1. TODAY IN GEOMETRY…
Learning Goal 1: 2.4 You will use postulates involving points, lines, and planes
Independent Practice – 20 minutes!
Learning Goal 2: 2.5 Use Algebraic properties to form logical arguments
Test Retakes before next Friday

2. Through any two points there exists one line.
POSTULATE 5:
POSTULATE 6:
Through any two points there exists one line.
A line contains at least two points. 𝐵 𝐴 𝐶 𝐷 𝐵 𝐴 𝐶

3. If two lines intersect, then their intersection is exactly one point.
POSTULATE 7:
POSTULATE 8:
If two lines intersect, then their intersection is exactly one point.
Through any three noncollinear points there exists exactly one plane. 𝐴 𝐴 𝐶 𝐵

4. A plane contains at least three noncollinear points.
POSTULATE 9:
POSTULATE 10:
A plane contains at least three noncollinear points.
If two points lie in a plane, then the line containing them lies in the plane. 𝐻 𝐸 𝐼 𝐻 𝐺 𝐹 𝐺

5. POSTULATE 11:
If two planes intersect, then their intersection is a line.

6. PRACTICE: State the postulate illustrated by the diagram.
If two lines intersect, then their intersection is exactly one point.
POSTULATE 11:
If two planes intersect, then their intersection is a line.

7. PRACTICE: Use the diagram to write examples of each postulate:
A plane contains at least three noncollinear points.
Plane P contains points noncollinear points B, A, C
If two points lie in a plane, then the line containing them lies in the plane.
Points A and B lie on Plane P, line n also lies on Plane P.

8. PRACTICE: Use the diagram to determine if the statement is true or false.
All points are
coplanar.
2. G, F, and E are
collinear.
3. 𝐵𝐹 and 𝐶𝐸 intersect.
4. ∠𝐴𝐻𝐹 and ∠𝐵𝐻𝐷 are vertical angles.
5. 𝐴𝐷 and 𝐵𝐹 intersect at H.
6. ∠𝐵𝐻𝐴≅∠𝐶𝐼𝐴 T F F T T F

9. 20 minutes HOMEWORK #2: Pg. 99: 3-8, 11-23
If finished, work on other assignments:
HW #1: Pg. 82: 3-18, 26-28, 47-54

10. 2.5 Algebraic Properties
Addition property
Subtraction property
Multiplication property
Division Property
Substitution property
Distribution property
Combine like terms

11. ADDITION PROPERTY:
SUBTRACTION PROPERTY:
Use when you must add to solve an algebraic problem.
Use when you must subtract to solve an algebraic problem.
EXAMPLE:
If 𝑥−3=12,
then 𝑥=15
EXAMPLE:
If 𝑥+5=8,
then 𝑥=3

12. MULTIPLICATION PROPERTY:
DIVISION PROPERTY:
Use when you must multiply to solve an algebraic problem.
Use when you must divide to solve an algebraic problem.
EXAMPLE:
If 3𝑥=24,
then 𝑥=8
EXAMPLE:
If 𝑥 2 =5,
then 𝑥=10

13. SUBSTITUTION PROPERTY:
DISTRIBUTIVE PROPERTY:
COMBINE LIKE TERMS:
Use when you must substitute to solve an algebraic problem.
Use when you must distribute to solve an algebraic problem.
Use when you must combine like terms to solve an algebraic problem.
EXAMPLE:
If 𝑥=4,
then 2𝑥=8
EXAMPLE:
If 6(𝑥−2),
then 6𝑥−12
EXAMPLE:
If 9𝑥+𝑥−4,
then 10𝑥−4

14. Given: 3𝑥+12=30 ALGEBRAIC PROOF: Prove: 𝑥=6 Statement Reason 1. Given.
1. 3𝑥+12=30
1. Given.
− 12 − 12
2. 3𝑥=18
2. Subtraction Property
3. 𝑥=6
3. Division Property

