Grade by Grade
E. Spatial Sense · Shapes, Maps and Measurement

Congruent Shapes: Same Size, Same Shape

E1.3About 30 minutes
Learning goals
  • I can tell if two sides are the same length by matching them.
  • I can tell if two angles are the same size by matching them.
  • I can decide if two shapes are congruent, even when one is turned or flipped.
  • I can explain why two shapes are or are not congruent.

Words to know

congruent
Exactly the same size and the same shape. Congruent shapes fit perfectly on top of each other.
match
To place one side or angle on top of another to see if they are the same.
turn
To rotate a shape around, like a wheel. Turning does not change its size or shape.
flip
To turn a shape over, like a pancake. Flipping does not change its size or shape.

Twins!

Take two cookies from the same box, or two crackers. Put one on top of the other. Do they fit exactly? Then they are congruent!

Congruent (say it: con-GROO-ent) means the same size and the same shape.

Two round, plain butter cookies side by side on a white plate.
These two cookies are the same size and the same shape. If you put one on top of the other, they would fit exactly, so they are congruent. AI-generated image

How to check

To find out if two shapes are congruent, try to put one exactly on top of the other.

  • You can turn it.
  • You can flip it over.
  • You cannot stretch it, shrink it or bend it.

If it covers the other shape exactly, with nothing sticking out, the shapes are congruent.

ABCA and B: congruentC: not congruent
Shape B is shape A turned. They are congruent. Shape C has the same shape but is smaller, so it is not congruent.

Matching sides and angles

Congruent shapes have all the same parts:

  • every side matches a side of the same length, and
  • every angle matches an angle of the same size.

You can match sides by laying a strip of paper along one side, marking its length, and checking it on the other shape. You can match angles by tracing an angle on thin paper and laying it on the other angle. Or you can just picture it in your head: imagine turning the shape until it matches.

all sides 5 cmsquare cornersall sides 5 cmno square cornersSame side lengths, different angles: NOT congruent
Both shapes have 4 sides that are 5 cm long. But the angles are different, so they are not congruent.

Check sides AND angles

Matching sides is not enough. A square and a pushed-over rhombus can have the very same side lengths. Their angles are different, so they do not fit on top of each other.

Worked example

Example 1: Are they congruent?

Look at triangles P, Q and R. P and Q look like mirror images. R looks like P but bigger. Which triangles are congruent?

  1. Compare P and Q. Imagine flipping Q over. It lands exactly on P. Every side and angle matches.
  2. Compare P and R. The angles look the same, but R's sides are longer. If you put P on R, R sticks out.
  3. Flipping is allowed. Making bigger is not.

P and Q are congruent. R is not congruent to them because it is bigger.

PQRWhich triangles are congruent?
Triangles P, Q and R. P and Q match when Q is flipped. R is larger.
Worked example

Example 2: Matching on a grid

Ava draws a rectangle 3 squares long and 2 squares tall. Ben draws a rectangle 2 squares long and 3 squares tall. Are they congruent?

  1. Count Ava's sides: 3, 2, 3, 2.
  2. Count Ben's sides: 2, 3, 2, 3.
  3. Turn Ben's rectangle on its side. Now it is 3 long and 2 tall, just like Ava's. All the corners are square corners in both.

Yes, they are congruent. Ben's rectangle is just Ava's rectangle turned.

Just for fun

What did one cracker say to its congruent twin?

"Stop copying me!"

Hands-on activity

Congruent or not?

You need: paper, scissors, a pencil, and some flat things from home (crackers, coins, sticky notes, playing cards, puzzle pieces).

  1. Collect pairs of things. For each pair, guess: congruent or not? Then check by stacking them.
  2. Fold a sheet of paper in half and cut out a shape through both layers. You made 2 congruent shapes! Turn one over. Do they still match?
  3. Draw a triangle. Ask a grown-up to draw a triangle congruent to yours without tracing. Check by cutting out both and stacking them.
  4. Try coins: a loonie and a toonie. Are they congruent? (No. The toonie is bigger and the loonie has 11 sides.)

Math talk: "How do you know they are congruent? Which parts did you match?"

Your turn

1

What does congruent mean?

2

Two squares have sides of 4 cm. Are they congruent?

3

A big triangle and a small triangle have the same shape. Are they congruent?

4

If you turn a shape upside down, is it still congruent to the shape it was before?

5

Two sheets of paper come from the same pack. Are they congruent?

6

Two rectangles both have long sides of 6 cm. One has short sides of 2 cm and one has short sides of 3 cm. Are they congruent?

More practice: warm-up

1

Are two loonies congruent?

2

Are a toonie and a dime congruent?

3

Can you flip a shape when you check for congruence?

4

Can you stretch a shape to make it match?

5

How can you check if two cookies are congruent?

More practice: on your own

1

Two triangles each have sides of 3 cm, 4 cm and 5 cm. One is pointing up and one is pointing down. Are they congruent? Explain.

2

A square and a rhombus both have all sides 5 cm. Are they congruent? Explain.

3

Leo cut 2 shapes from folded paper (both layers at once). Are the 2 shapes congruent? Why?

4

On grid paper, shape A covers 6 squares and shape B covers 8 squares. Could they be congruent?

5

Two shapes both cover 6 squares on grid paper. Are they for sure congruent?

Challenge

1

How many congruent triangles do you get if you cut a square in half from corner to corner? How do you know they are congruent?

2

Draw a rectangle on grid paper that is 4 squares long and 1 square tall. Draw a different rectangle that covers the same number of squares. Are they congruent?

3

Is every hexagon congruent to every other hexagon? Explain.

Check your understanding

Quiz

1.

Congruent shapes are...

2.

Which can you do when checking if shapes are congruent?

3.

A triangle is turned upside down. Is it congruent to how it was before?

4.

Two squares have sides of 3 cm and 5 cm. Are they congruent?

5.

A square and a rhombus both have 4 sides of 5 cm. Why are they not congruent?

6.

Which pair is congruent?

For parents and teachers

Big idea: Congruent shapes are identical in size and shape. They can be turned or flipped but not stretched or shrunk. Children check by matching sides and angles, physically (stacking, tracing) or in their heads.

Common difficulties:

  • Thinking a turned shape is different. Cut out the shape and physically turn it so they can see it is the same.
  • Thinking shapes that look alike are congruent even if sizes differ. Stack them so the extra part shows.
  • Checking only the sides. Show a square and a "squashed" square (rhombus) made of the same 4 straws.

How to help: Stack crackers, coins, cards and puzzle pieces. Ask "Are these congruent? How could we check?" Fold-and-cut paper shapes always make congruent pairs.