Advanced Geometry Section 2.6 Multiplication and Division Properties

Advanced Geometry

Section 2.6 Multiplication and Division Properties

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

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

Opener: Given: ST ≅ SM, TP ≅ MN Prove: SP ≅ SN

S

T

P

M

N

Statement Reason

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

How would this proof be different if the 2nd given and the statement to be

proved were switched?


In this figure, if B, C, F and Gare trisection points,what does that tell us?

If the length of AB is 5 , what is the length of the following? Why? AC = ______ why? AD = ______ why?

 BC = ______ why? BD = ______ why?

 EG = ______ why? EH = ______ why?


Remember:In this figure, if B, C, F and Gare 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 = ____?


Why is this? Because if two segments are congruent, then multiplying each of them by the same value gives us congruent segments.

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


In this figure, AD and GH areangle bisectors. What pairsof 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?


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)


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.


This fact also applies to angles.

If AD and GH are angle bisectorsand BAC ≅ FGE, then whatpairs of angles can we concludeare congruent by dividing the original angles?


This leads us to another important theorem:

THEOREMIf two segments (or angles) are congruent, then their like divisions are congruent (Division Property)


Using the Multiplication and Division Properties in Proofs1. 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.






HW

Pg. 92 # 1,3-6,10















Advanced Geometry Section 2.6 Multiplication and Division Properties

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Advanced Geometry Section 2.6 Multiplication and Division Properties