1. Advanced Geometry
Section 2.6 Multiplication and Division Properties
Learner Objective: Students will apply the multiplication and division 
 properties of segments and angles.

2. Proof Opener: Given: ST ≅ SM, TP ≅ MN Prove: SP ≅ SN T N Statement
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
Opener:
Given: ST ≅ SM, TP ≅ MN
Prove: SP ≅ SN S M T P N
Statement
Reason
How would this 
proof be 
different if the 
2nd given and 
the statement 
to be proved 
were switched?
1. ST ≅ SM
1. given
2. TP ≅ MN
2. given
3. SP ≅ SN
3. If ≅ seg's added to
≅ seg's, the sums
are ≅ (Add Prop)
Proof

3. In this figure, if B, C, F and G are trisection points,
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
In this figure, if B, C, F and G
are trisection points,
what does that tell us?
If the length of AB is , what is the length of the following? Why?
 AC = ______ why? AD = ______ why?
 BC = ______ why? BD = ______ why?
 EG = ______ why? EH = ______ why?

4. In this figure, if B, C, F and G are trisection points, and AB = 5.
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
Remember:
In this figure, if B, C, F and G
are trisection points, and AB = 5.
What if we also knew that
AB ≅ EF, would we now know
the lengths of the following?
EG = ____? FH = ____? EH = ____?

5. AD ___ EH, AC ___ EG, BD ___ FH, BD ___ EG, AC ___ FH
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
So, what does knowing that
AB ≅ EF and that B, C, F, and G are all trisection points allow us 
to conclude about the following 
pairs of segments?
AD ___ EH, AC ___ EG, BD ___ FH,
BD ___ EG, AC ___ FH
Why is this?
Because if two segments are congruent, then multiplying each 
of them by the same value gives us congruent segments.

6. In this figure, AD and GH are angle bisectors. What pairs
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
In this figure, AD and GH are
angle bisectors. What pairs
of angles do we know are
congruent? Why?
If we are also given that BAD ≅ FGH. What additional pairs of angles do we now know are congruent? Why?

7. These facts lead us to the following important theorem:
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
These facts lead us to the following important theorem:
THEOREM:
If two segments (or angles) are congruent,
 then their like multiples are congruent.
 (Multiplication Property)

8. B, C, F and G are trisection points.
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
B, C, F and G are trisection points.
If we are given that AD ≅ EH,
are the following pairs of
segments congruent? Why?
AB ___ EF AB ___ FG
AC ___ EG BD ___ FH  
If two segments are congruent, then dividing them both by the 
same value results in congruent segments.

9. This fact also applies to angles.
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
This fact also applies to angles.
If AD and GH are angle bisectors
and BAC ≅ FGE, then what
pairs of angles can we conclude
are congruent by dividing the 
original angles?

10. This leads us to another important theorem:
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
This leads us to another important theorem:
THEOREM
If two segments (or angles) are congruent,
then their like divisions are congruent
(Division Property)

11. Using the Multiplication and Division Properties in Proofs
Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.
Using the Multiplication and Division Properties in Proofs
1. Look for a double use of the word midpoint, trisects, or 
 bisects in the given information.
2. The Multiplication Property is used when the segments or 
 angles in the conclusion are greater than those in the 
 given information.
3. The Division Property is used when the segments or 
 angles in the conclusion are smaller than those in the 
 given information.

12. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

13. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

14. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

15. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

16. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles. HW
Pg. 92 # 1,3-6,10

17. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

18. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

19. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

20. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

21. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

22. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

23. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

24. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

25. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

26. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

27. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

28. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

29. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.

30. Learner Objective: Students will apply the multiplication and division properties of segments
 and angles.