PROOF 148

Prove.

Translations and reflections preserve lengths and angles

On the Euclidean plane.

What you may use

A1d² = (Δx)² + (Δy)²
distance between two points (Proof 21) — stands on the Pythagorean theorem
A2three equal sides ⟹ congruent
SSS congruence condition (Euclid I.8)
A3congruent ⟹ corresponding angles equal
definition of congruence
A4Q between ⟺ PQ+QR = PR
betweenness and addition of segments
A5an angle is determined by two rays
definition of an angle — the same angle whichever points on the rays are chosen
A6collinear ⟺ one is between
collinearity test for three distinct points
A7d,e ≥ 0, d² = e² ⟹ d = e
uniqueness of the square root
A8(a+b)+c = a+(b+c)
associativity of addition
A9a + b = b + a
commutativity of addition
A10a + (−a) = 0, a + 0 = a
additive inverse and identity
A11a−b=a+(−b), −(−a)=a
convention of subtraction and signs
A12−(a+b) = (−a) + (−b)
distribution of the sign
A13(−a)(−b) = ab
sign rule (Proof 1)
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

Fix two transformations of the coordinate plane. For real numbers a, b, call T(x, y) = (x + a, y + b) a translation, and call σ(x, y) = (x, −y) the reflection in the x-axis.

Proof.

Related labs

Symmetry & Transformations lab →