What is an Angle?
An angle is formed when two straight lines or rays meet at a common endpoint, called the vertex. Angles are usually measured in degrees (°).
Key types of angles include:
- Acute Angle: An angle that is less than 90°. Think of a sharp corner.
- Right Angle: An angle that is exactly 90°. It forms a perfect corner, like the corner of a square, and is often marked with a small square symbol (∟).
- Obtuse Angle: An angle that is greater than 90° but less than 180°. It's wider than a right angle.
- Straight Angle: An angle that is exactly 180°. It forms a straight line.
- Reflex Angle: An angle that is greater than 180° but less than 360°. It's the larger angle on the "outside" of a bend.
- Full Angle (or Full Rotation): An angle that is exactly 360°. It represents a complete circle.
Basic Angle Rules (2D)
1. Angles on a Straight Line
Angles that lie on a straight line add up to 180°.
If A, B, and C are angles on a straight line, then A + B + C = 180°
Example:
If angle A = 70° and angle B = 50° on a straight line, find angle C.
C = 180° - (70° + 50°) = 180° - 120° = 60°.
2. Angles Around a Point
Angles around a central point add up to 360° (a full circle).
A + B + C + D = 360°
Example:
If three angles around a point are 100°, 120°, and 80°, find the fourth angle (D).
D = 360° - (100° + 120° + 80°) = 360° - 300° = 60°.
3. Vertically Opposite Angles
When two straight lines intersect, they form four angles. The angles opposite each other at the vertex are called vertically opposite angles, and they are equal.
The two angles marked 'A' are equal. The two angles marked 'B' are equal.
Example:
If angle A = 110°, then the vertically opposite angle A is also 110°.
Angle B would be 180° - 110° = 70° (angles on a straight line), and the other angle B would also be 70°.
Angles in Triangles (2D)
A triangle is a 2D shape with three straight sides and three angles.
Key Fact: The sum of the interior angles in any triangle always adds up to 180°.
Angle A + Angle B + Angle C = 180°
Example: Finding a Missing Angle in a Triangle
A triangle has two angles measuring 60° and 70°. What is the size of the third angle?
Workings:
1. Sum of known angles: 60° + 70° = 130°.
2. Subtract from 180°: 180° - 130° = 50°.
Answer: The third angle is 50°.
Isosceles Triangles
An isosceles triangle has two sides of equal length and two equal base angles (the angles opposite the equal sides).
The two base angles (marked X°) are equal.
Example 1 (Isosceles): Apex Angle Given
An isosceles triangle has one angle of 40°. This is the angle between the two equal sides (the apex angle). What are the sizes of the other two angles?
Workings:
1. The sum of angles in a triangle is 180°.
2. Subtract the known apex angle: 180° - 40° = 140°. This is the sum of the two equal base angles.
3. Divide by 2 to find each base angle: 140° ÷ 2 = 70°.
Answer: The other two angles are both 70°.
Example 2 (Isosceles): Base Angle Given
An isosceles triangle has one of its base angles measuring 55°. What are the sizes of the other two angles?
Workings:
1. In an isosceles triangle, the two base angles are equal. So, the other base angle is also 55°.
2. Sum of the two base angles: 55° + 55° = 110°.
3. The sum of all angles in a triangle is 180°. Subtract the sum of base angles to find the apex angle: 180° - 110° = 70°.
Answer: The other base angle is 55°, and the apex angle is 70°.
Equilateral Triangles
An equilateral triangle has all three sides of equal length and all three angles are equal.
Since the angles in a triangle add up to 180°, each angle in an equilateral triangle is 180° ÷ 3 = 60°.
Angles in Quadrilaterals (2D)
A quadrilateral is a 2D shape with four straight sides and four angles.
Key Fact: The sum of the interior angles in any quadrilateral always adds up to 360°.
Angle A + Angle B + Angle C + Angle D = 360°
Example: Finding a Missing Angle in a Quadrilateral
A quadrilateral has three angles measuring 90°, 110°, and 75°. What is the size of the fourth angle?
Workings:
1. Sum of known angles: 90° + 110° + 75° = 275°.
2. Subtract from 360°: 360° - 275° = 85°.
Answer: The fourth angle is 85°.
Specific Quadrilaterals:
- Square: All four angles are 90°. All sides are equal.
