Angle Reasoning

This chapter explores angle reasoning. It covers using the symmetries of a square, an equilateral triangle and an isosceles triangle to work out triangles. No prior knowledge is required for this chapter.

Folding a square in half

Like all squares such as that shown below have equal sides and four 90° angles.
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If we folded the square along the diagonal as shown below we make two equal triangles. The triangles are identical or congruent because the square is symmetrical and also because the triangle have the same lengths.
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The short sides are all part of a square. They share the longest side (hypotenuse). The angle shown below is 45°.
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Folding an equilateral triangle in half

Below is an equilateral triangle.
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The triangle has three sides and three equal angles. An equilateral triangle has all angles equal to 60°, and add up to 360°.
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Suppose we fold the equilateral triangle in half.
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The red line in the middle of the triangle is the line of symmetry. There are two triangles formed once the equilateral triangle is folded in half. The angle BAD is 30° and ABD is 60°.
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As we can see above each triangle has an angle of 90° and they are right angled triangles.

Folding an Isosceles triangle in half

The triangle below is an Isosceles triangle.
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The Isosceles triangle has two equal sides and two equal angles at the bottom. The angles inside the triangle must add up to 180°. That must mean that the rest of the angle is 30°.
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Suppose we folded the triangle in half as shown below.
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The two triangles formed above are congruent or identical. The vertical line in half is the line of symmetry.
The size of the angle CAD is 15° because the top angle BAC is 30° and we have just halved it.
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Each triangle has a 90° angle and they are right angled triangles.