Transcript PR = QS
Chapter 2.7
Proving Segment Relationships
Objective: Practice using proofs for geometric relationships
by starting with segments
Spi.1.4
Check.4.3
Use definitions, basic postulates, and theorems about points, lines, angles, and planes to
write/complete proofs and/or to solve problems.
Use definitions, basic postulates, and theorems about points, lines, angles, and planes to
write/complete proofs
Add to your
listing
Geometric Properties
Postulate 2.8 (Ruler Postulate)
Postulate 2.9 (Segment Addition Postulate)
The points on any line or line segment can be paired with real
numbers so that given any two points A and B on a line, A
corresponds to zeros and B corresponds to a positive real number.
If B is between A and C, then AB + BC = AC or
If AB + BC = AC, then B is between A and C
AB
A
Theorem 2.2
BC
B
AC
Congruence of segments is reflexive, symmetric, and transitive.
Reflexive Property
AB AB
Symmetric Property
If AB CD, then CD AB
Transitive Property
If AB CD, and CD EF, then AB EF
It's not what you look at that matters, it's what you see. Henry David Thoreau
C
Use paper to solve
Given BC = DE
Prove AB + DE = AC
1.
2.
3.
Statements
BC = DE
AB + BC = AC
AB + DE = AC
1.
2.
3.
Reasons
Given
Segment Addition Postulate
Substitution
Use paper to solve
Given PR QS
Prove PQ RS
1.
2.
3.
4.
5.
6.
7.
8.
Statements
PR QS
PR = QS
PQ + QR = PR
QR + RS = QS
PQ + QR = QR + RS
PQ = RS
PQ RS
1.
2.
3.
4.
5.
6.
7.
Reasons
Given
Definition of Congruence
Segment Addition Postulate
Segment Addition Postulate
Substitution
Subtraction
Definition of Congruence
Proof with Segment Addition
process
Prove the following:
Given: PQ = RS
Prove: PR = QS
Statements
1. PQ = RS
P
1.
Q
R
Reasons
Given
2.
PQ + QR = QR + RS
2.
Addition Property
3.
PQ + QR = PR and
3.
Segment Addition Postulate
4.
Substitution
QR + RS = QS
4.
PR = QS
S
Proof with Segment Addition
process
Prove the following:
Given: PR = QS
Prove: PQ = RS
Statements
1. PR = QS
P
1.
Q
R
Reasons
Given
2.
PR - QR = QS - QR
2.
Subtraction Property
3.
PR - QR = PQ and
3.
Segment Addition Postulate
4.
Substitution
QS - QR = RS
4.
PQ = RS
S
Proof with Segment Congruence
process
J
Prove the following:
Given: JK KL, HJ GH, KL HJ
Prove: GH JK
Statements
1. JK KL, KL HJ
1.
Reasons
Given
K
L
H
2.
JK HJ
2.
Transitive Property
3.
HJ GH
3.
Given
4.
JK GH
4.
Transitive Property
5.
GH JK
5.
Symmetric Property
G
Prove the following.
Given: AC = AB
AB = BX
CY = XD
Prove: AY = BD
Which reason correctly completes the proof?
Proof:
Statements
Reasons
1. AC = AB, AB = BX
1. Given
2. AC = BX
2. Transitive Property
3. CY = XD
3. Given
4. AC + CY = BX + XD
4. Addition Property
5. AC + CY = AY;
BX + XD = BD
? Addition Postulate
Segment
5. ________________
6. AY = BD
6. Substitution
Practice Assignment
Page 145, 4-16 even