Becoming a geometry lawyer
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Becoming a ‘Geometry Lawyer’. Lesson 1. Similar Triangles. Last week we learned what similar triangles are, and how we can use them to find side lengths. What is the minimum amount of information you need to know to declare two triangles SIMILAR?

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Similar triangles
Similar Triangles

  • Last week we learned what similar triangles are, and how we can use them to find side lengths.

  • What is the minimum amount of information you need to know to declare two triangles SIMILAR?

    • TWO corresponding angles must be shown congruent—that automatically implied the third set of angles would be congruent as well. (Call this technique, ‘AA’).


For example
For example,

  • PROVE ∆PQR ~ ∆TSR.

Pretend you are a lawyer. Your job is to convince a Geometry judge that these two triangles are similar.

First, you need to know what to show the judge! We just saw that AA will be enough to convince the judge, since if two corresponding angles are congruent, then the triangles are similar.

SO, show that two corresponding angles ARE congruent.

Q

P

AIA

VA

R

T

S

∠PRQ is congruent to ∠SRT by VA. (There is one set of congruent angles)

It is given that PQ and ST are parallel, therefore,

∠QPR is congruent to ∠RTS by AIA (The second set of congruent angles)

Thus, ∆PQR ~ ∆TSR! □


Paragraph proof
Paragraph Proof

  • The previous example was simply telling a story and explaining the diagram in whole sentences. These sentences lead the reader to an undeniable conclusion--the exact conclusion that was asked of you as the lawyer.


Congruent triangles
Congruent Triangles

  • Not only is there a shortcut to showing that two triangles are similar (AA), there also are shortcuts that will prove that a triangle is CONGRUENT.

  • Construction worker example

  • How can we make it easy to show that triangles are congruent? We don’t want to have to prove all 6 corresponding parts of triangles are the same!


SSS

  • The first shortcut is Side-Side-Side which may be abbreviated as SSS.

    • If two different triangles each have three congruent sides, then this automatically implies all 3 angles will also be congruent, and the two triangles are congruent!!


SAS

  • In this shortcut, you simply need to show that two sides are congruent between the two triangles. If the INCLUDED angle (between these two sides) is also congruent, then the triangles are congruent.

By SAS, ∆ABC is congruent to ∆XYZ

Y

Z

A

93°

93°

X

B

C


Classwork
CLASSWORK

  • P.222, #1-6, 9-14


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