The Timpson Lyceum · Geometry (2024)

Similar Triangles & Ratios – Lecture 11 (Geometry)

Overview

This lecture opens the study of similarity by building on the parallel postulate from the previous chapter. Two figures are similar when one is a scaled copy of the other: their corresponding angles are congruent and their corresponding sides are proportional. Both conditions matter — congruent angles alone (as in two differently-shaped rectangles) do not guarantee similarity. To make "proportional" precise we revisit ratios, recall how the rationals are built from undefined arithmetic, prove that the ratio of two segments is independent of the measuring unit, and discuss when two segments are commensurable. The payoff is the central theorem: in triangles, congruent angles alone force proportional sides, which is why the familiar "AA" check works.

Key Concepts

• Similar figures — Two figures are similar when (1) all corresponding angles are congruent and (2) all corresponding sides are proportional. Written $\triangle ABC \sim \triangle A'B'C'$. One figure is a uniform scaling (enlargement or reduction) of the other.

• Opposite sides of a parallelogram are congruent — Drawing a diagonal splits the parallelogram into two triangles; each pair of alternate interior angles formed by the parallel sides and the diagonal (the shared transversal) is congruent, so the triangles are congruent by ASA, making opposite sides congruent.

• Equal spacing on parallels (Theorem 91) — Given three parallel lines, if a transversal cuts off congruent segments between them, then every transversal does, regardless of its slope. The proof draws auxiliary lines parallel to one transversal to create congruent triangles (by ASA) and parallelograms (whose opposite sides are congruent).

  • Corollary (midsegment, one direction): If $D$ and $E$ are the midpoints of sides $AB$ and $AC$ of a triangle and $DE \parallel BC$, the parallel divides the sides equally.
  • Corollary (midsegment converse): If $D$ bisects $AB$ and $E$ bisects $AC$, then $DE \parallel BC$.
  • Application: A segment can be divided into any number of equal parts with straightedge and compass — mark off equal arbitrary lengths along an auxiliary ray, connect the last mark to the endpoint, and draw parallels through the other marks. No measurement or rounding needed.

• Ratio (etymology aside) — "Ratio" is the Latin for reason (Greek logos), the same root as rational. A ratio of two numbers is their quotient $x/y$.

• Building numbers — Starting from undefined terms $0$, $1$, $+$, $\times$, the naturals are defined recursively ($2 := 1+1$, $3 := 2+1$, …). Subtraction is defined: $x - y = z$ means $x = y + z$. Division is defined: $x / y = z$ means $x = y\cdot z$ with $y \neq 0$. These definitions produce the integers and then the rationals.

• Proportionality — Quantities $a, c$ are proportional to $b, d$ iff $\dfrac{a}{b} = \dfrac{c}{d}$. More generally, $x_1,\dots,x_n$ are proportional to $x_1',\dots,x_n'$ iff $\dfrac{x_1}{x_1'} = \dfrac{x_2}{x_2'} = \cdots = \dfrac{x_n}{x_n'}$. The chosen "numerators" must be consistent, but it does not matter which set you call numerators.

• Fraction rules (proved, used later)

  • Multiply top and bottom: for $b,c \neq 0$, $\dfrac{a}{b} = \dfrac{a\cdot c}{b\cdot c}$ (so you may also cancel a common factor).
  • Cross-multiplication: for $b,d \neq 0$, $\dfrac{a}{b} = \dfrac{c}{d} \iff a\cdot d = b\cdot c$.

• Ratio of segments — Since a segment is not a number, the ratio of segment $AB$ to segment $CD$ is defined as the quotient of their lengths, $\dfrac{|AB|}{|CD|}$, requiring $C \neq D$ (a point has length $0$).

• Ratio is unit-independent (Theorems 92–94) — The ratio of two segments is the same number no matter what unit measures them. If unit $EF$ gives lengths $m$ and $n$, and one $EF$ equals $k$ of a finer unit $GH$, then $GH$ gives lengths $mk$ and $nk$; the common factor $k$ cancels:

A ratio is a pure number — it carries no units.

• Commensurable segments — $AB$ and $CD$ are commensurable iff their ratio is rational (expressible as an integer over an integer). Equivalently, there is a unit $EF$ that divides both an integer number of times: $|AB| = m$, $|CD| = n$ with $m, n \in \mathbb{Z}$. Not all segments are commensurable — the side and diagonal of a unit square give the ratio $1 : \sqrt{2}$, and $\sqrt{2}$ is irrational. (The Greeks, for whom every "number" was rational, found this deeply unsettling.)

• AA similarity for triangles — If the three angles of one triangle are congruent to the three angles of another, the triangles are similar (corresponding sides proportional). For triangles, congruent angles alone suffice — a special property that does not generalize to arbitrary polygons.

Worked Example

Proving cross-multiplication ($b, d \neq 0$): show $\dfrac{a}{b} = \dfrac{c}{d} \implies a\cdot d = b\cdot c$.

State the rule, then carry out the steps. Let $x = \dfrac{a}{b} = \dfrac{c}{d}$. By the definition of division, $a = x\cdot b$ and $c = x\cdot d$. Substitute into $a\cdot d$:

The reverse direction starts from $a\cdot d = b\cdot c$, multiplies top and bottom of $\tfrac{a}{b}$ by $d$, substitutes $a\cdot d = b\cdot c$, then cancels the common $b$ to recover $\tfrac{a}{b} = \tfrac{c}{d}$.

Summary

• Two figures are similar when corresponding angles are congruent and corresponding sides are proportional; both conditions are required.
• Opposite sides of a parallelogram are congruent, and equally spaced parallel lines cut every transversal into congruent segments — the tools behind dividing a segment into any number of equal parts by construction.
• "Proportional" rests on ratios: a ratio of segments is the quotient of their lengths, a pure number with no units, and it does not depend on the measuring unit.
• Segments are commensurable exactly when their ratio is rational; the side and diagonal of a square ($1 : \sqrt{2}$) show that incommensurable segments exist.
• The key fraction facts — multiplying/cancelling top and bottom, and cross-multiplication $\tfrac{a}{b} = \tfrac{c}{d} \iff ad = bc$ — are the algebraic backbone for working with proportions.
• For triangles only, congruent corresponding angles force proportional sides (AA similarity), justifying the familiar angle-only test for similar triangles.