1/13/16- Similarity Investigation

Warm Up

  1. a. You get a job that makes $10/hr. You work 40 hrs per week and every other week, your paycheck is for $575.60. How much would you expect your paycheck to be if you work for 30 hours instead?
  1. b. Over the holidays you put in 5 extra hours of work per week (on top of your 40 hour weeks) at time-and-a-half pay. How much would you expect your paycheck to be?

Similarity Investigation

Recall: Definition of Similarity- If all of the corresponding angles between two shapes are congruent, and the ratios of their corresponding sides are proportional, then the two figures are similar.

Definition of Congruence- If all of the corresponding angles between two shapes are congruent, and all of the corresponding sides between two shapes are congruent, then the two shapes are congruent.

Notice the differences and similarities between these definitions: Both talk about corresponding angles being congruent. For sides, they are only congruent when the figures are congruent- when the figures are similar, we have to talk about the ratios of corresponding sides being proportional.

Then, recall that although all corresponding parts need to be congruent to say that two shapes are congruent, there were some shortcuts, or structures, that we could use to show that triangles were congruent, without knowing everything about all six angles and sides.

These structures were SSS, SAS, ASA, AAS.

When we see “SAS” we often say “Side-Angle-Side,” but it is important to remember there is much more to “SAS” as a congruence structure. What “SAS” really stands for is:

IF two triangles have two pairs of corresponding sides that are congruent and the Angles between those sides are also congruent, then the two triangles are congruent.

Hopefully it seems reasonable that similar “shortcuts” or “structures” would exist for similar triangles.

What would “SAS” look like or be described as in the context of similarity?

SAS Similarity- IF two triangles have two pairs of corresponding sides whose ratios are proportional and the angles between those sides are congruent, then the two triangles are similar.

Notice the difference between the similarity version and the congruence version: Now when we talk about sides, we aren’t talking about congruent sides, but proportional  ratios of corresponding sides. The angles are still identified as being congruent between the two shapes.

We then used rulers and protractors to create pairs of triangles that started off with pairs of sides that were proportional to each other and angles connecting those sides that were exactly the same (congruent). After creating the two triangles with the 3 specific parts for each, we then measured the remaining parts (2 other angles and 1 other side). What should happen, if this SAS scenario creates similar shapes, is that the remaining two pairs of angles should be congruent, and the remaining pair of sides should create a ratio that is proportional to the other ratios from pairs of corresponding sides (by the definition of what it means to be similar).

We then did the same process using AAA structure. First off, what would AAA say?

AAA- IF two triangles have three pairs of congruent angles, then the two triangles are similar.

So create two triangles that both have the same angle measures. Once again, the question is does leave us with similar triangles. To find out, measure all of the sides in each triangle, and then check the ratios of corresponding sides and see if they are proportional. If they are, the triangles are similar!

The big idea, is that the structures that we used for talking about congruent triangles, can be altered slightly to apply to talking about similar triangles. The big change has to do with sides- when talking about similarity, we talk about sides being proportional (the ratios of corresponding sides are proportional).

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