# 6th Class Mental Ability Related to Competitive Exam Geometry

Geometry

Category : 6th Class

Geometry

Learning Objectives

• Geometry
• Types of angles
• Parts of Circle

Geometry

The branch of mathematics which deals with mathematical objects like points, lines, panes and space is called geometry.

Point: A point is to be thought of as a location in space. In other words, a point determines location in a space.

Line segment: Let A and B be two point on a plane. Then the straight path between points a and B is known as Sine segment AB.

The above line segment is denoted as$\overline{AB}$.

Ray: A line segment extended endlessly in one direction Is. called a ray.

The above ray Is denoted as $\overrightarrow{AB}$.

Line: A line is a straight path that extends on and on in both directions endlessly.

The above ray Is denoted as $\overrightarrow{AB}$.

Line: A line is a straight path that extends on and on in both directions endlessly.

 Parallel lines: If two or more lines do not meet each other however far they are extended, then they are called parallel lines Intersecting lines: If two or more lines meet each other at one point they are called intersecting lines. Concurrent lines: Three or more lines in a place are concurrent if all of them pass through the same point. The common point is called point of concurrence. Collinear points: Three or more points which lie on the same line are collinear, and the line is called the line of collinearity for the given points. Angle: The figure formed by two rays with the same initial point is called an angle. Types of angles Acute angle: An angle which measures more than $=30{}^\circ \times 3=90{}^\circ .$but less than $90{}^\circ$is called an acute angle. Right angle: An angle which measure $90{}^\circ$ is called a right angle. Obtuse angle: An angle which measures more than $90{}^\circ$ but less than $180{}^\circ$ is called an obtuse angle. Straight angle: An angle which measures $180{}^\circ$ is called a straight angle. Reflex angle: An angle which measures more than $180{}^\circ$ but less than $360{}^\circ$ is called a reflex angle. Complete angle: An angle which measures $360{}^\circ$ is called a complete angle. Zero angle: An angle which measures $0{}^\circ$is called zero angle. Triangle: A triangle is a closed plan figure bound by three lines segments. In fact triangle is a polygon. Types of trianlges Equilateral triangle: A triangle whose all sides are of equal length is known as equilateral triangle. Isosceles triangle: A triangle whose all sides are of equal length is known as isosceles triangle. Scalene triangle: A triangle whose all sides are of different length is known as scalene triangle. Acute angled triangle: If all the three angles of a triangle are acute then the triangle is known as acute angled triangle. Obtuse angled triangle: The triangle having an obtuse angle in known as obtuse angled triangle. Right angled triangle: The triangle having a right angle is known as right angled triangle. Quadrilateral: A quadrilateral is a closed plane figure bounded by four lines segments or a quadrilateral is a four sided polygon. Types of quadrilatreral Trapezium: A quadrilateral, in which one and only one pair of opposite sides are parallel, is known as trapezium. Parallelogram: A quadrilateral in which both pair of opposite sides are parallel is called a parallelogram. Rhombus: A parallelogram, whose all sides are of equal length, is known as rhombus. Rectangle: A parallelogram, in which each of the angles is a right angle, is known as rectangle. Square: A parallelogram, in which all sides are equal and each angle is a right angle, is known as square. Circle: A circle is simple closed curve all of whose points are at the same distance from a given point in the same plane.

Parts of a circle

Centre: The fixed point in the plane which is equidistant from every point lying on the boundary of the circle is called centre of the circle.

Radius: The line segment joining the centre of the circle to any point on the circle is the radius of that circle.

Chord: A line segment joining any two points on a circle is called chord of the circle.

Diameter: A chord which passes through the centre of a circle is called a diameter of that circle.

Secant: A line which intersects the circles at two distinct points is called a secant.

Arc: An arc is a part of a circle.

Segment: A region in the interior of the circle enclosed by a chord and an arc is called a segment of the circle.

Sector: A region in the interior of the circle enclosed by two radii and an arc is called sector of the circle.

Semicircle: A diameter divides a circle into two equal parts. Each part is called a semicircle.

Circumference: The length of the boundary of the interior of a circle is called circumference of the circle.

