Flashcards in Chapter 3: Similarity Deck (18):

1

## Chapter 3 notation:

###
-[ABC] = area of △ABC = bxh/2

-~ denotes Similarity

-△ABC ~ △DEF means ∠A==∠D, ∠B==∠E, ∠C==∠F and AB/DE = BC/EF = AC/DF = k (proportionality constant/magnification factor)

-k>1 => △ABC is larger than △DEF ; 0 △ABC is smaller than △DEF ; k=1 => △ABC == △DEF

2

## Thm 3.2.2:

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Given parallel lines l and m and 2 △s with bases on m and opposite vertices on l, the ratio of areas of the triangles is the ratio of the lengths of their bases.

[ABC] / [DEF] = BC/EF

Pf provided.

3

## Thm 3.2.3 (Parallel Lines Preserve Ratios):

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Suppose l, m, and n are parallel lines, met by transversals t, t' at A, B, C and A', B', C' resp. Then, AB/BC = A'B'/B'C'

Pf provided.

4

## Fact (Useful in proofs, like Lemma 3.3.6):

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If P and Q are pts on AB, then

AP/PB = AQ/QB => P=Q

Pf provided.

5

## Lemma 3.3.6 (Partial Converse to Parallel Lines Preserve Ratios):

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Let P and Q be pts on AB and AC, resp. If AP/PB = AQ/QC, then PQ||BC.

Pf provided.

6

## Note:

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Parallel Lines Preserve Ratios is not necessarily reversible.

AB/CB = DE/EF does not mean BE||CF.

Ex provided.

7

## Thm 3.4.7 (Angle Bisector Thm):

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Let D be a pt on side BC of △ABC

1. For D ∈ BC, AD is the internal bisector of ∠BAC iff AB/AC = DB/DC

2. For D ∉ BC, AD is the external bisector of ∠BAC iff AB/AC = DB/DC

Pf provided with aside.

8

## Define Similar.

### Similar describes objects with the same shape, but possibly different sizes.

9

## Thm 3.2.4 (Useful in proofs):

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In △ABC, suppose DE||BC. If D and E are on lines AB and AC, resp, and D≠A, D≠B, D≠C, then △ABC ~ △ADE.

Pf provided.

10

## Both congruency and similarity are 'equivalence relations'. That is, they are both:

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-reflexive: △ABC ~ (==) △ABC

-symmetric: △ABC ~ (==) △DEF iff △DEF ~ (==) △ABC

-transitive: If △ABC ~ (==) △DEF and △DEF ~ (==) △GHI then △ABC ~ (==) △GHI

-Congruency => Similarity

11

## Thm 3.3.1 (AAA):

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Two triangles are similar iff all 3 corresponding angles are congruent.

Note: Fails for quads.

12

## Thm 3.3.5 (AA):

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Two triangles are similar iff 2 corresponding angles are congruent (equivalent to Thm 3.3.1).

Pf provided.

13

## Thm 3.3.3 (sAs) (small s for side ratios as opposed to big S for congruency/same length):

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If in △ABC and △DEF, AB/DE = AC/DF and ∠A = ∠D, then △ABC ~ △DEF.

Pf provided.

14

## Thm 3.3.4 (sss):

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△ABC ~ △DEF iff AB/DE = BC/EF = AC/DF.

Pf provided.

15

## Thm 3.4.1 (Pythagoras Thm):

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If the hypotenuse of a right triangle has length C, and the other 2 sides have length a and b, then c^2 = a^2 + b^2.

Pf provided.

16

## Thm 3.4.2 (Converse to Pythagoras):

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In △ABC, if (AB)^2 = (BC)^2 + (AC)^2, then ∠C = 90.

Pf left as exercise.

17

## Ptolemy's Thm:

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A quad. ABCD is cyclic iff (AC)(BD) = (AB)(CD) + (AD)(BC).

Pf provided (long).

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