0580 · 4.6Angles

Basic angle rules

0:00A complicated geometry diagram can contain many unknown angles, yet only a few facts are needed to solve it. The challenge is recognising how those facts connect. An angle measures a turn, and that idea explains many familiar rules. By the end, you will understand the rules and use them to calculate unknown angles with clear geometrical reasons. An angle is formed by two rays meeting at a vertex. The vertex is the point where the rays meet.

0:25In angle PQR, the middle letter tells us that Q is the vertex. The angle lies between the ray from Q through P and the ray from Q through R. We measure angles in degrees. A right angle is ninety degrees, a straight angle is one hundred and eighty degrees, and a complete turn is three hundred and sixty degrees. A reflex angle is greater than one hundred and eighty degrees but less than three hundred and sixty degrees.

0:50Always check which part of the turn the marked angle represents. Angles at a point add up to three hundred and sixty degrees because together they complete one full turn. In this diagram, the three marked angles fill the space around O without overlapping. Two are ninety-five degrees and one hundred and forty degrees. The remaining angle is whatever is needed to complete the turn. First, add ninety-five degrees and one hundred and forty degrees to obtain two hundred and thirty-five degrees.

1:17Subtract two hundred and thirty-five degrees from three hundred and sixty degrees. This gives x equal to one hundred and twenty-five degrees. The geometrical reason is that angles at a point add up to three hundred and sixty degrees. A straight line represents half a full turn. Therefore, angles on a straight line add up to one hundred and eighty degrees. Only add angles that fill the same side of that line.

1:40Two angles whose sum is one hundred and eighty degrees are called supplementary angles. When two straight lines cross, vertically opposite angles are equal. These are the angles directly across the intersection, not neighbouring angles. The word vertically does not mean that the diagram must stand upright. Rotating the entire diagram leaves this relationship unchanged because the same straight lines still cross. Here, angle y is angle POQ, and it lies beside the angle of forty-seven degrees on the straight line through P and R.

2:10Subtracting forty-seven degrees from one hundred and eighty degrees gives y equal to one hundred and thirty-three degrees. The reason is that angles on a straight line add up to one hundred and eighty degrees. Angle z is angle POS, and it equals forty-seven degrees because vertically opposite angles are equal. This explanation reveals an important difference between neighbouring and opposite angles. Neighbouring angles at this intersection add up to one hundred and eighty degrees.

2:35Opposite angles are equal because each one supplements the same neighbouring angle. Do not assume that every pair of angles meeting at a point is equal.

Angles in parallel lines

2:43Parallel lines stay the same distance apart and never meet. A transversal is a line that crosses two or more other lines. Because parallel lines have the same direction, the transversal makes matching angles at both intersections. These matching positions explain corresponding angles, rather than leaving us with a rule to memorise. Corresponding angles are equal when the lines are parallel. They occupy the same relative position at the two intersections.

3:09Here, angle c matches the marked angle of sixty-four degrees, because both sit below their parallel line and left of the transversal. Therefore, angle c is sixty-four degrees, because corresponding angles are equal. Alternate angles are also equal when the lines are parallel. The interior pair lies between the parallel lines, on opposite sides of the transversal. Here, angle d and the marked angle of sixty-four degrees form that pair.

3:34So angle d is sixty-four degrees, because alternate angles are equal. The same value also follows from angle c, since vertically opposite angles are equal. Co-interior angles lie between the parallel lines on the same side of the transversal. They add up to one hundred and eighty degrees, so they are supplementary. Angle e and the marked angle of sixty-four degrees are co-interior angles. Subtracting sixty-four degrees from one hundred and eighty degrees gives angle e equal to one hundred and sixteen degrees.

4:02Some students recognise these arrangements as letter shapes, but the letters are only visual reminders. A rotated or crowded diagram may hide the familiar shape. Instead, identify the parallel lines, locate the transversal, and compare the angle positions. In an explanation, write corresponding angles, alternate angles or co-interior angles, rather than naming a letter shape.

Triangle angles and worked example

4:25The angle sum of a triangle is one hundred and eighty degrees. We can understand why by drawing a line through the top vertex parallel to the opposite side. Alternate angles move the two bottom angle sizes to the top. Together with the original top angle, they fill a straight angle. This argument works for any triangle, not just a symmetrical one. Its shape may change, but the three interior angles still make one hundred and eighty degrees.

4:50An interior angle is an angle inside a shape at one of its vertices. Once two triangle angles are known, subtract their sum from one hundred and eighty degrees. Consider an exam-style question asking us to calculate x and y, giving geometrical reasons. The line XY is parallel to the line QS. Angle XPQ is sixty-eight degrees, and angle YPR is fifty-two degrees. We first transfer those known angles into the triangle.

5:17Angle PQR is sixty-eight degrees because alternate angles are equal. Angle PRQ is fifty-two degrees for the same reason. Adding these two interior angles gives one hundred and twenty degrees. Subtracting that total from one hundred and eighty degrees gives x equal to sixty degrees. The reason is that angles in a triangle add up to one hundred and eighty degrees. Angle y lies outside the triangle, beside the angle of fifty-two degrees at R.