Concentric circles: Two or more circles with the same centre are called concentric circles.

Three Dimensional Shapes

Cuboid: Cuboid is a three dimensional solid shape. A rectangular wooden box is an example of a cuboid. A cuboid has 6 rectangular faces, 12 edges and 8 vertices.

Cube: A cuboid whose length, breadth and height are equal is called a cube, A cube has 5 square faces, 12 edges and 8 vertices.

Cylinder: A cylinder is three dimensional geometric figure that has two congruent and parallel bases. It has no vertex, two circular faces, one curved face and two curved edges.

Sphere: An object which is in the shape of a ball is said to have the shape of sphere. It has a curved surface but no vertex and no edge.

Cone: A cone is a three dimensional geometric shape that tapers smoothly from a fiat to a point called the apex or vertex. It has one vertex, one curved edge, one curved face and one flat face.

Pyramid: A pyramid is a solid whose base is a plane rectilinear figure and whose side faces are triangles having a common vertex called the vertex of the pyramid.

Symmetry

A figure drawn on the paper is symmetric, if it can be folded in such a way that the two halves of the figure exactly cover each other. The line of fold is called axis of the symmetry,

Symmetry of some geometrical shapes

Line segment: A line segment is symmetrical about its perpendicular bisector.

Angle: An angle with equal arms is symmetrical about the bisector of the angle,

Scalene triangle: A scalene triangle has no line of symmetry.

Parallelogram: A parallelogram has no line of symmetry.

Table of Symmetry of Geometrical Shapes

 Shape Figure Number of lines of symmetry Equilateral triangle 3 Isosceles triangle 1 Square 4 Rectangle 2 Rhombus 2 Semicircle 1 Circle Infinite

Which one of the following is the complementary angle of $\mathbf{30}{}^\circ$ ?

(a) $30{}^\circ$          (b) $60{}^\circ$

(c) $90{}^\circ$           (d) All of these

(e) None of these

Explanation: Complementary angle of $30{}^\circ =90{}^\circ -30{}^\circ =60{}^\circ .$

In the options given below the pair of angles are given. Find the complementary pair.

(a)$51{}^\circ ,25{}^\circ$

(b) $61{}^\circ ,29{}^\circ$

(c) $45{}^\circ ,55{}^\circ$

(d) All of these

(d) None of these

Explanation: If the sum of pair of angles is $90{}^\circ$ then the pair is called cornplementary pair of angles. The sum of angles $61{}^\circ +29{}^\circ =90{}^\circ .$ Hence the pair of angles $61{}^\circ ,29{}^\circ$ is a complementary pair of angles,

Choose the pair of supplementary angle from the options given below?

(a) $100{}^\circ ,200{}^\circ$

(b) $105{}^\circ ,75{}^\circ$

(c) $30{}^\circ ,180{}^\circ$

(d) All of these

(e) None of these

Explanation: The sum of angles of the pair $105{}^\circ ,75{}^\circ =105{}^\circ +75{}^\circ =180{}^\circ$ Hence, the pair of angles $105{}^\circ ,75{}^\circ$ is a supplementary pair of angles.

Find the measurement of A if B and C are given when A, B and C are the angle of the triangle.

(a)$A=180{}^\circ -\left( B+C \right)$    (b) $A=180{}^\circ +\left( B-C \right)$

(c) $A=180{}^\circ +C-B$

(d) All of these

(e) None of these

Explanation: The sum of angles of the triangle $=180{}^\circ$. Hence, the angle $A=180{}^\circ -\left( B+C \right).$

If the base of a triangle is 18 cm and distance between base and opposite vertex is 10 cm, what will be the area of triangle?

(a) $206\text{ }c{{m}^{2}}$

(b) $10\text{ }c{{m}^{2}}$

(c) $90\text{ }c{{m}^{2}}$

(d) All of these

None of these

Explanation: The distance between base and opposite vertex is height,

$The\,area={{\frac{1}{2}}^{~}}\times base\times height$

$=\frac{1}{2}18\times 10=90c{{m}^{2}}.$

#### Other Topics

##### Notes - Geometry

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