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Specific Objectives
By the end of the topic the learner should be able to:
- Name and identify types of angles
- Solve problems involving angles on a straight line
- Solve problems involving angles at a point
- Solve problems involving angles on a transversal cutting parallel lines
- State angle properties of polygons
- Solve problems involving angle properties of polygons
- Apply the knowledge of angle properties to real life situations.
Content
- Types of angles
- Angles on a straight line
- Angles at a point
- Angles on a transversal (corresponding, alternate and allied angles)
- Angle properties of polygons
- Application to real life situations.
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.
Acute angle. Reflex angle.
Obtuse angle Right 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 .If any two angles X and Y are such that:
- Angle X + angle Y =, the angles are said to be complementary angles. Each angle is then said to be the complement of the other.
- Angle X + angle Y =, the angles are said to be supplementary angles. Each angle is then said to be the supplement of the other.
In the figure below < POQ and < ROQ are a pair of complementary angles.
In the figure below Angles on a straight line. The below shows a number of angles with a common vertex 0.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 to. 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 and AOC are adjacent angles on a straight line. We can now show that a = c as follows: (Angles on a straight line) (Angles on a straight line) So, a + b + c + d =+ = This shows that angles at a point add up to 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: Angle properties of polygons A polygon is a plan figure bordered by three or ore straight lines Triangles A triangle is a three sided plane figure. The sum of the three angles of a triangle add up to 18.triangles are classified on the basis of either angles sides. Exterior properties of a triangle Angle DAB = p + q. Similarly, Angle EBC = r + q and angle FCA = r + p. But p + q + r = But p + q + r = Therefore angle DAB + angle EBC + angle FCA = 2p +2q + 2r =2(p +q +r) = 2 x = 36 In general the sum of all the exterior angles of a triangle is . Quadrilaterals A quadrilateral is a four –sided plan figure. The interior angles of a quadrilateral add put .Quadrilaterals are also classified in terms of sides and angles. PROPERTIES OF QUADRILATERALS Properties of Parallelograms In a parallelogram, Properties of Rectangles Properties of a kite Properties of Rhombuses In a rhombus, Properties of Squares In a square, Properties of Isosceles Trapezoids In an isosceles trapezoid, Proving That a Quadrilateral is a Parallelogram Any one of the following methods might be used to prove that a quadrilateral 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. One can also show that a quadrilateral is a rectangle without first showing that it is a parallelogram. Proving That a Quadrilateral is a Kite To prove that a quadrilateral is a kite, either of the following methods can be used. 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. One can also prove that a quadrilateral is a rhombus without first showing that it is a parallelogram. 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. Note: 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. End of topic Did you understand everything? If not ask a teacher, friends or anybody and make sure you understand before going to sleep! 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)The parallel sides are parallel by definition.
In a rectangle,
The diagonals are perpendicular.
The diagonals divide the rhombus into four congruent right triangles.