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Maths

Congruent and similar shapes

Congruent shapes

If two shapes are congruent, they are identical in both shape and size.

Remember: Shapes can be congruent even if one of them has been rotated or reflected.

image: various shapes labelled with letters

Question

Which of the shapes in the illustration above are congruent?

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Answer

Did you get the following pairs?

  • A and G
  • D and I
  • E and J
  • C and H

Remember: Shapes can be congruent even if one of them has been rotated (as in A and G) or reflected (as in C and H).

The symbol identical means 'is congruent to'.

Two triangles are congruent if one of the following conditions applies:

1. Three sides are the same

The three sides of the first triangle are equal to the three sides of the second triangle (the SSS rule: Side Side Side).

Two triangles, left-side and right-side of both triangles are equal in length. Bottom lengths is of both triangles are equal in length

2. Two sides and one angle are the same

Two sides of the first triangle are equal to two sides of the second triangle, and the included angle is equal (the SAS rule: Side Angle Side).

image: Two triangles, the right-side lengths of both triangles are equal to each other. The left-sides of both triangles are equal to each other. The top corner angle of both triangles equal to 80 degrees.

3. Two angles and one side are the same

Two angles in the first triangle are equal to two angles in the second triangle, and one (similarly located) side is equal (the AAS rule: Angle Angle Side).

image: Two triangles, left-side of both triangle are equal to each other. Top corners of both triangles both equal 105 degrees.

4. Two sides in right-angled triangle are the same

In a right-angled triangle, the hypotenuse and one other side in the first triangle are equal to the hypotenuse and corresponding side in the second triangle (the RHS rule: Right-angled, Hypotenuse, Side).

image: two triangles, bottom corners of both triangles equal 90 degrees, right-side lengths of both triangle equal to each other. Left-side lengths of both triangles equal to each other.

Question

For each of the following pairs of triangles, state whether they are congruent. If they are, give a reason for your answer (SSS, SAS, AAS or RHS).

Pair 1

image: two triangle, top corner of left triangle is 90 degress, right side of triangle: 30mm, bottom of triangle: 58mm. Right triangle: right side: 5.8cm, left side: 3cm, bottom left corner: 90 degress

Pair 2

image: two triangles, left triangle: right-side length: 4cm, bottom length: 5.5cm, left-side length: 3cm. Right triangle: top length: 5.5cm, right-side length: 3cm, left-side: 4cm

Pair 3

image: two triangles: left triangle: bottom right corner: 60 degrees, bottom length: 7cm, bottom left corner: 40 degrees. Right triangle: top left corner: 60 degrees, left length: 7cm, top right corner: 40 degrees.

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Answer
  1. Yes. RHS
  2. Yes. SSS
  3. No. The side of length 7cm is not in the same position on both triangles. Therefore, it is not AAS.

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