Angle Proofs

This chapter explores understanding and proving angles. At the end of this chapter you should have good understanding and be able to prove that; ‘angles in a triangle add up to 180°‘, ‘angles in a quadrilateral add up to 36°’ and an exterior angle of a triangle equals the sum of the opposite interior angles.
This chapter offers the explanations and proof as to why some angle theorems work. Below is some of the angle theorems this chapter explores.
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Proof of angles in a triangle

Below we shall show proof of “the angles in a triangle add up to 180°”
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First extend the 3 edges to form 3 exterior angles as shown below;
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Imagine reducing the size of the triangle such that it disappears, you will get;
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Above the triangle has been reduced to 3 angles around a point.
You must know that angles around a single point add up to 360°. If we look back at the original triangle to have a look at the interior angles.
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The pair of angles labelled add up to 180°. This is the same for angles at B and C.
So the total for all 6 angles;
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We used 360° for the exterior angles.
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…that leaves 180° for the interior angles. We have proved that the angles in a triangle add up to 180°.
Above we have managed to prove that the angles in a triangle add up to 180°.

Analysis

Let us have a quick summary of what we have explored above. Here is the same triangle below.
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First we extend the lines and label the angles.
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Let’s add up all the angles around a point.
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…angles on a straight line…
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…that means…
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That proves that the angles in a triangle add up to 180°

Angles in a triangle PROOF 2

This second proof also proves that “angles in a triangle add up to 180°” Below is the triangle in the previous example.
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Let’s extend the bottom edge.
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Next we draw a line through the top corner parallel to the base line.
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The unknown angle above must equal to angle a because they are alternate angles on the parallel lines.
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The unknown angle above must be equal to angle c because they are alternate angles on the parallel lines.
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The angles a, b, and c now lie on a straight line that must mean that they add up to 180°, so we have;
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The angles in the triangle add up to 180°.

Angles in a quadrilateral

Below we shall prove that “the angles in a quadrilateral add up to 360°” Below is the quadrilateral.
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First we extend the 4 sides and label all the angles as shown below.
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The angles shaded below add up to 360° since angles around a point add up to 360°. That is;
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Each pair of angles on the straight line add up to 180°.
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The total of all 8 angles on the straight line is 4 lots of 180°, that is;
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If we subtract off 360° for the exterior angles 360° is left. The exterior angles are;
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That is proof that the angles in a quadrilateral add up to 360°.

Interior and exterior angles in a triangle

This section explores the theorem which states;
“the exterior angle of a triangle is equal to the sum of the opposite interior angles” Below is a triangle with labelled angles.
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The angles a+b is equal to angle at b.
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The angles at c and d are on a straight line that must mean.
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…that must also mean that;
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The angles in a triangle add up to 180°, that is;
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…so that;
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a+b is also equal to 180°-c
…that must mean that d is equal to a+b as shown below.
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The above proves that “the exterior angle of a triangle is equal to the sum of the opposite interior angles”