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5.2 Explaining angle properties

5.2 Explaining angle properties

The angles of a triangle add up to 180 °. How do you know that is true?
You can measure the angles of a triangle and add them up. But that will
only show it is true for one triangle. How can you show it is true
for all triangles?
You can prove it by using the properties of parallel lines that you learned in
the previous topic.
Proof 1
To prove that the angles of triangle ABC add up to 180°. A

B C
A Y Extend BC to X and draw CY parallel to BA.
a ° Then a = d (alternate angles)
and b = e (corresponding angles)
c + d + e = 180 ° (angles on a straight line)

d ° ⇒ c + a + b = 180 °. ⇒ means ‘therefore’
b ° c ° e °
B C X This is the required result.

This is a proof that the angles of any triangle add up to 180 °. It gives a reason
for each line. You do not need to be able to write proofs in this way, but you
should understand them.
A
The example also proves another important fact.
Angle ACX is called the external angle of the triangle at C.
The proof showed that d + e = a + b.
This means that the external angle at C is equal to the sum of the interior
angles at A and B. This result is true for any triangle. B C X
The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

54 5 Angles