**Lines and Angles**

**Fundamental terms and Definitions:**

**Line segment and ray:**The part of a straight line whose both ends are fixed is called a line segment. If one point of a line is fixed, it is called a ray.**Collinear points and Non-collinear points:**If three or more points lie on a straight line, they are called collinear point. If three or more points do not lie on a straight line, they are called non-collinear points.**Types of angles:**According to measurement, angles are of following types.**Acute angle:**If an angle lies between 0° and 90°, it is called an acute angle.- Right angle: An angle whose measurement is 90° is called a right angle.
- Obtuse angle: If an angle lies between 90° and 180°, it is called obtuse angle.
- Straight angle: An angle whose measurement is 180° is called a straight angle.
- Reflex angle: If an angle lies between 180° and 360°, it is called Reflex angle.

**Complementary angles and Supplementary angles:**If sum of two angles is equal to 90°, they mutually formed a set of complementary angles; e.g., Complementary angle of 30° is 60° and Complementary angle of 60° is 30°.

If sum of two angle is 180°, they are called supplementary to each other; e.g. Supplementary angle of 60° is 120° and supplementary angle of 120° is 60°.

- Adjacent angles: Two angles are said to be adjacent angles if they have a common vertex, a common side and their uncommon sides are situated at two different sides of the common side.

In the adjacent figure ∠EBC and ∠DBC are adjacent angles because point B is common to both of them and their uncommon sides BD and BE are opposite to the common side BC.

Similarly ∠DBC and ∠DBA are adjacent angles, but ∠EBC and ∠DBA are not adjacent angles as they donot have a common side.

**Linear pair of angles:**In the adjacent figure ∠AOC and ∠COB are adjacent angles and AOB is a straight line i.e., uncommon sides of adjacent sides form a straight line. Such angles are called linear pair of angles.**Vertically opposite angles:**If two straight lines AB and CD intersect each other at point O, then angles facing each other is called vertically opposite angles.

In the adjacent figure, ∠AOD and ∠BOC are one pair of vertically opposite angles, while ∠AOC and ∠BOD are another pair of vertically opposite angles.

**Transversal line:**A straight line intersecting two or more lines at different points is called a transversal line.

In the given figure straight line *n* intersects two different lines *l* and *m* respectively at point P and Q, so line n is a transversal line.

**Exterior angles and Interior angles:**In the figure given below, a transversal line*n*intersects two straight lines*l*and*m*respectively at*P*and*Q*. Around each point P and Q, four angles are formed, among these angles ∠1, ∠2, ∠7, ∠8 are called exterior angles while ∠3, ∠4, ∠5, ∠6 are called interior angles.

**Corresponding angles and Alternate angles:**In the figure drawn above- ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7 and ∠4 and ∠8 are called pair of corresponding angles.
- “∠4 and ∠6” and “∠3 and ∠5” are called pair of alternate interior angles.
- ∠1 and ∠7″ and “∠2 and ∠8” are called alternate exterior angles.
- ∠4 and ∠5″ and “∠3 and ∠6” are called consecutive interior angles or Alternate interior/exterior allied angles or co-interior angle.

All type of alternate angles are commonly known as alternate angles.

**Exterior angle and Interior opposite angle of a triangle:**In the adjacent figure sides BC, CA and AB of triangle ABC are respectively produced to points D, E and F. ∠ACD, ∠BAE and ∠CBF thus formed are called exterior angles of the triangle.

Interior angles ∠A and ∠B are called interior opposite angles to the exterior angle ∠ACD. Similarly ∠B and ∠C are interior opposite angles to the exterior angle BAE etc.

**Types of triangles according to sides:****Equilateral triangle:**When all the sides of a triangle are equal, it is called an equilateral triangle.**Isosceles triangle:**If any two sides of a triangle are equal, it is called an isosceles triangle.**Scalene triangle:**If sides of a triangle are unequal, it is called a scalene triangle.

**Types of triangles according to their angles:****Acute angle triangle:**If all the three angles of a triangle are acute, then the triangle is called an acute angle triangle.**Right angle triangle:**If one of the angle of a triangle is right angled (= 90°) then it is called a right angle triangle. A triangle has at most one right angle**Obtuse angle triangle:**If one of the angle of a triangle is obtuse (lies between 90° and 180°) then it is called an obtuse angle triangle. A triangle has at most one obtuse angle.

