Lines and Angles Class 9: notes and important questions

Lines and Angles: revision notes

1. Rays and angles

A ray \(OA\) starts at \(O\) and goes on for ever through \(A\). An angle \(\angle AOB\) is formed by two rays \(OA\) and \(OB\) with the same starting point \(O\) (the vertex). A straight angle is \(180^\circ\), two right angles; a right angle is \(90^\circ\).

• Acute: between \(0^\circ\) and \(90^\circ\); right: \(90^\circ\); obtuse: between \(90^\circ\) and \(180^\circ\); straight: \(180^\circ\); reflex: between \(180^\circ\) and \(360^\circ\).
• The reflex angle of \(\theta\) is \(360^\circ - \theta\).

2. Pairs of angles

• Complementary: add to \(90^\circ\). Supplementary: add to \(180^\circ\).
• Adjacent angles share a vertex and an arm and lie on opposite sides of it.
• Linear pair: two adjacent angles whose outer arms form a straight line.

3. Intersecting lines

Linear pair theorem. If a ray stands on a line, the two adjacent angles it makes add to \(180^\circ\). Converse. If two adjacent angles add to \(180^\circ\), their outer arms lie on one straight line.

Vertically opposite angles are equal. If lines \(AB\) and \(CD\) cross at \(O\): \(\angle AOC + \angle AOD = 180^\circ\) and \(\angle AOD + \angle BOD = 180^\circ\) (linear pairs), so \(\angle AOC = \angle BOD\) (equals subtracted from equals).

4. Proof by contradiction

To prove a statement, suppose it is false and show that this leads to something impossible. Example: two distinct lines cannot meet in two points, because then two lines would pass through the same two points, but two points fix only one line.

5. Parallel lines and a transversal

If two parallel lines are cut by a transversal, then corresponding angles are equal, alternate interior angles are equal, and co-interior angles (interior angles on the same side) add up to \(180^\circ\). The converses are also true: if corresponding (or alternate) angles are equal, or co-interior angles are supplementary, the lines are parallel.

Transitivity. Lines parallel to the same line are parallel to each other: if \(l \parallel m\) and \(m \parallel n\), then \(l \parallel n\).

6. Angles of a triangle

• The angles of a triangle add up to \(180^\circ\): draw a line through one vertex parallel to the opposite side and use alternate angles.
• An exterior angle equals the sum of the two interior opposite angles.
• A triangle cannot have two right angles or two obtuse angles (two such angles already make \(180^\circ\) or more), so it has at least two acute angles.
• Its angles cannot all be less than \(60^\circ\) (the sum would be under \(180^\circ\)), and cannot all be more than \(60^\circ\).

Worked example 1

Lines \(AB\) and \(CD\) meet at \(O\), and \(\angle AOC = 48^\circ\). Find \(\angle BOD\), \(\angle AOD\) and \(\angle BOC\).

\(\angle BOD = 48^\circ\) (vertically opposite); \(\angle AOD = 180^\circ - 48^\circ = 132^\circ\) (linear pair); \(\angle BOC = 132^\circ\) (vertically opposite).

Worked example 2

Two parallel lines are cut by a transversal; one co-interior angle is \(3x^\circ\) and the other is \(2x^\circ\). Find \(x\).

\(3x + 2x = 180 \Rightarrow x = 36\).

Worked example 3

The angles of a triangle are in the ratio \(1 : 3 : 5\). Is the triangle acute, right or obtuse?

\(9k = 180^\circ\), \(k = 20^\circ\): the angles are \(20^\circ\), \(60^\circ\), \(100^\circ\). One angle is obtuse, so the triangle is obtuse.

Common errors

• Calling co-interior angles equal: they are supplementary.
• Using parallel-line facts when the lines are not given as parallel.
• Taking the reflex angle of \(\theta\) as \(180^\circ + \theta\) instead of \(360^\circ - \theta\).
• Writing angle facts without the reason in brackets.

Exam tips

• Write each step with its reason: "(vertically opposite)", "(linear pair)", "(alternate angles, \(AB \parallel CD\))".
• When a point lies between two parallel lines, draw a third line through it parallel to both.