These questions ask you to find missing angles or identify statements that must be true. Your first job is to recognize the configuration, because that tells you which total or equality to use.
Match the configuration to its fact
Read the spatial words before calculating. Decide whether the angles lie along a line, surround a point, sit inside a polygon, or come from two crossing lines. Add the known angles once, then subtract that sum from the correct total. If equal angles share a remainder, divide only after finding the remainder.
| Signal | Rule and tiny example |
|---|---|
| Angles lie along one straight line | Use 180 degrees. If one part is 112 degrees, the other is 180 minus 112, which is 68 degrees. |
| Angles completely surround one point | Use 360 degrees. If two parts are 140 degrees and 80 degrees, the remaining part is 360 minus 220, which is 140 degrees. |
| Two straight lines cross | Facing angles are equal, while neighbouring angles on a line total 180 degrees. Across from 74 degrees is 74 degrees; beside it is 106 degrees. |
| Angles are inside a triangle | Use 180 degrees. With angles of 48 degrees and 67 degrees, the third is 180 minus 115, which is 65 degrees. |
| Angles are inside a quadrilateral | Use 360 degrees. If three angles total 285 degrees, the fourth is 75 degrees. |
| A remainder is shared equally | Find the remainder first, then divide by the number of equal angles. A 36 degree apex leaves 144 degrees, so two equal base angles are 72 degrees each. |
Where angle work goes wrong
| Trap | How to reject it |
|---|---|
| Using 360 degrees whenever several angles are mentioned | The number of angles does not choose the total. A straight line and a triangle use 180 degrees; a full turn and a quadrilateral use 360 degrees. |
| Reporting the sum already used | After adding the known angles, ask whether the question wants that subtotal or the missing part. A missing angle requires subtracting the subtotal from the full total. |
| Treating opposite and adjacent as the same relationship | At a crossing, the angle across the point repeats the measure. An angle beside it shares a straight line and is found by subtracting from 180 degrees. |
| Assuming two angles match because their positions look similar in words | Equality needs a stated reason, such as vertically opposite angles or equal base angles. Without such a reason, calculate from the total instead of copying a measure. |
Try it
Three angles lie side by side along a straight line with no gaps between them. Two of them measure 40 degrees and 90 degrees. Enter the number of degrees in the third angle.
Two straight lines cross each other at one point, forming four angles. One of those four angles measures 63 degrees. Select all the statements about this crossing that must be true.
- All four angles at the crossing measure 63 degrees.
- The angle vertically opposite the 63 degree angle also measures 63 degrees.
- The four angles at the crossing add up to 360 degrees.
- The angle vertically opposite the 63 degree angle measures 27 degrees.
A learner is asked for the base angles of an isosceles triangle with an apex angle of 40 degrees and two equal base angles. The learner writes: 180 minus 40 is 140, so each base angle is 140 degrees. Which statement names the mistake in this working and gives the correct size?
- Nothing is wrong, and each base angle really is 140 degrees.
- The 140 degrees covers both base angles, so each one is 70 degrees.
- All three angles here are equal, so each base angle is 60 degrees.
- The apex should have been subtracted from 360, so each base angle is 160 degrees.