In Mathematics, equal angles are often shown by using the same mark in the angles (as shown below), usually single or double dashes, or dots.
P O M N M N
A theorem is a statement that can be proven to be true in a series of logical steps using axioms and/or other theorems.
A proof contains a series of logical mathematical arguments (steps) used to show the truth of a theorem.
In a proof, we can use: ● ●
axioms such as ’a straight angle is equal to 180°’ existing theorems
The result of a proof is often called a theorem. Theorem (1) Vertically opposite angles are equal in measure
Proof of the theorem: Vertically opposite angles are equal in measure Supplementary angles (two angles that sum to 180 ° ) are used in this proof. Line 1 |∠A| + |∠B| = 180 ° (straight angle)
Similarly, it can be show that |∠B| = |∠C| (after subtracting line 2 from line 1) |∠A| = |∠D| (which are vertically opposite)
For Junior Cycle Mathematics, students should display an understanding of the proofs of theorems (full formal proofs are not examinable).
Worked example
Using the diagram on the right, and given that |∠C| is equal to 120 ° , fi nd the value of the following and give the axiom or theorem used in each part.
( i ) ( i i )
|∠A| |∠D|
Solution (ii)
( i i i ) ( i v )
|∠B| |∠A| + |∠B| + |∠C| + |∠D| B 120°
A C
D P O
(iii) (iv)
Section A Introducing concepts and building skills