Are All Congruent Figures Similar

All congruent figures are similar, but not all similar figures are congruent. Congruence means 2 objects (whether two dimensional or 3 dimensional) are identical in size and shape. Everything nigh them -- their angles, lengths of sides, overall dimensions -- are identical. Similar figures accept the aforementioned shape and proportions but are not necessarily the same size.

Congruence
Try It!
Geometry Try It!
Similarity
Using Congruence and Similarity

Congruence

Two objects tin can be the same size and shape but not be oriented the same way. They are withal coinciding, similar these bounding main stars:

[insert 2 copyright-costless images of the same sea star {starfish} merely rotate one moving picture 36° and then its spines are oriented differently]

Or take these chess pieces. I knight is on a high shelf, the other on a depression shelf. But because they are in different planes in three dimensions does non dominion out their congruence. They are still congruent:

[insert drawing of bookshelf with knights {or pawns or whatever chess pieces} on two different shelves]

In geometry, congruent figures have three properties:

Aforementioned size
Aforementioned shape
Corresponding parts are congruent

That last part is why, in geometry proofs, we sometimes see CPCFC, which means, "Respective parts of congruent figures are congruent."

Effort It!

Which pairs of figures below are coinciding?

[in a ii-by-two array, insert two identical copyright-gratis images of a body of water urchin; ii pictures, one smaller than the other, of a sand dollar; two identical images of a clownfish; and pictures of a lobster and a crab]

The bounding main urchins and clownfish are congruent. The sand dollars, though the aforementioned shape, are not the same size. They are similar. The crustaceans are two completely different animals then they are not congruent or similar.

Geometry Try It!

Which pairs of figures beneath are coinciding?

[In a two-by-two assortment: a square and a rectangle; ii images of identical regular pentagons; {2d row} two identical equilateral triangles but one rotated 90°; and two identical circles]

The square and rectangle are not congruent. The pairs of pentagons, triangles and circles are the same size and shape. They accept congruence. They are congruent.

Similarity

Similarity means the same shape and proportions, but not necessarily the aforementioned size. Angles of similar figures volition be equal, but lengths of sides ordinarily are not equal.

Pairs of shapes that are coinciding are automatically similar, simply this relationship does not work in opposite. All congruent figures are similar, but not all like figures are congruent:

[insert a three-by-one array: cartoon of two like rectangles one larger than the other; two similar circles 1 larger than the other; and two congruent squares]

Both rectangles have the same proportions. All circles are similar. Neither pair of rectangles or circles is congruent, though. Only the squares, being congruent, are besides similar to each other.

Using Congruence and Similarity

Knowing the backdrop of congruence and similarity allows you to use them in proofs. You can establish ratios between corresponding parts of ii similar figures, like this:

Congruence and Similarity Figure 1

Here the ratios of width to length are the same:

three cm/5 cm = i.five cm/2.5 cm

The ratios of corresponding parts are also the same:

five cm/5 cm = 1.5 cm/3 cm

Using these ratios and the congruence of angles, yous know the shapes are similar. They are the same shape and are proportional to each other. You lot can utilise similarity in more challenging proofs:

Congruence and Similarity Figure 2