By the end of this section you should: ● understand and defi ne the terms axiom and theorem ● be able to fi nd the measure of vertically opposite angles
An axiom is a statement that is taken to be true without being proven. B 360° 180 ° A
Axiom: A full angle is equal to 360°. Axiom: A straight angle is equal to 180°, also called a straight line.
Discuss and discover
Working with a classmate, answer the following questions. (i) What does |∠A| + |∠B| = ? (ii) What does |∠A| + |∠C| = ? (iii) What does this tell you about |∠B| and |∠C|? (iv) How would you use a protractor to verify that your ideas about |∠B| and |∠C| are correct?
What have you discovered about angles during this activity?
Axiom: There is only one straight line through two points.
A
B C
D
Vertically opposite angles are the angles opposite each other when two lines cross.
The word ‘vertical’ in this case means they share the same vertex (corner point), not the usual meaning of up-down.
Vertex P C M N
∠A is vertically opposite to ∠C Recall from naming angles:
∠ OQN is another way of naming ∠ A using points In the diagram above:
|∠A | = |∠C| or
|∠OQN | = |∠PQM | 170 Linking Thinking 1 Q A O
Vertex P
M Q B N
∠B is vertically opposite to ∠D ∠ PQM is another way of naming ∠ C using points
In the diagram above:
|∠B | = |∠D | or
|∠MQN | = |∠PQO | D O
For vertically opposite angles look for an ‘X’ shape.