Angles and Plane Figures
Geometry · Angles and Plane Figures
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 20 Angles and Plane Figures
Lesson objectives
By the end of this topic, you should be able to:
- Solve problems involving angles on a straight line and at a point.
- Solve problems involving angles formed when a transversal cuts parallel lines.
- State and apply the angle properties of triangles and quadrilaterals.
- Calculate interior and exterior angles of polygons.
Angles and Plane Figures
This topic collects the angle rules. They let you find an unknown angle from the ones you are given.
a) On a line and at a point
Angles on a straight line sum to 180°.
Angles at a point sum to 360°.
Vertically opposite angles, formed where two lines cross, are equal.
b) A transversal cutting parallel lines
Corresponding angles occupy matching positions at the two intersections, on the same side of the transversal. They are equal. Look for the F shape.
Alternate angles lie between the parallels on opposite sides of the transversal. They are equal. Look for the Z shape.
Co-interior angles lie between the parallels on the same side. They are supplementary, adding to 180°. Look for the C or U shape.
That last pair is the one that adds rather than matches. Assuming they are equal is the commonest error in the topic.
c) Triangles
The angles of any triangle sum to 180°.
An equilateral triangle has three equal sides and three 60° angles.
An isosceles triangle has two equal sides, and the angles opposite them are equal.
A scalene triangle has all sides and all angles different.
The exterior angle of a triangle equals the sum of the two interior opposite angles. That follows directly from the 180° rule and often saves a step.
d) Quadrilaterals
The angles of any quadrilateral sum to 360°.
| Shape | Properties |
|---|---|
| Square | 4 equal sides, 4 right angles |
| Rectangle | opposite sides equal, 4 right angles |
| Rhombus | 4 equal sides, opposite angles equal |
| Parallelogram | opposite sides parallel and equal |
| Trapezium | one pair of parallel sides |
| Kite | two pairs of adjacent sides equal |
In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Both follow from the transversal rules, since its opposite sides are parallel.
e) Polygons
The exterior angles of any polygon sum to 360°, whatever the number of sides.
An interior angle and its exterior angle lie on a straight line, so they sum to 180°.
The interior angles sum to 180(n − 2), because an n-sided polygon splits into n − 2 triangles.
For a regular polygon with n sides, one exterior angle is 360/n. One interior angle is 180 minus that.
| Polygon | n | Exterior | Interior | Interior sum |
|---|---|---|---|---|
| Triangle | 3 | 120° | 60° | 180° |
| Quadrilateral | 4 | 90° | 90° | 360° |
| Pentagon | 5 | 72° | 108° | 540° |
| Hexagon | 6 | 60° | 120° | 720° |
| Octagon | 8 | 45° | 135° | 1080° |
f) Working a problem
Identify the configuration first. Is it a line, a point, parallels with a transversal, a triangle, a quadrilateral, or a polygon?
State the rule before substituting. It keeps the reasoning visible and earns method marks.
Check at the end that all your angles sum correctly for the figure.
g) Where this is used
Setting out buildings. Cutting joints in carpentry. Surveying. Tiling patterns that must fit without gaps.
Words to know
- Transversal -- a line crossing two or more other lines.
- Vertically opposite angles -- the equal angles formed opposite each other when two lines cross.
- Supplementary angles -- two angles totalling 180°.
- Regular polygon -- one with all sides and all angles equal.
- Exterior angle -- the angle between a side extended and the adjacent side.
:::checkpoint Check yourself
- Which pair of angles on parallel lines is supplementary rather than equal?
- Find the interior angle of a regular decagon.
- The angles of a quadrilateral are 80°, 95° and 110°. Find the fourth.
- What is the sum of the interior angles of a heptagon? :::
Bridge to practice
The exercises begin with angle types and angles on lines and at points, move through the parallel-line relationships and triangle and quadrilateral properties, and finish with polygons. For every angle you write down, write the reason beside it — that discipline is what the marking scheme rewards.