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Angles
Angle questions rest on a handful of facts: a straight line is 180°, a triangle is 180°, crossing lines make equal opposite angles. Know them and each question is one subtraction.
A straight line is 180°
An angle measures a turn, in degrees. A quarter turn is a right angle (90°), half a turn is a straight line (180°), and a full turn is 360°. Most angle questions are just one of these totals minus the angles you know.
Find the total, subtract what you know
Ask which total the angles belong to: a straight line (180°), a triangle (180°), around a point (360°) or a right angle (90°). Then the missing angle is that total minus the others.
How to solve them
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1
Kinds of angles
Questions use these names. The size of the angle tells you which one it is.
Acuteless than 90°a sharp corner, like 40°Right90°a square cornerObtuse90° to 180°an open corner, like 130°Straight180°a straight lineNext step: Crossing linesComplementaryTwo angles that make 90°
The complement of 38° is 52°
SupplementaryTwo angles that make 180°
The supplement of 38° is 142°
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2
Two crossing lines: opposite angles are equal
When two lines cross, they make four angles. The angles across from each other (vertical angles) are equal, and the angles next to each other add up to 180°.
Next step: TrianglesExampleTwo lines cross and one angle is 112°. Find the other three.
- The angle across from it is also 112°.
- The two next to it are 180 − 112 = 68° each.
- Check: 112 + 68 + 112 + 68 = 360.
Answer 112°, 68°, 68°
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3
The three angles of a triangle add up to 180°
In an isosceles triangle the two angles at the base are equal; in an equilateral triangle all three are 60°.
Next step: Parallel linesThird angleTwo angles are 48° and 77°
180 − 48 − 77 = 55°
IsoscelesThe top angle is 36°
(180 − 36) ÷ 2 = 72° each
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4
A line across two parallel lines makes only two sizes of angle
Every angle is either the big one or the small one, and the big one plus the small one is 180°. Angles in the same position at each crossing (corresponding) and angles in a Z shape (alternate) are equal.
Next step: PolygonsExampleA line crosses two parallel lines. One angle is 118°. What sizes can the other angles be?
- Every angle that looks wide is 118°.
- Every angle that looks narrow is 180 − 118 = 62°.
Answer 118° or 62°
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5
Polygons: (n − 2) × 180°
A polygon with n sides can be cut into n − 2 triangles from one corner, so its interior angles add up to (n − 2) × 180°. In a regular polygon all the angles are equal: divide by n.
sum = (n − 2) × 180°
Next step: Around a pointExampleWhat is each interior angle of a regular hexagon?
- A hexagon has 6 sides: (6 − 2) × 180 = 720°.
- Six equal angles: 720 ÷ 6 = 120°.
Answer 120°
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6
Angles around a point add up to 360°
Angles that go all the way around one point make a full turn. The same goes for the four angles of any quadrilateral: they add up to 360°.
Around a point150°, 95° and x
360 − 150 − 95 = 115°
QuadrilateralThree angles 85°, 100°, 70°
360 − 255 = 105°
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Supplement and complement mixed up
The complement of 35°:
Wrong180 − 35 = 145°
Right90 − 35 = 55°
Using 360° for a triangle
A triangle with angles of 50° and 65°:
Wrong360 − 115 = 245°
Right180 − 115 = 65°
Vertical angles treated as supplementary
Two lines cross; the angle opposite a 40° angle:
Wrong140°
Right40°
n × 180 for a polygon
The angles of a pentagon add up to:
Wrong5 × 180 = 900°
Right3 × 180 = 540°
5 questions at test level
Pick an answer, then read why. No timer, no account.
What is the supplement of a 47° angle?
Show the answer
B. Supplementary angles add up to 180°: 180 − 47 = 133. 43° is the complement (90 − 47), 137° adds 47 to 90, and 313° subtracts from 360. The supplement is 133°.
Two angles of a triangle measure 52° and 69°. What is the third angle?
Show the answer
A. The angles add up to 180°: 180 − 52 − 69 = 59. 121° is the two known angles added, and 239° uses 360. The third angle is 59°.
Two lines cross. One of the angles is 104°. What is the angle directly across from it?
Show the answer
C. Vertical angles (directly across) are equal, so it is also 104. 76° is the angle next to it (180 − 104), and 52° halves it for no reason. The answer is 104°.
In an isosceles triangle, the angle between the two equal sides is 50°. What is each of the other two angles?
Show the answer
B. The other two angles are equal and share 180 − 50 = 130°, so each is 130 ÷ 2 = 65. 130° forgets to split it between the two. Each angle is 65°.
What is the sum of the interior angles of an octagon?
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D. An octagon has 8 sides: (8 − 2) × 180 = 6 × 180 = 1,080. 900° uses n − 3, and 540° is a pentagon. The sum is 1,080°.
11 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 11 cards known.
You finished angles.
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Next topic: Polygons & similaritySee this topic move.
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