[G.CO.3] Congruence #3

Objective

Common Core Text:

• [G.CO.3] Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Said Differently:

• • For rectangles, parallelograms, trapezoids, and regular polygons, determine any

  • rotational symmetry

  • line symmetry

Example

For this rectangle

a) draw all lines of symmetry

b) name the angle of rotational symmetry, if it has one

Answer:

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Explanation

There are two types of symmetry

• Line symmetry

  • A shape has line symmetry if there's a line you can draw that:

    • cuts the shape in half, and

    • reflecting one half across the line creates the other half

In this shape of a moon, you can see that our line cuts the picture in half. Also, reflecting one half across this line gives the other half. Therefore:

This moon has line symmetry

The line is called a "line of symmetry"

• Rotational Symmetry

  • A shape has rotational symmetry if you can rotate it around so that the new shape looks exactly like the original shape

In this video, you can see that if we rotate the star 90°, it looks the same as the original shape. Therefore:

This star has rotational symmetry

The angle (90°) is called a "angle of rotational symmetry"

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Now we'll go through all 4 shapes. Remember, instead of trying to memorize the lines and angles, try to understand why.

Rectangle

A rectangle has 2 lines of symmetry

Some people think rectangles have lines diagonal lines of symmetry. Let's use GeoGebra to reflect across a diagonal line and see what happens

Original Shape

Half

Here's what happens when you reflect a rectangle across the diagonal. You can see it does not create the original shape, so a diagonal line is not a line of symmetry in a rectangle.

A rectangle does have rotational symmetry

As you can see, the angle of rotational symmetry for any rectangle is 180°

Parallelogram

A parallelogram no lines of symmetry

Don't be fooled by this line. Let's use GeoGebra to reflect and see what happens

Original Shape

Half

Here's what happens when you reflect across the line. You can see it does not create the original shape, so this is not a line of symmetry. You can use GeoGebra to test other lines to see why they don't work.

A parallelogram does have rotational symmetry

As you can see, the angle of rotational symmetry for any parallelogram is 180°

Trapezoid

A trapezoid has 1 line of symmetry

A trapezoid does not have rotational symmetry

As you can see, one must turn a full 360° before returning to the original shape. That doesn't count. No angles of rotational symmetry.

Regular Polygons

A regular polygon is any shape with equal sides and equal angles. Let's look at the first 3 regular polygons

Let's look for a pattern.

See the pattern? However many sides a regular polygon has, that's how many lines of symmetry it has.

Rotational symmetry is a little harder. Again, let's look at the first three shapes.

Let's look for a pattern.

See the pattern? However many sides a regular polygon has, divide 360° by that number. That's the angle of symmetry for that shape.