Geometry

**Category : **5th Class

**Geometry**

**Line:**

A group of points in a straight path, extending on both sides infinitely form a line. Two points on the line denote it.

A line \[\overleftrightarrow{\text{AB}}\] is read as line AB'.

\[\overleftrightarrow{\text{AB}}\]=\[\overleftrightarrow{\text{BA}}\]

**Line Segment:**

A part of a line with two end points is called a line segment. It has a definite length.

Line segment

A line segment PQ is written as PQ.

\[\overline{\text{PQ }}\text{= }\overline{\text{QP}}\]

**Ray:**

A part of a line, which extends infinitely in one direction only, from a point, is a ray. The point is called the end-point of the ray.

A ray OP is written as 0?. A ray is denoted by writing the initial point first.

So, \[\overrightarrow{\text{OP}}\ne \overrightarrow{\text{PO}\text{.}}\]

**Angle:**

Two rays or line segments with a common end-point form an angle.

The common end-point is called the vertex of the angle and rays or line segments are called arms of the angle.

The unit of the angle is degree denoted by a small ° on the measure.

**e.g.,** \[\angle \text{AOB}\]=47°

\[\overrightarrow{\text{OA}}\] and \[\overrightarrow{\text{OB}}\] form an angle AOB. 0 is the vertex of \[\angle \text{AOB}\]and \[\overrightarrow{\text{OA}}\]and \[\overrightarrow{\text{OB}}\] are its arms or sides.

Z AOB is the same as Z BOA. Only vertex can also be used to denote an angle. Thus ZA means the angle whose vertex is A.

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**Types of Angles:**

(a) An angle whose measure is between 0° and 90° is called an acute angle.

(b) An angle whose measure is 90° is called a right angle.

(c) An angle whose measure is between 90° and 180° is called obtuse angle.

- Collinear Points: The points which lie on the same line, are called collinear points.
- Non-collinear points: The points which do not lie on the same line are called non-collinear points.

In the given figure, B, F, and E are collinear, while A, C and D are non-collinear points.

**Circle:**The set of points equidistant from a fixed point is called a circle. The fixed point is called the centre of the circle.

The centre of a circle is usually denoted as 0.

**Radius:**The distance between the centre and any point on the circle is called the radius. All the radii of the same circle have the same length. Infinitely many radii can be drawn in a circle.

In the figure, OA = OC = OD = radius

**Diameter:**A line segment that passes through the center of the circle and whose end-points lie on the circle is called a diameter. In the figure, CD is a diameter.

Diameter = 2 x Radius

All the diameters of a circle are of the same lengths.

Infinitely many diameters can be drawn in a circle.

**Chord:**A line segment which joins two points on the circumference of a circle is called a chord of the circle. A diameter is the longest chord of a circle.**Arc:**Any part of the circumference of a circle is called an arc.**Semicircle:**Half of the circumference of a circle is called a semicircle.**Triangle:**A closed figure formed by three line segments is called a triangle. A triangle is named using its vertices A, B, and C. It is denoted as AABC, read as triangle ABC.

**Note:** AABC can also be denoted ABCA or as ACAB.

**Characteristics of a triangle:**

(a) A triangle can be drawn only when the three given points are non-collinear.

(b) A triangle has three vertices, three sides and three angles.

(c) In the figure, the three line segments AB, BC and CA are the three sides of ?ABC.

\[\angle \]A, \[\angle \]B and \[\angle \]C are its three angles and A, B and C are its vertices.

**Properties of a triangle:**

(a) The sum of the lengths of any two sides of a triangle is greater than the length of its third side. AB + BC > AC

(b) The difference of the lengths of any two sides of a triangle is smaller than the length of the third side. AB - BC < AC

(c) The sum of the measures of three angles of a triangle is 180°.

(d) In a triangle ABC, \[\angle \]ABC + \[\angle \]BAC + \[\angle \]BCA = 180°.

**Quadrilateral:**

(a) A quadrilateral is a simple closed figure bounded by four line segments.

(b) A quadrilateral has four sides, four vertices, four angles and two diagonals (the line joining the opposite vertices.)

(c) The sum of measures of the four angles of a quadrilateral is 360°.

**Types of quadrilaterals:**

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**(a) Parallelogram:**

A parallelogram is a quadrilateral whose opposite sides are parallel and equal.

AB is parallel to DC. AC and BD are the diagonals.

BC is parallel to AD. AB = DC and BC = AD.

**(b) Rectangle:**

A rectangle is a parallelogram in which all the angles are right angles.

AC and BD are the diagonals. AB = DC and AD = BC. \[\angle \]A =\[\angle \] B = \[\angle \]C = 90°

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**(c) Square:**

A square is a rectangle in which all sides are equal. AB = BC = CD = DA

AC and BD are the diagonals.

\[\angle \]A= \[\angle \]B= \[\angle \]C= \[\angle \]D=90°

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**(d) Rhombus:**

A rhombus is a parallelogram in which all the sides are equal

AB = BC = CD = DA

AC and BD are the diagonals.

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**(e) Trapezium:**

A quadrilateral is called a trapezium if a pair of its opposite sides are parallel.

AB is parallel to DC.

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