Pythagorean theorem — proof (by similar triangles, rearrangement), converse: Foundations

Before you can read the parent note, you must own every piece of its vocabulary. This page builds each symbol from nothing: what it means in plain words, what picture it draws, and why the topic can't do without it. Read top to bottom — each idea leans on the one above it.

1. What is a triangle, and what is an angle?

An angle is the amount of "opening" or turn between two sides that meet at a vertex. Picture standing at a corner and looking down one side, then swinging your gaze to the other side — how far you swung is the angle.

We measure that swing in degrees (written with a little circle, like $9 0^{\circ}$ ). A full turn all the way around is $36 0^{\circ}$ ; a half-turn is $18 0^{\circ}$ ; a quarter-turn is $9 0^{\circ}$ .

Figure — Pythagorean theorem — proof (by similar triangles, rearrangement), converse

2. The right triangle and its parts

That one square corner gives the three sides special names:

The two sides that form the right angle are the legs. On the parent page these are called $a$ and $b$ .
The side opposite (across from) the right angle is the hypotenuse, always labelled $c$ . It is always the longest side — because it stretches across the widest opening.

Why the topic needs this: the whole theorem $c^{2} = a^{2} + b^{2}$ only works when you know exactly which side is $c$ . Mislabel it and the rule breaks (see Parent → Mistake 2).

3. What "length squared" means — the symbol $a^{2}$

The little raised $2$ means "multiply the thing by itself once": $a^{2} = a \times a$

Why the topic needs this: the parent's [!intuition] callout says the theorem is about areas of squares built on the sides. So $a^{2}$ , $b^{2}$ , $c^{2}$ are not abstract — they are three real squares, and the theorem claims the two small ones tile perfectly into the big one.

4. The square root symbol — undoing a square

Squaring turns a length into an area. To go back from an area to a length we need the opposite operation: the square root, written .

Because a square of area $N$ has side $N$ , taking a root literally reads the side-length off the area square in figure s03.

That is why the parent writes both forms: $c^{2} = a^{2} + b^{2} ⟺ c = a^{2} + b^{2}$

The first gives the area relationship; the second takes the root to recover the actual length of the hypotenuse. This is the distance formula in disguise, and it powers Vectors and Magnitude and the 3D Distance Formula too.

Recall Test yourself: why can't

$c = a^{2} + b^{2}$ ? Because $a^{2} + b^{2}$ has area units (it's the combined square), while $c$ is a length. You must take to turn area back into length. Skipping the root is Parent → Mistake 3.

5. The equals sign, ratios, and the fraction bar

The proof by similar triangles uses ratios like $\frac{b}{c}$ . A ratio just compares two lengths by division: $\frac{b}{c}$ means "how many times does $c$ fit into $b$ ", or " $b$ compared to $c$ ".

Cross-multiplying is the move that clears the fractions: if $\frac{b}{c} = \frac{x}{b}$ , multiply both sides by $c$ and by $b$ to get $b \times b = c \times x$ , i.e. $b^{2} = c x$ . Why do this? Fractions are awkward to add; whole products like $b^{2} = c x$ are easy to combine, which is exactly what Proof 1 does in Step 4.

6. Similar triangles — the engine of Proof 1

The parent uses the AA rule: if two triangles share two equal angles, they must share the third (all angles sum to $18 0^{\circ}$ ), so they are similar. Dropping the perpendicular from the right-angle vertex splits the big triangle into two smaller copies of itself — that is why the ratios in Step 2 hold.

7. Congruent triangles — the engine of the Converse

The SSS rule (Side-Side-Side) says: if all three sides of one triangle match all three sides of another, the triangles are congruent. The Converse proof builds a known right triangle with the same three sides as our mystery triangle, then uses SSS to conclude they're the same triangle — so ours must also have a right angle.

8. How it all feeds the theorem

Read it top-down: angles give you the right angle, which defines the right triangle; squaring plus square-root give meaning to $a^{2}$ and $c$ ; ratios build similar triangles for Proof 1; congruence builds the Converse. Every arrow is a prerequisite you now own.

9. Where these foundations reappear

$c = a^{2} + b^{2}$ becomes the Distance Formula in Coordinate Geometry and the 3D Distance Formula.
Special whole-number solutions like $3, 4, 5$ are Pythagorean Triples.
The right triangle underlies all Trigonometric Ratios.
When the angle is not $9 0^{\circ}$ , you upgrade to the Law of Cosines.
Length-from-components powers Vectors and Magnitude and Complex Numbers.
Everything here lives under the parent: Pythagorean Theorem — Proof & Converse.

Equipment checklist

Cover the right side and see if you can answer each before revealing.

What does a small square drawn in a triangle's corner mean?

That corner is a right angle, exactly

9 0^{\circ}

Which side of a right triangle is the hypotenuse?

The longest side, sitting opposite the right angle (never touching it).

What does

a^{2}

mean, both in numbers and as a picture?

a \times a

; the area of a square whose side has length

a

What question does

N

answer?

"Which positive number times itself equals

N

?" — it undoes squaring.

Why is

c = a^{2} + b^{2}

but not

c = a^{2} + b^{2}

a^{2} + b^{2}

gives the area

c^{2}

; you must take the root to recover the length

c

What makes two triangles similar?

Same angles (same shape), possibly different size — sides in equal proportion.

What makes two triangles congruent, and which rule uses all three sides?

Identical shape and size; the SSS (Side-Side-Side) rule.

What does cross-multiplying turn

\frac{b}{c} = \frac{x}{b}

into?

b^{2} = c x

Which rule handles a triangle whose angle is not

9 0^{\circ}

The Law of Cosines.