Angles-Angles Around a Point
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12-3-2017
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Angles around a point will always add up to 360 degrees.
The angles above all add to 360°
53° + 80° + 140° + 87° = 360°
Because of this, we can find an unknown angle.
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Example: What is angle "c"?
To find angle c we take the sum of the known angles and take that from 360°
Sum of known angles = 110° + 75° + 50° + 63°
Sum of known angles = 298°
Angle c = 360° − 298°
Angle c = 62°
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Angles On One Side of A Straight Line
Angles on one side of a straight line will always add to 180 degrees.
If a line is split into 2 and you know one angle you can always find the other one.
30° + 150° = 180°
Example: If we know one angle is 45° what is angle "a" ?
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Angle a is 180° − 45° = 135°
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This method can be used for several angles on one side of a straight line.
Example: What is angle "b" ?
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Angle b is 180° less the sum of the other angles.
Sum of known angles = 45° + 39° + 24°
Sum of known angles = 108°
Angle b = 180° − 108°
Angle b = 72°
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Interior Angle
An Interior Angle is an angle inside a shape.
Exterior Angle
The Exterior Angle is the angle between any side of a shape, and a line extended from the next side.
Interior Angles of Polygons
An Interior Angle is an angle inside a shape.
Triangles
The Interior Angles of a Triangle add up to 180°
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90° + 60° + 30° = 180°
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80° + 70° + 30° = 180°
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It works for this triangle!
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Let's tilt a line by 10° ...
It still works, because one angle went up by 10°, but the other went down by 10°
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Quadrilaterals (Squares, etc)
(A Quadrilateral has 4 straight sides)
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90° + 90° + 90° + 90° = 360°
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80° + 100° + 90° + 90° = 360°
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A Square adds up to 360°
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Let's tilt a line by 10° ... still adds up to 360°!
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The Interior Angles of a Quadrilateral add up to 360°
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Because there are Two Triangles in a Square
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The interior angles in this triangle add up to 180°
(90°+45°+45°=180°)
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... and for this square they add up to 360°
... because the square can be made from two triangles!
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Pentagon
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A pentagon has 5 sides, and can be made from three triangles, so you know what ...
... its interior angles add up to 3 × 180° = 540°
And if it is a regular pentagon (all angles the same), then each angle is 540° / 5 = 108°
(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)
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The Interior Angles of a Pentagon add up to 540°
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The General Rule
Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:
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If it is a Regular Polygon (all sides are equal, all angles are equal)
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Shape
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Sides
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Sum of
Interior Angles
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Shape
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Each Angle
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Triangle
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3
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180°
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60°
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Quadrilateral
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4
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360°
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90°
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Pentagon
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5
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540°
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108°
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Hexagon
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6
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720°
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120°
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Heptagon (or Septagon)
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7
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900°
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128.57...°
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Octagon
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8
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1080°
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135°
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Nonagon
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9
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1260°
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140°
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...
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...
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..
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...
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...
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Any Polygon
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n
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(n-2) × 180°
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(n-2) × 180° / n
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So the general rule is:
Sum of Interior Angles = (n-2) × 180°
Each Angle (of a Regular Polygon) = (n-2) × 180° / n
Perhaps an example will help:
Example: What about a Regular Decagon (10 sides) ?
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Sum of Interior Angles
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= (n-2) × 180°
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= (10-2)×180° = 8×180° = 1440°
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And it is a Regular Decagon so:
Each interior angle = 1440°/10 = 144°
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Exterior Angles of Polygons
The Exterior Angle is the angle between any side of a shape,
and a line extended from the next side.
Note: when you add up the Interior Angle and Exterior Angle you get a straight line, 180°.
Polygons
A Polygon is any flat shape with straight sides
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The Exterior Angles of a Polygon add up to 360°
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In other words the exterior angles add up to one full revolution
(Exercise: try this with a square, then with some interesting polygon you invent yourself.)
Note: This rule only works for simple polygons
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Here is another way to think about it:
Each lines changes direction until you eventually get back to the start:
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