- Rectangle: All four angles are 90°. Opposite sides are equal.
- Parallelogram: Opposite angles are equal. Adjacent angles (next to each other) add up to 180°. Opposite sides are parallel and equal. (e.g., if one angle is 70°, the opposite is 70°, and the other two are each
180° - 70° = 110°). - Rhombus: A special type of parallelogram where all sides are equal. Opposite angles are equal, and adjacent angles add up to 180°.
- Trapezium: Has one pair of parallel sides. Angles between a parallel side and one of the non-parallel sides (co-interior angles) add up to 180°.
- Kite: Has two pairs of equal-length sides that are adjacent to each other. One pair of opposite angles (between the unequal sides) are equal.
Angles in 3D Shapes
Understanding angles in 3D shapes often involves looking at the 2D faces of the shape or the angles formed where edges and faces meet.
Cuboids
A cuboid (like a rectangular box) has several important angle properties:
- All angles on each face are right angles (90°), because each face is a rectangle.
- The angle between any two adjacent faces is also a right angle (90°). For example, where a wall meets the floor.
- The angle between any three edges meeting at a vertex (corner) are all right angles (90°).
Each corner of a cuboid has three 90° angles where edges meet.
Example: Angle on a Face of a Cuboid
Consider one face of a cuboid. If you draw a diagonal line across this rectangular face, what type of angles are formed at the corners of the rectangle by the diagonal?
Workings:
1. Each face of a cuboid is a rectangle.
2. All angles in a rectangle are 90°.
3. When a diagonal is drawn, it divides two of these 90° angles into two smaller angles. These smaller angles will be acute (less than 90°).
Answer: The diagonal creates acute angles with the sides of the rectangle at the corners it connects. The original corner angles of the face remain 90°.
Other 3D Shapes
- Cube: A special cuboid where all faces are squares, so all angles on faces and between faces/edges are 90°.
- Pyramids: The angles on the triangular faces will depend on the shape of the base and the height of the pyramid. The sum of angles in each triangular face is 180°. The angles at the apex where the triangular faces meet also depend on the pyramid's dimensions.
- Prisms: Have two identical end faces (bases) and rectangular side faces (if it's a right prism). Angles on the rectangular faces are 90°. Angles on the end faces depend on the shape of those ends (e.g., a triangular prism has triangular ends, and the angles within those triangles sum to 180°).
For Functional Skills, understanding the 2D shapes that make up the faces of 3D shapes is often key to solving angle problems related to them. Complex calculations of angles within 3D space (e.g., angle between a diagonal of a cuboid and a face) typically require trigonometry beyond Level 2, but recognizing right angles is crucial.
Interactive: Find Missing Angle (2D)
Triangle Missing Angle
Enter two known angles of a triangle to find the third.
Quadrilateral Missing Angle
Enter three known angles of a quadrilateral to find the fourth.
Quick Quiz on Angles
1. Two angles on a straight line are 75° and X°. What is X?
Answer: 105°
Workings:
Angles on a straight line add to 180°. So, X = 180° - 75° = 105°.
2. An isosceles triangle has an apex angle of 80°. What is the size of each base angle?
Answer: 50°
Workings:
Sum of base angles = 180° - 80° = 100°. Since base angles are equal, each is 100° ÷ 2 = 50°.
3. Three angles around a point are 90°, 130°, and 60°. What is the fourth angle?
Answer: 80°
Workings:
Angles around a point add to 360°. Sum of known angles = 90° + 130° + 60° = 280°. Fourth angle = 360° - 280° = 80°.
4. What is the name for an angle that is exactly 90°?
Answer: Right Angle
5. A cuboid has rectangular faces. What is the angle at each corner of these faces?
Answer: 90°
Explanation:
Each face of a cuboid is a rectangle, and all angles in a rectangle are right angles (90°).
Key Angle Facts to Remember
- Angles in a triangle add up to 180°.
- Angles in a quadrilateral add up to 360°.
- An isosceles triangle has two equal angles (base angles).
- An equilateral triangle has three equal angles (each 60°).
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Vertically opposite angles are equal.
- Opposite angles in a parallelogram are equal, and adjacent angles add to 180°.
- In a cuboid, angles on faces and between adjacent faces/edges are typically 90°.