Advanced Geometry Section 2.6 Multiplication and Division Properties

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Presentation transcript:

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

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

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 5 , what is the length of the following? Why?  AC = ______ why? AD = ______ why?  BC = ______ why? BD = ______ why?  EG = ______ why? EH = ______ why?

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 = ____?

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.

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?

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)

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.

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?

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)

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.