Mathematics Year 6: Complementary and supplementary angles
Mathematics, Year 6 (Portugal): Two angles that add up to 90° are complementary and two that add up to 180° are supplementary; the sum stays the same even when the parts change.
Adding up to 90° or to 180°
Two angles are complementary when their amplitudes add up to 90° (a right angle) and supplementary when they add up to 180° (a straight angle). The complement of 30° is 60°; the supplement of 30° is 150°. When two angles sit side by side, the angle they form together is called the sum angle: for complementary angles it is a right angle, for supplementary angles a straight angle. Hint: C comes before S, just as 90 comes before 180.
Why the sum does not change when the parts change
Picture a right angle and, from its vertex, a ray inside it that splits it into two angles: 30° and 60°. Turn the ray a little: 45° and 45°; a little more: 70° and 20°. The two parts change, but they always add up to 90°, because the sum angle, the right angle, has not changed: what one gains, the other loses. The same happens with a straight angle split in any way: the parts always add up to 180°. That is why knowing one angle tells you the other.
A case: the complement and supplement of 35°
An angle measures 35°. Complement: 90° − 35° = 55°. Supplement: 180° − 35° = 145°. Check: 35° + 55° = 90° and 35° + 145° = 180°. Notice that 145° − 55° = 90°. Another problem: on a straight line, a ray makes two angles side by side, and one measures 128°. Together they form a straight angle, so the other measures 180° − 128° = 52°.
The trap: swapping 90° and 180°
For the complement of 40°, some people do 180° − 40° = 140°, but that is the supplement; the complement is 90° − 40° = 50°. Second slip: thinking the angles must sit side by side. They need not: 30° and 60° are complementary even in different corners of the page; they only form a visible sum angle when they are adjacent. Finally, an obtuse angle, such as 100°, has no complement (it is already past 90°), but it has a supplement: 80°.
Where you see this outside school
When you open a laptop to 110°, there are 180° − 110° = 70° left before the screen lies flat: those angles are supplementary. In a picture frame, each corner joins two strips cut at 45°: 45° + 45° = 90°, complementary. Whenever something turns and the other part “shrinks”, the total stays the same: that is what lets you find one angle without measuring it.
Test what you learned
Which of these pairs of angles is complementary?
- a) 20° and 160°
- b) 30° and 30°
- c) 25° and 65°
- d) 100° and 80°
Correct answer: c) 25° and 65° — Yes: 25° + 65° = 90°, a right angle. Each is the complement of the other.
Three angles side by side form a straight angle. Two of them measure 40° and 65°. How much does the third measure?
- a) 105°
- b) 75°
- c) 25°
- d) 140°
Correct answer: b) 75° — Yes: the sum angle is 180°, so the third measures 180° − (40° + 65°) = 180° − 105° = 75°.
How much does the complement of a 40° angle measure?
- a) 140°
- b) 90°
- c) 320°
- d) 50°
Correct answer: d) 50° — Yes: 90° − 40° = 50°, and 40° + 50° = 90°, a right angle.
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