Translation, reflection, and rotation preserve every length and every angle.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Combining them
Two reflections in parallel lines make a translation. Two in intersecting lines make a rotation.
Symmetry is a transformation that changes nothing
A figure has line symmetry if a reflection maps it onto itself.
Order matters
A reflection then a rotation is generally not the same as a rotation then a reflection.
What rigid motions preserve
Translations, reflections and rotations preserve distance and angle. Anything defined purely in terms of distances and angles is therefore unchanged by them, which is why congruence can be defined through these motions.
Orientation is the one thing that can flip
Reflections reverse orientation; translations and rotations do not. A figure and its mirror image are congruent but cannot be slid onto each other in the plane. This matters for chirality in chemistry as much as in geometry.
Symmetry is a motion that maps a figure to itself
A square has four rotational symmetries and four reflective ones. Symmetry is not a vague aesthetic property here — it is the precise set of rigid motions leaving the figure exactly where it was.
Compositions are still rigid
Any sequence of rigid motions is itself a rigid motion. In fact every rigid motion of the plane is a composition of at most three reflections, which is a surprisingly economical description of all possible movements.
Step 2: Try It Yourself
Tap and try it out.
Every length and every angle is exactly as it was. The shape moved without changing, so the two figures are congruent.
Step 3: Watch an Example
One step at a time.
Watch Sam Compose Two Reflections
Sam reflects a shape in one vertical line, then in a second parallel to it.
- Step 1
The first reflection flips the shape and moves it across the line.
Step 4: Your Turn
Practice makes it stick.
The Square
Problem 1 of 2
How many lines of symmetry does a square have?
The Rotation
Problem 2 of 2
A regular hexagon maps onto itself after a rotation of how many degrees, at minimum?
degrees
Move and Match
1 of 4
Lines of symmetry in an equilateral triangle?
2 of 4
A regular pentagon maps onto itself after how many degrees, at minimum?
3 of 4
Do rigid motions change side lengths?
4 of 4
Lines of symmetry in a rectangle that is not a square?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A regular octagon maps onto itself after how many degrees, at minimum?
Question 2 of 2
Two reflections in parallel lines compose to what?
What You Learned
- Rigid motions preserve lengths and angles.
- Two parallel reflections make a translation.
- Symmetry is a transformation that maps a figure onto itself.