ANGLES

Definition of an Angle- When two lines meet at a point they form an angle and the angle is a measure of the space between two straight lines (or sides or arms) that extends from the common point (or vertex).

Vertex- The point where two lines meet is called a vertex.

Measurement of an Angle- Angles are measured most commonly using degrees, where the

symbol ( ) means degree/s.

TABLE SHOWING TYPES OF ANGLES

TYPES OF ANGLES DIAGRAM OF THE ANGLES DEFINITON OF THE ANGLE

RIGHT ANGLE An angle of 90 is called a right angle.

STRAIGHT ANGLE An angle of 180 is called a straight angle

ACUTE ANGLE An angle is acute if its magnitude is greater than 0 but less than 90

OBTUSE ANGLE An angle is obtuse if its magnitude is greater than 90 but less than 180

REFLEX ANGLE An angle is reflex if its magnitude is greater than 180 but less than 360

COMPLEMENTARY ANGLE

Two angles are said to be complementary if their sum is equal to 90

SUPPLEMENTARY ANGLES

Two angles are said to be supplementary if their sum is equal to 180 . Example 140 + 40 =180Or 60 + 120 = 180

TABLE SHOWING PROPERTIES OF ANGLES FORMED BY INTERSECTING LINES

TYPES OF ANGLES DIAGRAM OF THE ANGLE PROPERTIES OF THE ANGLESADJACENT ANGLES Adjacent angles are two

angles which have a common vertex and lie on opposite sides of a common arm.

Angle ABC is adjacent to angle CBD

Because:

they have a common side (line CB)

they have a common vertex (point B)

VERTICALLY OPPOSITE ANGLES

When two straight lines intersect at a point, vertically opposite angles are formed. Furthermore vertically opposite angles are always equal. a = b

CORRESPONDING ANGLES When a transversal cuts two parallel lines then the corresponding angles formed are always equal. Therefore corresponding angles are angles that are in corresponding position.

In this example, these are corresponding angles:

a and e b and f c and g d and h

ALTERNATE INTERIOR ANGLE When a transversal cuts two parallel lines, then the alternate angles formed are always equal also alternate angles are angles enclosed by a Z.

In this example, these are Alternate Interior Angles:

c and f are Alternate Interior Angles

d and e are also Alternate Interior Angles

ALTERNATE EXTERIOR ANGLE When the pairs of angles are on the opposite sides of the transversal but outside the two lines they are called alternate exterior angles.

In the example:

a and h are alternate exterior angles

b and g are also

alternate exterior angles

2 DIMENSIONAL SHAPES( Flat shapes)-

A two-dimensional shape (polygon) is said to be "regular" (or equilateral) if all its sides are the same length and all its angles are the same angle.

TABLE FOR SUM OF INTERIOR ANGLES OF REGULAR TWO DIMENSIONAL POLYGON SHAPES:

SHAPE SIDES SUM OF INTERIOR ANGLES

EACH SIDE ANGLE

Triangle 3 180° 60°Quadrilateral 4 360° 90°Pentagon 5 540° 108°Hexagon 6 720° 120°Heptagon 7 900° 128.57...°Octagon 8 1080° 135°

TABLE FOR NAMES OF TWO DIMENSIONAL SHAPES

TWO DIMENSIONAL SHAPE NAMES

SHAPES PROPERTIES

Square Contrary (opposite) sides are congruent and each side has the equal length.

Rectangle In a rectangle two dimensional shape, contrary (opposite) sides are congruent and each angle is right angle (90 ).

Triangle Triangles are polygons that have 3 vertices and three faces and those are line segments also triangles are the 2-dimensional shape with 3 side plane.

Ellipse An ellipse is a two dimensional shape which is a smooth bounded curve that is symmetric about its centre.

Parallelogram A parallelogram two dimensional figure is one type of the quadrilaterals which has opposite (contrary) faces that is parallel and equal in length. Parallelograms opposite angles are congruent in measuring.

Trapezium Any one pair of congruent sides is parallel in the trapezium two dimensional shapes. In the isosceles trapezium shapes the nonparallel faces are equal and the base angles of the trapezium are also congruent.

Quadrilateral The Quadrilateral two dimensional pictures is a plane figure having 4 straight sides. The word quadrilateral is made up of the words “quad” means four and then the word “lateral” means the faces. The sums of interior angles of any quadrilateral are equal to 360 degree.

TABLE SHOWING TYPES OF TRIANGLES

TRIANGLE BY SIDES PROPERTIES

Scalene triangle

No two sides are equal

No two angles equal

Isosceles triangle

Two sides are equal Two angles are equal The vertex where the two equal side

meet is called an apex An equal angle is formed by an equal

side and the side opposite the apex.

Equilateral triangle

All three sides are equal All three angles are equal, hence each

angle is equal to 60

TRIANGLE BY ANGLE PROPERTIES

Acute- angled triangle

Each angle is an acute angle

Obtuse-angled triangle

One angle is an obtuse angle

Right- angled triangle

One angle is a right angle

TABLE SHOWING TYPES OF QUADRILATERALS

QUADRILATERAL PROPERTIES

Trapezium Isosceles trapezium

One pair of opposite sides are parallel for both trapezium

Pair of non-parallel sides are equal and 2 pairs of equal adjacent angles are properties of the Isosceles trapezium

Kite

2 pairs of equal adjacent sides

One pair of equal opposite angles

Diagonals intersect at right angles

One diagonal is bisected by the other diagonal

2 pairs of congruent triangles are formed by diagonals.

