Chapter 3: Similarity

Question 1

Chapter 3 notation:

Answer 1

• [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

Question 2

Thm 3.2.2:

Answer 2

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.

Question 3

Thm 3.2.3 (Parallel Lines Preserve Ratios):

Answer 3

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.

Question 4

Fact (Useful in proofs, like Lemma 3.3.6):

Answer 4

If P and Q are pts on AB, then
AP/PB = AQ/QB => P=Q
Pf provided.

Question 5

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

Answer 5

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

Question 6

Note:

Answer 6

Parallel Lines Preserve Ratios is not necessarily reversible.
AB/CB = DE/EF does not mean BE||CF.
Ex provided.

Question 7

Thm 3.4.7 (Angle Bisector Thm):

Answer 7

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.

Question 8

Define Similar.

Answer 8

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

Question 9

Thm 3.2.4 (Useful in proofs):

Answer 9

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.

Question 10

Both congruency and similarity are ‘equivalence relations’. That is, they are both:

Answer 10

• reflexive: △ABC ~ (==) △ABC
• symmetric: △ABC ~ (==) △DEF iff △DEF ~ (==) △ABC
• transitive: If △ABC ~ (==) △DEF and △DEF ~ (==) △GHI then △ABC ~ (==) △GHI
• Congruency => Similarity

Question 11

Thm 3.3.1 (AAA):

Answer 11

Two triangles are similar iff all 3 corresponding angles are congruent.
Note: Fails for quads.

Question 12

Thm 3.3.5 (AA):

Answer 12

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

Question 13

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

Answer 13

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

Question 14

Thm 3.3.4 (sss):

Answer 14

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

Pf provided.

Question 15

Thm 3.4.1 (Pythagoras Thm):

Answer 15

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.

Question 16

Thm 3.4.2 (Converse to Pythagoras):

Answer 16

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

Pf left as exercise.

Question 17

Ptolemy’s Thm:

Answer 17

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

Question 18

Thm 3.4.12:

Answer 18

In △ABC, suppose D and E are pts on AB and AC s.t. DE||BC. Let P be a pt on BC (between B and C) and Q a pt on DE (between D and E). Then A, P, and Q are collinear (lie on same line segment) iff BP/PC = DQ/QE.
Pf provided.