Congruent and similar shapes - Maths Part 2 (Similar triangles)

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We know that two shapes are similar if their corresponding angles are equal, and their corresponding sides in the same proportion. The same is true for similar triangles.

To prove that two triangles are similar, we have to show that one (not all) of the following statements is true:

1. The three sides are in the same proportion.

image: two triangle: left triangle: left-side: 3cm, right-side: 5cm, bottom: 4cm. Right triangle: left-side: 4.5cm, right-side: 7.5cm, bottom: 6cm.  (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_5.gif)

2. Two sides are in the same proportion, and their included angle is equal.

image: two triangles: left triangle: right-side: 5cm, bottom-right corner: 30 degree, bottom: 3cm. Right triangle: right-side: 10cm, bottom-right corner: 30 degree, bottom: 6cm  (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_6.gif)

3. The three angles of the first triangle are equal to the three angles of the second triangle.

image: two triangles: left triangle: top Y corner: 85 degrees, right Z corner: 40 degrees, left corner: X. Right triangle: same labels: Y: 85 degrees, X: 55 degrees. (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_7.gif)

We have only shown two angles in the above triangles. If we calculate the missing angles, we find

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