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#Sas similarity how to#

Thirdly, since the corresponding angles are equal, each side of the image triangle is parallel to the matching side of the original triangle:Ī ′ B ′ || AB, B ′ C ′ || BC and C ′ A ′ || CA. Secondly, we can verify that each angle of A ′ B ′ C ′ is equal to the matching angle of ABC.ī ′ A ′ C ′ = BAC, A ′ C ′ B ′ = ACB and C ′ B ′ A ′ = CBA. We can use this diagram to verify three important properties of enlargements.įirst, it is easy to verify with compasses that each side of the image triangle A ′ B ′ C ′ is twice the length of the matching sides of the original triangle ABC.

Join up the triangle A ′ B ′ C ′, which is called the image of ABC.

Join OC, then extend OC to OC ′ so that OC ′ = 2 × OC.

Join OB, then extend OB to OB ′ so that OB ′ = 2 × OB.

Join OA, then extend OA to OA ′ so that OA ′ = 2 × OA.

Because the construction only involves doubling, it can be done with straight edge and compasses.

#Sas similarity how to#

The next figure below shows how to construct the image A ′ B ′ C ′ of ABC under the enlargement with centre O and enlargement factor 2. For example, the diagram below shows a point O and a triangle ABC. In this module, we will only deal with positive enlargement factors. The distances of all points from the enlargement centre increase or decrease by this factor.

An enlargement factor k (or enlargement ratio 1 : k).

The centre of enlargement stays fixed in the one place, while the enlargement expands or shrinks everything else around it. To specify an enlargement, we need to specify two things: This section introduces a fourth type of transformation of the plane called an enlargement, in which all lengths are increased or decreased in the same ratio. Indeed, we defined two figures to be congruent if one could be mapped to the other by a sequence of these transformations. These three transformations are examples of congruence transformations, because the image of a figure under one of these transformations is congruent to the original. We have already dealt with three transformations of the plane − translations, rotations and reflections. In Section 4−6, the discussion of similar triangles begins with the AAA similarity test, which is usually considered the most straightforward test to use. Section 3 can then introduce similarity in terms of enlargement transformations.

Scale drawings and enlargements are usually discussed a year or so earlier than similarity, and these topics therefore receive a self-contained treatment in Sections 1−2. The treatment of similarity and enlargements in this module has been guided by well-established classroom practice. Secondly, just as congruence was used to prove many basic theorems about triangles and special quadrilaterals, so similarity will allow us to establish further important theorems in geometry. First, most situations involving similarity can be reduced to similar triangles, and we shall establish four similarity tests for triangles, corresponding to the four congruence tests for triangles. The theory of similarity develops in the same way as congruence.

This constant ratio is the same ratio that appears in scale drawings and enlargements. Matching angles in similar figures are equal, but matching lengths in two similar figures are all in the same ratio. Thus two figures are similar if an enlargement of one is congruent to the other.Īny two figures that have the same shape are similar. Figures that can be mapped one to the other by these transformations and enlargements are called similar. The module, Congruence studied congruent figures, which are figures that can be mapped one to the other by a sequence of translations, rotations and reflections.

This is different from the three transformations that we have already introduced − translations, rotations and reflections all produce an image that is the same size and shape as the original figure. An enlargement transformation preserves the shape of the figure, but increases or decreases all distances by a constant ratio. The transformation that produces a scale drawing is an enlargement. It is usually expressed in terms of a ratio, so the topic of scale drawings is closely related to ratios and fractions. The proportional increase or decrease in lengths is called the scale of the drawing.

For example, we would want to reduce the size when drawing:Īnd we would want to increase the size when drawing: Scale drawings are used when we increase or reduce the size of an object so that it fits nicely on a page or computer screen.