Angles
4.0 Geometry · 4.1 Angles
Syllabus tag: Kenya CBC | Grade 7 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.1 Angles (8 lessons)
Lesson objectives
By the end of this topic, you will be able to:
- Relate angles on a straight line and at a point
- Relate angles on a transversal cutting parallel lines
- Relate angles in a parallelogram
- Identify angle properties of polygons up to hexagons, and solve for missing angles and sides
Angles
An angle measures turning. This topic collects the rules for finding an angle you were not given, using the ones you were.
a) On a straight line and at a point
Angles on a straight line add to 180°.
Angles at a point, going right round, add to 360°.
Vertically opposite angles, formed where two lines cross, are equal.
These three facts solve a surprising number of questions on their own. If two angles sit on a straight line and one is 130°, the other must be 50°.
b) A transversal cutting parallel lines
A transversal is a line that cuts across two others. When the two lines are parallel, three useful relationships appear.
Corresponding angles sit in matching positions at the two crossings, both on the same side of the transversal. They are equal.
Look for the letter F in the shape of the lines. Corresponding angles sit in the corners of an F, which may be back to front or upside down.
Alternate angles sit between the parallel lines, on opposite sides of the transversal. They are also equal.
Look for the letter Z. Alternate angles sit in the corners of a Z.
Co-interior angles sit between the parallel lines on the same side of the transversal. They are not equal. They add to 180°.
Look for the letter C or U. This is the one pair that adds rather than matches. Assuming they are equal is the commonest error in the topic.
c) Angles in a parallelogram
A parallelogram has two pairs of parallel sides, so the transversal rules apply directly to it.
Opposite angles are equal.
Angles next to each other add to 180°, because they are co-interior between a pair of parallel sides.
All four angles add to 360°.
So one angle is enough. Given 70°, the opposite angle is also 70°. The other two are each 110°.
d) Angles in polygons
The exterior angles of any polygon add to 360°, whatever the number of sides.
An interior angle and its exterior angle lie on a straight line, so they add to 180°.
For a regular polygon with n sides, one exterior angle is 360 ÷ n.
| Polygon | Sides | Exterior | Interior |
|---|---|---|---|
| Triangle | 3 | 120° | 60° |
| Square | 4 | 90° | 90° |
| Pentagon | 5 | 72° | 108° |
| Hexagon | 6 | 60° | 120° |
The interior angles of a triangle add to 180°. For any polygon they add to 180 × (n − 2). That is because the shape splits into n − 2 triangles.
e) Working out a missing angle
Name what you know and what you want.
Look for a straight line, a point, a pair of parallel lines, or a polygon. Each brings its own rule.
Write the rule down before substituting. It keeps the reasoning visible and earns method marks even if the arithmetic slips.
Check at the end. Angles on a line should still total 180°. Angles in your polygon should still total 180 × (n − 2).
f) Where this is used
A carpenter cutting a mitre joint. A tiler fitting hexagons without gaps. A surveyor setting out a plot. Anyone reading a plan where only some angles are marked.
Words to know
- Transversal — a line that crosses two (often parallel) lines.
- Corresponding angles — angles in matching positions at two crossings of a transversal; equal when the lines are parallel.
- Alternate angles — angles on opposite sides of a transversal, between the parallel lines; equal when the lines are parallel.
- Co-interior (allied) angles — angles on the same side of a transversal, between the parallel lines; sum to 180° when the lines are parallel.
- Interior angle — an angle inside a polygon, at one of its vertices.
- Exterior angle — the angle formed outside a polygon, between one side and the extension of the adjacent side.
:::checkpoint Check yourself
- Two angles sit on a straight line. One is 115°. What is the other?
- Which pair of angles on parallel lines adds to 180° rather than being equal?
- Find one interior angle of a regular octagon.
- One angle of a parallelogram is 65°. Give the other three. :::
Bridge to practice
Try the exercises below — on angles at a point, on a transversal, in a parallelogram, and in polygons.