Angle chasing
Tonight's idea
An angle measures a turn. A full turn, all the way round, is 360° (° means degrees). Half a turn leaves you facing the opposite way, along a straight line, so a straight line is 180°. Two facts follow.
On a line: angles that sit side by side on a straight line add to 180°. At a point: angles that go all the way round a point add to 360°. At a crossing: when two straight lines cross, opposite angles are equal.
The third fact comes from the first. In the crossing, \(a\) and \(b\) sit on one straight line, so \(a + b = 180^\circ\). But \(b\) and \(c\) sit on the other straight line, so \(b + c = 180^\circ\) too. Both \(a\) and \(c\) are "180° take away \(b\)", so they are equal.
Angle chasing means starting from the angles you know and using these facts, one step at a time, until every angle is found. Give a reason for each step.
Watch one: two lines cross
In the crossing above, suppose \(a = x + 20^\circ\) and \(b = 3x\). Find all four angles.
On a line: \(x + 20^\circ + 3x = 180^\circ\). So \(4x = 160^\circ\) and \(x = 40^\circ\).
Then \(a = 60^\circ\) and \(b = 120^\circ\).
At a crossing: \(c = a = 60^\circ\) and \(d = b = 120^\circ\).
Check at a point: \(60 + 120 + 60 + 120 = 360\). It fits.
Your turn
- Two straight lines cross. One of the four angles is 72°. Find the other three, with a reason for each.
- (a) Three angles side by side on a straight line are 40°, 85° and \(x\). Find \(x\). (b) Four angles go all the way round a point: 90°, 110°, 75° and \(y\). Find \(y\).
- Three angles side by side on a straight line are \(x\), \(2x\) and \(3x\). Find \(x\). What is special about the largest angle?
- Three straight lines cross at one point and make six angles. Two of them are 35° and 80°. Find \(p\), \(q\), \(r\) and \(s\). Say which fact you used each time.
Stretch: halve them both
A ray stands on a straight line and makes two angles, \(a\) and \(b\). Draw a line that cuts \(a\) exactly in half, and another that cuts \(b\) exactly in half. Show that these two new lines always meet at a right angle, whatever \(a\) and \(b\) are.
Answers
- The opposite angle is 72° (at a crossing). Each neighbour is \(180^\circ - 72^\circ = 108^\circ\) (on a line). Check: \(72 + 108 + 72 + 108 = 360\).
- (a) \(x = 180^\circ - 40^\circ - 85^\circ = 55^\circ\). (b) \(y = 360^\circ - 90^\circ - 110^\circ - 75^\circ = 85^\circ\).
- \(x + 2x + 3x = 6x = 180^\circ\), so \(x = 30^\circ\). The angles are 30°, 60° and 90°. The largest is a right angle.
- 35°, 80° and \(p\) sit side by side on the horizontal line, so \(p = 180^\circ - 35^\circ - 80^\circ = 65^\circ\) (on a line). Then \(q = 35^\circ\), \(r = 80^\circ\) and \(s = 65^\circ\), each equal to the angle opposite it (at a crossing). Check: \(35 + 80 + 65 + 35 + 80 + 65 = 360\).
Stretch. On a line, \(a + b = 180^\circ\). The angle between the two new lines is half of \(a\) plus half of \(b\). That is half of \(a + b\), which is half of 180°, so it is 90° every time.