**Geometry:**

**Lines & Angles:**

When we joint the two different points straight-way it makes a line. A line can be of different directions i.e., *horizontal lines* and *vertical lines*. Horizontal lines are flat in direction, like surface of water; while vertical lines are steep in direction, like one of the world steepest and tallest cliff of Boltoro Glacier in Gilgit-Baltistan, Pakistan.

**Intersection of lines:**

When two lines cross each other, in other words, they are intersecting, as shown bellow:

**Angles:**

When two lines intersect each other, they make an angle as shown in the figure bellow:

**Parallel lines:**

Two lines are parallel when they never intersect each other. Green and red lines are parallel as shown bellow:

**Prependicular lines:**

When two lines intersect each other such that they make an angle of 90 degree, these lines are said to be perpendicular lines, i.e., these lines are perpendicular to each other. *Line l* and *line m* are perpendicular lines given bellow:

Similarly, red line and blue line are perpendicular to each others, as given bellow:

**Angle of a lines:**

The angle of a line is always 180. For instance, when a line l intersect with another line m, the sum of adjacent angles are always equals to 180 degree, as shown bellow:

Also,

Remember that opposite angles are always equal, when the two lines are intersect with each others, as shown bellow:

**Coordinate Plane:**

When a horizontal line and a vertical line are placing such that they intersect at a point *(origin)*; it is said to make coordinate plane, also known as xy-plane. In coordinate plane / xy-pane, the horizontal line is called x-axis, while the vertical line is known as y-axis, as shown bellow:

The point of intersection of x-axis and y-axis is called origin (usually represented by ‘O’) whose coordinates are (0,0), where first 0 from left represent x-coordinate and second 0 from left represent y-coordinate, as shown bellow:

X-axis is similar to a number-line that gradually increase when we move towards right and decrease when moving left; but here x-axis always have center on 0, on right side of center of x-axis there exist positive numbers, and on left side of the center of x-axis there exist negative numbers.

Similarly, y-axis is also having the same trend but on vertical perspective. The center of y-axis is always 0. At top it has positive numbers and at bottom of the center of y-axis, it has negative numbers.

**Important Note:** The center of x-axis and that of y-axis is on the same point which is called *Origin*, that we have discussed earlier.

**Coordinates of a point on xy-plane:**

To find the coordinates of a point in xy-plane, one must draw an intelligent drawing (dotted line as shown bellow, that must pass through the point and one must be vertical (touching x-axis) and other must be horizontal (touching y-axis). the dotted line that passes through x-axis tells the x-coordinate, while the dotted line that passes through y-axis tells the y-coordinate of the point, as shown bellow:

After making intelligent drawing as explained earlier,

Now, remember that the coordinates of a point should be of such a form (x-coordinate, y-coordinate), i.e., x-coordinate on left-side of parenthesis (bracket) and y-coordinate on right-side in bracket.

Therefore, the coordinates of point A will be (3,5), as shown bellow:

**Circle:**

A circle is a set of points that are all having the same distance from a point *(known as center of the circle)*. By definition, every circle has a center, usually written as ‘O’.

**Radius of Circle:**

The radius of a circle is the distance between the center of the circle and a point on the circle. every circle also has radius, usually written as ‘r’, as shown bellow:

**Diameter of Circle:**

Twice the radius of a circle is its diameter, in other words,. It shows how much width of a circle is, as shown bellow:

Mathematically,

Diameter of Circle = d = 2r

**Circumference of Circle:**

The outer boundary length of a circle or distance around a circle is termed as circumference.

Let us consider a 100 meter wire is bend to make circle, this length is called circumference of that circle. i.e.,

Mathematically,

Circumference of Circle = 2 *π r*

**Area of Circle:**

The space inside the circle is known as area of that circle. i.e.,

Mathematically,

Area of Circle = *π r*^{2}

**Triangle:**

Any three sided figure is called triangle. In other words, when three lines are placing together so that each line intersect with the other two lines; the shape they form is known as triangle, as shown bellow:

**Important Note:** When the two sides of a triangle is given, the length of third side can’t be find, BUT possible range of third side can be find.

**Important Note:** Sum of all interior angles in any triangle is always 180 degree. So if two angles are given, we can find the third angle of the triangle.

**Right-angle Triangle:**

The triangle that has one angle of ’90’ is known as Right-angle Triangle, as shown bellow:

**Important Note:** The smallest angle of any triangle has smallest side in front of that angle, in other words, a smallest side of any triangle has smallest angle in front of that side. Similarly, the largest angle of any triangle has a largest side in front of it, in other words, the largest side has a largest angle in front of that side.

**Area of Triangle:**

Area of any triangle can be obtained by using following formula:

**Pythagoras Theorem:**

This theorem applies only to right-angle triangle. The theorem holds that, the sum of the square of the smaller sides of right-angle triangle is always equals to the square of the largest side of that right-angle triangle, as shown in figure bellow:

The horizontal side is called ‘Base’, and the height or the vertical side is called ‘perpendicular, while the greatest side which is in front of greatest angle (90) is called ‘Hypotenuse’, as shown bellow:

**Quadrilateral:**

Any four sided figure is called quadrilateral. There are many types of quadrilateral:

1.) Square

2.) Rectangle

3.) Rhombus

4.) Parallelogram

5.) Kite

6.) Trapezoid

7.) Other Quadrilateral.

**1.) Square:**

A quadrilateral, in which all sides are equal; also all angles are equal & 90 degree is known as square, as shown bellow:

Area of square = Length × Width

= *a × a*

= *a*^{2}

**2.) Rectangle:**

A quadrilateral, in which opposite sides are equal & parallel; also all angles are equal & 90 degree is known as rectangle, as shown bellow:

Area of rectangle = length × width

= a × b

= ab

**3.) Rhombus:**

A quadrilateral, in which all sides are equal, and opposite sides are are parallel; while all angles are not equal. If you change as shown bellow:

Area of rhombus = base × height

**4.) Parallelogram:**

A quadrilateral, in which opposite sides are equal and parallel, and also opposite angles are equal. But here all sides and angles are not equal, as shown bellow:

Area of parallelogram = base × height

Here complete syllabus relate to geometry and algebra has not been written like geometri sequence,tangents chords projection of a triangle arithmetic sets functions etc

Hi Bareera,

The complete syllabus is in advance level study plan. This is just basics.