1
Given: EF = GH
Prove: E苶G
苶 ⬵ F苶H
苶
E
SOLUTION
a. LK = 5
b. JK = 5
Æ
c. Definition of congruent segments d. LK £ JL
Æ
R
In the diagram, Q is the midpoint of PR.
P
1
2
Show that PQ and QR are each equal to ᎏᎏPR.
q
W
Æ
GIVEN 䉴 Q is the midpoint of PR .
1
2
PROVE 䉴 PQ = ᎏᎏPR and QR = ᎏᎏPR.
Statements
STUDENT HELP
Study Tip
The distributive property
can be used to simplify a
sum, as in Step 5 of the
proof. You can think of
PQ + PQ as follows:
1(PQ) + 1(PQ) =
(1 + 1) (PQ) = 2 • PQ.
1. Q is the midpoint of PR.
1. Given
2. PQ = QR
2. Definition of midpoint
3. PQ + QR = PR
3. Segment Addition Postulate
4. PQ + PQ = PR
4. Substitution property of equality
5. 2 • PQ = PR
1
6. PQ = ᎏᎏPR
2
1
7. QR = ᎏᎏPR
2
5. Distributive property
T
Y
(Def. of congruent segments)
6. Division property of equality
Extra Example 3 and Checkpoint
Exercises on next page.
7. Substitution property of equality
2.5 Proving Statements about Segments
X
3. RT = RS + ST; WY = WX + XY
(Segment Addition Post.)
4. RS + ST = WX + XY
(Substitution prop. of equality)
5. ST = WX
(Given)
6. RS = XY (– prop. of =)
7.
(Def. of congruent
segments) R苶S苶 • X苶Y苶
Reasons
Æ
S
Given: R苶T苶 ⬵ W
苶Y苶, ST = WX
Prove: R苶S苶 ⬵ X苶Y苶
Statements (Reasons)
1.
(Given) R苶T苶 • W
苶Y苶
2. RT = WY
Decide what you know and what you need to prove. Then write the proof.
•
H
R
SOLUTION
1
2
G
EXTRA EXAMPLE 2
Complete the proof.
Using Segment Relationships
EXAMPLE 3
Proof
Æ
F
Statements (Reasons)
1. EF = GH (Given)
2. EF + FG = GH + FG (+ prop. of =)
3. EG = EF + FG, FH = GH + FG
(Segment Addition Post.)
4. EG = FH (Subs. prop. of =)
5. E苶G
苶 • F苶H
苶 (Def. of • seg.)
103
103
ACTIVITY
Copy a Segment
Construction
EXTRA EXAMPLE 3
S
M
Æ
Use the following steps to construct a segment that is congruent to AB.
X
R
N
Given: X is the midpoint of M
苶N
苶,
and MX = RX.
Prove: XN = RX
1
X
C
Use a straightedge
to draw a segment
Æ
longer than AB.
Label the point C
on the new segment.
2
Set your compass
Æ
at the length of AB.
A
B
C
D
Place the compass
point at C and mark
a second point, D,
on the new segment.
Æ
CD is congruent
Æ
to AB.
3
Y
You will practice copying a segment in Exercises 12–15. It is an important
construction because copying a segment is used in many constructions
throughout this course.
Given: RS = XY, ST = WX
Prove: RT = WY
Statements (Reasons)
1. RS = XY, ST = WX (Given)
2. RS + ST = XY + WX (Addition
prop. of equality)
3. RT = RS + ST (Segment
Addition Post.)
4. WY = XY + WX (Segment
Addition Post.)
5. RT = WY (Substitution prop.
of equality)
CLOSURE QUESTION
In the diagram, if A
苶B
苶⬵ B
苶C苶 and
B
苶C苶 ⬵ C苶D
苶, find BC. 11
A
D
3x ⫺ 1
GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
2. Using the Transitive
Property of Segment
Congruence, we can
only assume that
Æ
Æ
SR £ QR .
1. An example of the Symmetric Property of Segment Congruence is
Æ
Æ
Æ Æ
?㛭㛭㛭, then CD £ 㛭㛭㛭㛭㛭㛭
?㛭㛭㛭.” CD ; AB
“If AB £ 㛭㛭㛭㛭㛭㛭
Æ
Æ
Æ
Æ
2. ERROR ANALYSIS In the diagram below, CB £ SR and CB £ QR.
Explain what is wrong with Michael’s argument.
Æ
Æ
Æ
Æ
Because CB £ SR and CB £ QR ,
Æ
Æ
then CB £ AC by the Transitive
Property of Segment Congruence.
A
C
Q
B
R
S
2x ⫹ 3
B
C
Skill Check
✓
3. By the definition of
midpoint, Point D is
halfway between
BÆ
and
Æ
F. Therefore BD £ FD .
5. By the Transitive
Property of Segment
Æ
Congruence, if CE £
Æ
Æ
Æ
BD and
BDÆ
£ FD ,
Æ
then CE £ FD .
104
104
B
A
C
CHECKPOINT EXERCISES
For use after Examples 1–3:
1. R
S
T
W
B
A
Statements (Reasons)
1. X is the midpoint of M
苶N
苶. (Given)
2. XN = MX (Def. of midpoint)
3. MX = RX (Given)
4. XN = RX (Transitive prop.
of equality)
BRIDGES The diagram below shows a portion of a trestle bridge,
Æ
Æ
Æ
where BF fi CD and D is the midpoint of BF .
Æ
Æ
3. Give a reason why BD and FD are
congruent.
4. Are ™CDE and ™FDE complementary?
Explain. See margin.
Æ
Æ
5. If CE and BD are congruent, explain
Æ
Æ
why CE and FD are congruent.
Chapter 2 Reasoning and Proof
A
C
E
B
D
F