Similarity Updates

Similar figures and congruent figures both have the same shape.

Two triangles are congruent if they are the same size and shape.

Congruency:They are exactly same in length, width, shape, size etc.

They are similar if they are the same shape (and maybe or maybe not the same size).

Also remember:

i. Every congruent pair of shapes is similar also.

ii. Not all similar figures are congruent though.

iii. The areas of two similar figures may be different.

iv. However, the areas of two congruent figures are equal.

v. Ratio of Corresponding Sides:

a. The ratios between the corresponding sides of two similar figures are equal.

b. The ratios between the corresponding bases of two congruent figures are always one.

vi. Symbol:

a. The symbol ≅ is for Congruent.

b. The symbol ~ is for Similar.

vii. Examples:

You went to a garment store to buy shirts. You liked one with black strips. You saw they were kept in various sizes: S, M, L, XL, XXL…

Now these shirts of different sizes are similar because they have the same shape.

But 2 shirts of the same size are congruent. They are exactly same in length, width, shape, size etc.

Two triangles are said to be similar to each other when they have same shape, proportion and angles but different sizes

Congruence on the other hand means when two things are exactly identical to each other.

The difference between similar figures and congruent figures is that congruent figures also have the same size. So, corresponding sides and corresponding angles of congruent figures have the same measures.

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The SSS rule states that:- If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ. Side - Side - Side (SSS) Rule Side-Angle-Side (SAS) Rule Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent. The SAS rule states that :- If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent. An included angle is an angle formed by two given sides. Included and Non-included angle:- Included Angle Non-included angle For the two triangles below, if AC = PQ, BC = PR and angle C = angle P , then using the SAS rule, triangle ABC is congruent to triangle QRP congruent triangles Angle-Side-Angle (ASA) Rule Angle-side-angle is a rule used to prove whether a given set of triangles are congruent. The ASA rule states that If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent. Angle-Angle-Side (AAS) Rule Angle-angle-side is a rule used to prove whether a given set of triangles are congruent. The AAS rule states that If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent. In the diagrams below, if AC = QP, angle A = angle Q, and angle B = angle R, then triangle ABC is congruent to triangle QRP. read less

There are the following steps to prove that two triangles are similar:- 1. AAA Test (angle angle angle): All three pairs of corresponding angles are the same. 2. SSS Test (side side side): All three pairs of corresponding sides are in the same proportion. 3. SAS Test (side angle side): Two pairs of sides are in the same proportion and the angle between them is equal.

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