Angle Chase

Angle Chase

Angle questions are marked in two halves, and most candidates collect only the first. One mark is for the number. The second is for the reason, written in the accepted wording: not "because I worked it out" but "angles on a straight line add to 180°". The reason is not decoration and it is not showing off. On a multi-step chase it is the only way an examiner can see which rule you used at which point, and it is the half most often thrown away. Every question in this module asks for both.

The facts, and how to say them

Learn the right-hand column as carefully as the left. That phrasing is what earns the reason mark:

Find the vertically opposite angle

Two straight lines cross, forming four angles. Tap the angle that is vertically opposite to angle x, the one labelled at the top.

Where the second mark is

A question shows two angles on a straight line, one of them 125°, and is worth 2 marks. A student writes "x = 55°" and nothing else, and is given 1 mark. What single addition would have earned the other one?

• "Angles on a straight line add to 180°": the geometrical rule that makes the answer follow, quoted in its accepted form
• "Because 180 − 125 = 55", which shows the calculation that produced the answer
• "I measured it with a protractor and it came to 55°"
• "It is obvious from the diagram that the angles are supplementary"

Across parallel lines

A line crossing a pair of parallel lines makes three named pairs. Two of them are equal and one is not, and telling them apart is most of the work:

Two steps, one answer

A straight line crosses two parallel lines. At the upper crossing an angle of 65° is marked. Its alternate angle at the lower crossing sits on a straight line with angle y. How many degrees is y?

Justify the chase

For full marks on that question you must give the reasons as well as the answer. Which set of reasons is correct and complete?

• "Alternate angles are equal", so the lower angle is also 65°; then "angles on a straight line add to 180°", so y = 180 − 65 = 115°
• "Angles on a straight line add to 180°", so y = 115°
• "Corresponding angles are equal", so the lower angle is 65°; then "angles on a straight line add to 180°", so y = 115°
• "Co-interior angles add to 180°", so y = 180 − 65 = 115°

What you may not assume

Diagrams are not drawn to scale, and they say so. Anything you take from how the picture looks is worth nothing, and can be actively wrong, because the diagram is often distorted deliberately. You may only use what is stated in the question or marked on the diagram: a right-angle square, matching tick marks on equal sides, or arrows showing lines are parallel. No arrows means you may not use the parallel-line rules at all, however parallel the lines look. And 360 ÷ n only works for a regular polygon. Every polygon has exterior angles summing to 360°, and only a regular one has them all equal, so only for a regular one can you divide. For an irregular polygon use the interior sum instead: (n − 2) × 180°, which holds for any polygon at all, regular or not.

What can you conclude?

A diagram shows a triangle. Two of its sides look about the same length, but there are no tick marks on them and the question does not describe the triangle. One angle is marked 40°. What may you conclude about the other two?

• Only that they add to 140°, because the angles in a triangle add to 180°. Nothing permits treating it as isosceles, so the two cannot be found individually
• That it is isosceles, so the other two are 70° each
• That they can be measured off the diagram to whatever accuracy is needed
• Nothing at all, since not enough information has been given

Polygon angles

Why exterior angles always add to 360°. Imagine walking once round the outside of the polygon. At each corner you turn through the exterior angle, and when you arrive back where you started you are facing the way you began, having turned through exactly one full rotation. So the turns must total 360°, and that argument works for any polygon of any shape. For a regular polygon, all the corners are identical, so each exterior angle is 360 ÷ n. A regular pentagon: 360 ÷ 5 = 72°. Its interior angle sits on a straight line with the exterior one, so it is 180 − 72 = 108°. For any polygon at all, regular or not, the interior angles add to (n − 2) × 180°. A pentagon: (5 − 2) × 180 = 540°. That formula never needs the polygon to be regular, which is why it is the one to reach for when the shape is irregular.

The fifth angle

An irregular pentagon has four interior angles of 100°, 110°, 120° and 95°. How many degrees is the fifth interior angle?

Which reason would you quote?

A complete answer

A question gives a regular hexagon and asks for one interior angle, with 3 marks available. Put a full-mark answer into order.

• State that the exterior angles of any polygon add to 360°
• Since the hexagon is regular, all six exterior angles are equal, so each is 360 ÷ 6 = 60°
• State that an interior angle and its exterior angle lie on a straight line, so they add to 180°
• Therefore the interior angle is 180 − 60 = 120°
• Check it against the other route: (6 − 2) × 180 = 720°, and 720 ÷ 6 = 120°

Quote the reasons

These questions are marked in two halves, so an answer that gives only the number throws away the _____ mark. Alternate and corresponding angles are equal, but co-interior angles _____ to 180°, which is the pair most often copied across by mistake. Nothing may be assumed from how a diagram looks: you may use only what is stated or _____ on it. And the shortcut 360 ÷ n gives an exterior angle only when the polygon is _____, while (n − 2) × 180 works for any polygon at all.