**Some Theorems (Results) based on angles and straight line**

- If two straight lines intersect each other then D vertically opposite angles are equal. In the given figure,

∠BOC = ∠AOD (Vertically opposite angles)

∠AOC = ∠BOD (Vertically opposite angles)

- If a ray is inclined on a line then the sum of linear pair of angles thus formed is equal to 180° and its converse is also true.

In the given figure ray OC is standing (inclined) on the line AB,

Therefore 𝜃 + 𝛼 = 180°

Conversely, if 𝜃 + 𝛼 = 180° then AOB will be a straight line.

In general, in the adjacent figure many rays are coming out from a point O on the straight line AB,

∴𝜃_{1} + 𝜃_{2}+ 𝜃_{3}+𝜃_{4} + 𝜃_{5}= 180°

Converse of this statement is also true.

- If a transversal line intersects two parallel lines then the pair of corresponding angles thus formed are equal and its converse is also true.

In the adjacent figure, two parallel lines l_{1} and l_{2} are intersected by a transversal line l_{3} and thus the following pair of corresponding angles are equal—

∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7 and ∠4 = ∠8

Conversely, it at least one of the pair of corresponding angles are equal (say ∠1 = ∠5) the lines l_{1} and l_{2} are parallel. If ∠2 = ∠6 then the two lines are parallel etc.

- If a transverse line intersects two parallel lines then pair of alternate angles are equal and its converse is also true. In above figure,

∠3 = ∠5 and ∠4 = ∠6(Alternate interior angles)

∠1 = ∠7 and ∠2 = ∠8(Alternate exterior angles)

- When a transversal line intersects two parallel lines, sum of consecutive interior angles (or allied angles or co-interior angles is equal to 180° (i.e., consecutive interior angles are supplementary), its converse is also true In the above figure,

∠4 + ∠5 = 180° and ∠3 + ∠6 = 180°

- The sum of all the three angles of a triangle is equal to 180° (i.e., two right angle)

∠A +∠B+ C = 180° or, ∠BAC + ∠ABC + ∠BCA = 180°

- If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of two interior opposite angles.

In fig, ∠ACX = ∠A + ∠B ∠CAY = ∠B + ∠C ∠ABZ = ∠A + ∠C

**Some Important Points to Solve Objective Questions.**

- Sum of all angles around a point is 360°

In the figure given below 0X + 02 + 03 + 04 + 05 = 360°

- If OB and OC are angle bisector of base angles ZB and LC of AABC, then LBOC = 90° +

Proof: In the adjacent figure,

∠OBC = and ∠OCB =

∠BOC = 180° –

= 180° –

= 180° –

(∴∠A + ∠B + ∠C = 180° ∠B + ∠C = 180° – ∠A)

= 90° +

- (i) Sum of all the internal angles of a Quadrilateral is 360°

(ii) Sum of all the internal angles of a pentagon (five sides) is 540°.

(iii) Sum of all the internal angles of a hexagon (six sides) is 720° etc.

It is due to the fact that a quadrilateral can be divided into two triangles, a pentagon can be divided into three triangles, a hexagon can be divided into four triangles etc.

- From the above result we can conclude that sum of all the internal angles of a n sided polygon is (n – 2) × 180°.
- Each angle of a n-sided regular polygon is (n – 2) × 180°/n
- In the above figure, side of a polygon are produced in the same order.

Angle 𝜃_{1} + 𝜃_{2} + 𝜃_{3} + … + 𝜃_{n} thus formed are called exterior angles.

The sum of all the exterior angles of a polygon is 360°.

i.e., 𝜃_{1} + 𝜃_{2} + 𝜃_{3} + … + 𝜃_{n}= 360°

If polygon is a regular polygon then each angle = 360°/n.

From the above discussion we can conclude that if side of a triangle or a quadrilateral or a pentagon or a hexagon etc. are produced in the same order, sum of exterior angles in all cases = 360°

- In sided polygon has

n sided polygon has diagonals.

at n = 4, Quadrilateral has diagonals.

at n = 5, Pentagon has = 5 diagonals,

at n = 6, Hexagon has = 9 diagonals etc.

- The angle between angle bisector of two adjacent angles of a quadrilateral is equal to half the sum of remaining angles.

In the given figure OD and OA are internal bisector of ∠D and ∠A respectively.

∠AOD = 180° –

= 180° – (∠A + ∠D)

= 180° – (360° – ∠B – ∠C)

= (∠B + ∠C)