Opposite sides are parallel

Opposite sides are equal Opposite angles are

equal Diagonals bisect each

other

2 pairs of congruent triangles are formed by diagonals.

Rhombus

Opposite sides are parallel

All four sides are equal Opposite angles are

equal Diagonals bisect each

other at right angles Diagonals bisect the

angles at the vertices Four congruent triangles

are formed by diagonals

Rectangle

Opposite sides are parallel

Opposite sides are equal All four angles are right

angles Diagonals bisect each

other Diagonals are equal in

length Two pairs of congruent

triangles are formed by the diagonals

Opposite sides are parallel

All four sides are equal All four angles are right

angles Diagonals bisect each

other at right angles Diagonals bisect the

angles at the verticies. Hence each angle is equal to 45

Diagonals are equal in length

Four congruent triangles are formed by diagonals.

THREE DIMENSIONAL SHAPES

Solids have both length and breath, it also has a thickness height or depth. Thus a solid is a three dimensional shape.

TABLE OF THREE DIMENSIONAL OBJECTS

SOLIDS DEFINTION OF THE SOLIDS

NET SHAPE

Cube

A cube is a polyhedron which is a solid shape that has sides or faces

Cuboid A cuboid is a solid figure bounded by six faces, forming a convex polyhedron.

Triangular prism

A prism is a solid that has two congruent parallel bases that are polygon, the polygons form the base of the prism and the length of the edge joining the two bases is called the height

Cylinder

A cylinder is a solid with two congruent circles joined by a curved surface

Sphere

A sphere is a solid with all its points the same distance from the centre

Cone

A circular cone has a circlular base, which is connected by a curve surface to its vertex. A cone is called a right circular cone if the line from the vertex of the cone to the centre of its base is perpendicular to the base.

Square- based pyramid

A pyramid is a polyhedron with a base in the shape of a polygon and the other faces are triangles with a common vertex called the apex.

SYMMETRY

Symmetry is the kind of pattern that a shape has, it deals with the exact matching of a position or form about a point, line or place. There are three types of symmetry that a plane figure can have:

Translational symmetry- A movement along a straight line without turning is called a translation. A plane figure is said to have translational symmetry if it can be translated and still look the same.

Line symmetry or reflective symmetry- The line of symmetry or the axis of symmetry of a plane figure is a line which can be used as a fold, so that one half of the shape covers the other half exactly. A plane figure can have one or more lines of symmetry or axes of symmetry.

Rotation symmetry- A plane figure is said to have rotational symmetry of a certain order, if the plane figure maps unto itself that is coincides with itself under rotations through stated angles about a common centre. All plane figures have a rotation symmetry of order1, since a rotation about its centre through 360 degrees will map it onto itself. Thus a scalene triangle does not possess rotational symmetry.

TRANSFORMATIONS

A transformation is said to describe the relation between any point and its image point. A transformation is a one-to-one relation of all points on the object onto corresponding points on the image. The object under a transformation is the plane figure that is undergoing a change in position. There are six basic types of transformations:

Translation or displacement, represented by the letter T Reflexion, represented by the letter M Rotation, represented by the letter R Enlargement, represented by the letter E

TRANSLATION

A translation is a transformation in which a plane figure slides along a straight line and changes its position without turning. Each point moves the same distance and in the same direction. Hence all points subjected to the same translation undergo the same displacement.

REFLECTIONS

A reflection is a way of transforming a shape as a plane mirror, in a plane the result of giving an object a reflection in a mirror line or an axis of reflection is called its mirror image. The object and image are symmetrical about the mirror line.

ROTATION

A rotation is a transformation in which every point turns through the same angle about the same centre. In a plane, rotation is about a single point called the centre of rotation. The centre of rotation is the only point which does not change its position after the rotation. A rotation can either be clockwise or anti-clockwise.

ENLARGEMENT

An enlargement is a transformation which maps a shape onto a similar shape, from a centre of enlargement using a scale factor k. Under an enlargement, the length of a side of the image is equal to the scale factor of the enlargement times the length of the corresponding side of the object.

PARALLEL LINES

Parallel lines are lines that never meet no matter how far they are extended. Parallel lines are always the same distance apart and they will never intersect.

The red line is parallel to the blue

Example 1 Example 2

PERPENDICULAR LINES

Two lines are said to be perpendicular if they meet or cross or intersect at right angles that is 90 .

REFERENCES

http://www.regentsprep.org/Regents/math/ALGEBRA/AC3/Lparallel.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut28_parpen.htm

http://www.mathsisfun.com/geometry/corresponding-angles.html

http://www.mathopenref.com/tocs/anglestoc.html

http://www.google.tt/search?hl=en&client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&channel=s&q=3+dimensional+shapes&aq=0&aqi=g2g-c1g4g-c3&aql=&oq=+3+dimensional+shapes

http://www.mathsisfun.com/quadrilaterals.html

Mathematics- A complete course with CXC Questions – RAYMOND TOOLSIE B.Sc.. MACP