- Introduction
- Representation of Points and Lines on a Plane
- Types of Angles
- Angles on a Straight Line.
- Angles at a Point
- Angles on a Transversal
- Angle Properties of Polygons
- Past KCSE Questions on the topic

## Introduction

- A flat surface such as the top of a table is called a plane. The intersection of any two straight lines is a point.

## Representation of Points and Lines on a Plane

- A point is represented on a plane by a mark labelled by a capital letter. Through any two given points on a plane, only one straight line can be drawn.
- The line passes through points A and B and hence can be labelled line AB.

## Types of Angles

- When two lines meet, they form an angle at a point. The point where the angle is formed is called the vertex of the angle. The symbol ∠is used to denote an angle.
- To obtain the size of a reflex angle which cannot be read directly from a protractor ,the corresponding acute or obtuse angle is subtracted from 3600.If any two angles X and Y are such that:
- Angle X + angle Y =900, the angles are said to be complementary angles. Each angle is then said to be the complement of the other.
- Angle X + angle Y =1800, the angles are said to be supplementary angles. Each angle is then said to be the supplement of the other.

- Angle X + angle Y =900, the angles are said to be complementary angles. Each angle is then said to be the complement of the other.
- In the figure below ∠ POQ and ∠ ROQ are a pair of complementary angles.
- In the figure below ∠ DOF and ∠ FOE are a pair of supplementary angles.

## Angles on a Straight Line.

- The below shows a number of angles with a common vertex O. AOE is a straight line.
- Two angles on either side of a straight line and having a common vertex are referred to as adjacent angles.
- In the figure above:
- ∠ AOB is adjacent to ∠ BOC
- ∠ BOC is adjacent to ∠ COD
- ∠COD is adjacent to ∠DOE

- Angles on a straight line add up to180
^{0}.

## Angles at a Point

- Two intersecting straight lines form four angles having a common vertex. The angles which are on opposite sides of the vertex are called vertically opposite angles.
- Consider the following:
- In the figure above ∠ COB and ∠AOC are adjacent angles on a straight line. We can now show that a = c as follows:
- a + b = 180
^{0}(Angles on a straight line) - c + d = 180
^{0}(Angles on a straight line)

- a + b = 180
- So, a + b + c + d =180
^{0}+180^{0}=360^{0} ^{}This shows that angles at a point add up to360^{0}

## Angles on a Transversal

- A transversal is a line that cuts across two parallel lines.
- In the above figure PQ and ST are parallel lines and RU cuts through them. RU is a transversal.
- Name:
- Corresponding angles are Angles b and e, c and h, a and f, d and g.
- Alternate angles a and c, f and h, b and d, e and g.
- Co-interior or allied angles are f and d, c and e.

## Angle Properties of Polygons

- A polygon is a plan figure bordered by three or more straight lines

### Triangles

- A triangle is a three sided plane figure. The sum of the three angles of a triangle add up to 180
^{0}.triangles are classified on the basis of either angles sides.- A triangle in which one of the angles is 900 is called a right angled triangle.
- A scalene triangle is one in which all the sides and angles are not equal.
- An isosceles triangle is one in which two sides are equal and the equal sides make equal angles with the third side.
- An equilateral triangle is one in which all the side are equal and all the angles are equal

#### Exterior properties of a triangle

Angle DAB = p + q.

Similarly, Angle EBC = r + q and angle FCA = r + p.

But p + q + r = 180^{0}

But p + q + r = 180^{0}Therefore angle DAB + angle EBC + angle FCA = 2p +2q + 2r

=2(p +q +r)

= 2 x 180^{0}= 360^{0}

^{}In general the sum of all the exterior angles of a triangle is 3600.

### Quadrilaterals

- A quadrilateral is a four –sided plan figure. The interior angles of a quadrilateral add put 3600.Quadrilaterals are also classified in terms of sides and angles.