15. PROOFS WITHOUT THE “WORK”
Given: 3𝑥+12=30
Prove: 𝑥=6
ALGEBRAIC PROOF:
Statement
Reason
1. 3𝑥+12=30
1. Given.
2. 3𝑥=18
2. Subtraction Property
3. 𝑥=6
3. Division Property
PROOFS WITHOUT THE “WORK”

16. Given: −4(11𝑥+2)=80 ALGEBRAIC PROOF: Prove: 𝑥=−2 Statement Reason
1. −4 11𝑥+2 =80
1. Given.
−4 11𝑥 + −4 2 =80
2. −44𝑥−8=80
2. Distribution Property
3. −44𝑥=88
3. Addition Property
− −44
4. 𝑥=−2
4. Division Property

17. PROOFS WITHOUT THE “WORK”
Given: −4(11𝑥+2)=80
Prove: 𝑥=−2
ALGEBRAIC PROOF:
Statement
Reason
1. −4 11𝑥+2 =80
1. Given.
2. −44𝑥−8=80
2. Distribution Property
3. −44𝑥=88
3. Addition Property
4. 𝑥=−2
4. Division Property
PROOFS WITHOUT THE “WORK”

18. Given: 2 𝑥+5 −7=19 ALGEBRAIC PROOF: Prove: 𝑥=8 Statement Reason
1. 2 𝑥+5 −7=19
1. Given.
2 𝑥 −7=19
2. 2𝑥+10−7=19
2. Distribution Property
2𝑥+ 10−7 =19
3. 2𝑥+3=19
3. Combine like terms
− 3 − 3
4. 2𝑥=16
4. Subtraction Property
5. 𝑥=8
5. Division Property

19. PROOFS WITHOUT THE “WORK”
Given: 4 𝑥+5 −7=19
Prove: 𝑥=8
ALGEBRAIC PROOF:
PROOFS WITHOUT THE “WORK”
Statement
Reason
1. 2 𝑥+5 −7=19
1. Given.
2. 2𝑥+10−7=19
2. Distribution Property
3. 2𝑥+3=19
3. Combine like terms
4. 2𝑥=16
4. Subtraction Property
5. 𝑥=8
5. Division Property

20. REFLEXIVE TRANSITIVE SYMMETRIC PROPERTIES REAL NUMBERS SEGMENTS ANGLES
𝐹𝑜𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙
𝑛𝑢𝑚𝑏𝑒𝑟 𝑎,
𝑎=𝑎
𝐹𝑜𝑟 𝑎𝑛𝑦
𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵 ,
𝐴𝐵 = 𝐴𝐵
𝐹𝑜𝑟 𝑎𝑛𝑦
𝑎𝑛𝑔𝑙𝑒 ∠𝐴,
𝑚∠𝐴=𝑚∠𝐴
“equals itself”
𝐼𝑓 𝑚∠𝐴=𝑚∠𝐵,
𝑡ℎ𝑒𝑛,
𝑚∠𝐵=𝑚∠𝐴
𝐼𝑓 𝑎=𝑏,
𝑡ℎ𝑒𝑛, 𝑏=𝑎
𝐼𝑓 𝐴𝐵=𝐶𝐷,
𝑡ℎ𝑒𝑛, 𝐶𝐷=𝐴𝐵
“same in reverse”
𝐼𝑓 𝑚∠𝐴=𝑚∠𝐵
& 𝑚∠𝐵=𝑚∠𝐶
𝑡ℎ𝑒𝑛,
𝑚∠𝐴=𝑚∠𝐶
𝐼𝑓 𝑎=𝑏
𝑎𝑛𝑑 𝑏=𝑐
𝑡ℎ𝑒𝑛 𝑎=𝑐
𝐼𝑓 𝐴𝐵=𝐶𝐷,
𝑎𝑛𝑑 𝐶𝐷=𝐸𝐹
𝑡ℎ𝑒𝑛, 𝐸𝐹=𝐴𝐵
“there’s a middle man”

21. SOLUTIONS:
SYMMETRIC PROPERTY
TRANSITIVE PROPERTY
REFLEXIVE PROPERTY