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Angles

Angles. Definitions: Angles are a way of measuring direction or turn One full turn is divided into 360 equal parts One degree is of a full turn. Why is it 360 parts? Why not 100? Or 6.2831853071...?. Angles around a point add up to 360°. Proof:

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Angles

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  1. Angles Definitions: Angles are a way of measuring direction or turn One full turn is divided into 360 equal parts One degree is of a full turn Why is it 360 parts? Why not 100? Or 6.2831853071...?

  2. Angles around a point add up to 360° Proof: Turning 360° is the same as turning one complete turn, so any angles that complete a full turn must have a total of 360°. Degrees are fractions of a turn. If you turn of a turn, then , then , you will have completed 1 full turn.

  3. Angles on a straight line add up to 180° Proof: If you turn until you are pointing in the exact opposite direction, you will have turned exactly half of a full turn, which is half of 360°. Remember ‘angles on a straight line’ means at a given point, not just any old angles that happen to be on the same line.

  4. Where straight lines cross, opposite angles are equal Proof: Each angle is on a straight line with the one next to it, so they add up to 180°. This means the one to the left and the one to the right must be the same as each other. Crossing lines are symmetrical, so angles exactly opposite each other must be the same.

  5. Corresponding angles on parallel lines are equal Proof: An angle is just a measure of direction. If two lines cut across your line at the same angle, they must be going in the same direction. This is basically the definition of parallel lines. Another way of saying “goes in the same direction” is “intersects any line at the same angle”

  6. Interior angles on parallel lines add up to 180° Proof: This is just a simple extension of the corresponding angles rule. The top angle and the one directly below must add up to 180° because they’re on a straight line, so the interior angles must also add up to 180° The red and blue angles add up to 180° (this is sometimes called supplementary or complementary)

  7. Alternate angles on parallel lines are equal Proof: Using the straight lines crossing rule we can see that the angles on alternate sides of the crossing line must be equal. When a straight line crosses parallel lines, you only need 1 angle to calculate all the other 7.

  8. Angles in a triangle add up to 180° Proof: Angles on a straight line add up to 180° and alternate angles on parallel lines are equal. This is a fundamental rule that will be extended for polygons, so it’s important that we are convinced it is always true.

  9. Base angles in an isosceles triangle are equal Proof: An isosceles triangle has two sides the same length. If you cut it exactly in half from the top the two halves would be the mirror image of each other. This means the angles at the base of the equal sides must be equal. Once you know one angle in an isosceles triangle, you can work out the other two.

  10. All angles in an equilateral triangle are equal Proof: In an equilateral triangle, all the sides are the same length. This means it must be the same no matter which way round it goes, so all the angles must be the same. Since angles in a triangle always add up to 180°, each angle in an equilateral triangle must be 60°

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