5:43The reason here is that angles on a straight line add up to one hundred and eighty degrees. Subtracting fifty-two degrees from one hundred and eighty degrees gives y equal to one hundred and twenty-eight degrees. Every step uses a specific relationship, rather than the apparent size of the angle. An exterior angle is formed between a polygon side and the extension of an adjacent side. Here, y is an exterior angle of the triangle.

6:08It also equals the sum of the two opposite interior angles. Sixty-eight degrees plus sixty degrees gives one hundred and twenty-eight degrees, because both calculations subtract the remaining interior angle from a straight angle.

Polygon interior angle sums

6:19A polygon is a closed plane shape made from straight sides. A quadrilateral has four sides, and its interior angles add up to three hundred and sixty degrees. Drawing a diagonal divides it into two triangles. Each triangle contributes one hundred and eighty degrees, so their combined angle sum is three hundred and sixty degrees. This applies to irregular quadrilaterals as well as squares and rectangles. Equal sides are not needed for the angle sum.

6:45An irregular shape may have very different individual angles while keeping the same total. The number of sides determines that total, not how balanced the drawing looks. A pentagon can be divided into three triangles, giving an interior angle sum of five hundred and forty degrees. Each additional side allows one additional triangle. In general, a polygon with n sides divides into n − 2 triangles. Its interior angle sum is n − 2, multiplied by one hundred and eighty degrees.

7:14The subtraction of two is therefore counting triangles, not performing a mysterious adjustment. A hexagon contains four triangles, so its interior angles sum to seven hundred and twenty degrees. The same formula applies to simple concave polygons, which have an indentation. Their triangles must be drawn inside the shape using suitable diagonals. This irregular pentagon has an inward notch at S. Calculate its unknown interior angle using the same pentagon angle sum.

7:41The known angles are ninety degrees, ninety degrees, sixty degrees and sixty degrees. Notice that the interior angle at the notch is the larger angle inside the polygon, not the angle outside the shape above it. Subtracting two from five gives three triangles. Multiplying three by one hundred and eighty degrees gives five hundred and forty degrees. The four known angles total three hundred degrees. Subtracting three hundred degrees from five hundred and forty degrees gives x equal to two hundred and forty degrees.

8:11That answer is reasonable because the marked interior angle is reflex. An interior angle does not always have to be less than one hundred and eighty degrees. A convex polygon has no inward indentation, so each interior angle is smaller than a straight angle. Always inspect which angle the arc actually marks.

Exterior angles and regular polygons

8:29Exterior angles have another useful meaning: they describe the turns made while travelling around a polygon. For a convex polygon, choose one exterior angle at each vertex in the same direction. After travelling all the way around, your direction has completed one full turn. Those exterior angles therefore add up to three hundred and sixty degrees. This exterior angle sum does not depend on the number of sides. It also does not require a regular polygon.

8:54However, you must select one consistent set of exterior angles, not add every angle outside the shape. At each vertex, the interior angle and its adjacent exterior angle sum to one hundred and eighty degrees. A regular polygon has all sides equal and all interior angles equal. Its exterior angles are equal too, because each supplements the same interior angle. Divide three hundred and sixty degrees by the number of sides to find each exterior angle.

9:20Subtract that result from one hundred and eighty degrees to find each interior angle. Alternatively, divide the total interior angle sum by the number of sides. Each interior angle then equals n − 2, multiplied by one hundred and eighty degrees, divided by n. Both methods describe the same geometry. Division into equal individual angles is valid only when the polygon is regular or equality is otherwise established. For our final worked example, a regular polygon has an interior angle of one hundred and fifty-six degrees.

9:52Subtracting this from one hundred and eighty degrees gives an exterior angle of twenty-four degrees. Divide three hundred and sixty degrees by twenty-four degrees to find the number of equal turns. The result is fifteen, so the polygon has fifteen sides.

Exam technique and recap

10:05In an examination, calculate means finding the requested value, while a geometrical explanation must name the property used. Write that alternate angles are equal because the lines are parallel. For a polygon calculation, state its interior angle sum or the exterior angle sum. Avoid vague reasons such as angle rules or because the diagram shows it. These methods must work without a calculator on Paper two, and they also apply on calculator Paper four.

10:31The examples here give exact answers, so rounding is unnecessary. If a question produces an angle that needs rounding, follow its instructions and retain full values during working. Unless otherwise specified, angles are normally given to one decimal place, rather than three significant figures. The main traps are using parallel-line rules without established parallel lines, dividing an irregular polygon into equal angles, and confusing interior with exterior angles. Another mistake is trusting a diagram instead of its labels.

11:01A drawing may not be to scale, so an angle that looks acute could represent something else. Check the marked arc and justify each calculation. The central idea is that angles measure turns. A full turn gives three hundred and sixty degrees, while a straight turn gives one hundred and eighty degrees. Parallel lines create matching angle relationships. Triangles explain polygon interior angle sums, and a complete journey around a convex polygon explains its exterior angle sum.

11:28When solving a diagram, identify the relevant lines and vertices before choosing a rule. Calculate one angle at a time, then use each result to unlock another. Keep interior angles separate from exterior turns, and check whether the polygon is regular. Clear geometrical reasons make your solution understandable as well as correct.