#### Properties of quadrilaterals

**Properties of Parallelograms**

In a parallelogram,

- The parallel sides are parallel by definition.
- The opposite sides are congruent.
- The opposite angles are congruent.
- The diagonals bisect each other.
- Any pair of consecutive angles are supplementary.

**Properties of Rectangles**

In a rectangle,

- All the properties of a parallelogram apply by definition.
- All angles are right angles.
- The diagonals are congruent.

**Properties of a kite**

- Two disjoint pairs of consecutive sides are congruent by definition.
- The diagonals are perpendicular.
- One diagonal is the perpendicular bisector of the other.
- One of the diagonals bisects a pair of opposite angles.
- One pair of opposite angles are congruent.

**Properties of Rhombuses**

In a rhombus,

- Allthe properties of a parallelogram apply by definition.
- Two consecutive sides are congruent by definition.
- All sides are congruent.
- The diagonals bisect the angles.
- The diagonals are perpendicular bisectors of each other.
- The diagonals divide the rhombus into four congruent right triangles.

**Properties of Squares**

In a square,

- All the properties of a rectangle apply by definition.
- All the properties of a rhombus apply by definition.
- The diagonals form four isosceles right triangles.

**Properties of Isosceles Trapezoids**

In an isosceles trapezoid,

- The legs are congruent by definition.
- The bases are parallel by definition.
- The lower base angles are congruent.
- The upper base angles are congruent.
- The diagonals are congruent.
- Any lower base angle is supplementary to any upper base angle.

### Proving That a Quadrilateral is a Parallelogram

Any one of the following methods might be used to prove that a quadrilateral is a parallelogram.

- If both pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram (definition).
- If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.
- If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it is a parallelogram.
- If the diagonals of a quadrilateral bisect each other, then the it is a parallelogram.
- If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram.

### Proving That a Quadrilateral is a Rectangle

One can prove that a quadrilateral is a rectangle by first showing that it is a parallelogram and then using either of the following methods to complete the proof.

- If a parallelogram contains at least one right angle, then it is a rectangle (definition).
- If the diagonals of a parallelogram are congruent, then it is a rectangle.

One can also show that a quadrilateral is a rectangle without first showing that it is a parallelogram. - If all four angles of a quadrilateral are right angles, then it is a rectangle.

### Proving That a Quadrilateral is a Kite

To prove that a quadrilateral is a kite, either of the following methods can be used.

- If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it is a kite (definition).
- If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then it is a kite.

### Proving That a Quadrilateral is a Rhombus

To prove that a quadrilateral is a rhombus, one may show that it is a parallelogram and then apply either of the following methods.

- If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus (definition).
- If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus.

One can also prove that a quadrilateral is a rhombus without first showing that it is a parallelogram. - If the diagonals of a quadrilateral are perpendicular bisectors of each other, then it is a rhombus.

### Proving That a Quadrilateral is a Square

The following method can be used to prove that a quadrilateral is a square:

- If a quadrilateral is both a rectangle and a rhombus, then it is a square.

### Proving That a Trapezoid is an Isosceles Trapezoid

Any one of the following methods can be used to prove that a trapezoid is isosceles.

- If the nonparallel sides of a trapezoid are congruent, then it is isosceles (definition).
- If the lower or upper base angles of a trapezoid are congruent, then it is isosceles.
- If the diagonals of a trapezoid are congruent, then it is isosceles.

**Note:**

- If a polygon has n sides, then the sum of interior angles (2n -4) right angles.
- The sum of exterior angles of any polygon is 360
^{0}. - A triangle is said to be regular if all its sides and all its interior angles are equal.

The figure below is a hexagon with interior angles g ,h ,I ,k and I and exterior angles a, b ,c ,d ,e ,and f.

## Past KCSE Questions on the Topic

- In the figure below, lines AB and LM are parallel.

Find the values of the angles marked x, y and z (3 mks)

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