13. Point, Segment, Line, Ray, Plane

 

POINT, SEGMENT, LINE, RAY, PLANE, ANGLE

(Straight angle, Zero angle, Complete angle, Reflex angle, Vertically opposite angles, Adjacent angles, Supplementary angles, Complementary angles)

♦ Learn and Remember:

1. Point : When a small dot is made with a very fine tip on a sheet of paper, we get a point.

e.g. P (Point P).

2. Segment : In the figure, AB is a segment. A segment has two end points. In the figure, A and B are the end points of segment AB. It is written in brief as seg AB. The length of seg AB is written as l(AB). If ((AB) = 5 cm, l(PQ)=5 cm, then seg AB ≅ seg PQ.

3. Line : Observe the line AB passing through the points A and B.

(i) Arrows on either side indicate that a line has infinite

length. It has infinite number of points.

(ii) Two distinct points determine a line. This means that one and only one line can be

drawn passing through two distinct points.

(iii) Infinite number of lines can be drawn from a point.

(iv) Two distinct lines intersect each other at a point. In the

figure, lines l and m intersect each other at the point P.

(v) If there exists a line containing all the given points, then

those points are called collinear points. In the figure, A, B, C and D are collinear points.

(vi) If a line cannot be drawn passing through three or more points, then those points

are called non-collinear points.

(vii) If a number of lines intersect each other at the same point,

then those lines are said to be concurrent. In the figure, lines a, b, c and d

intersect each other at the point O. Hence a, b, c and d are called concurrent lines.

O is the point of concurrence.

(viii) Coplanar lines which do not intersect each other are said to be

parallel lines. In the figure, line l and line m are parallel lines.

(ix) Lines perpendicular to the same line are parallel to

each other.

Line m and line n are perpendicular to line l.

∴ line m || line n.

4. Ray :

Observe the ray AB shown in the figure.

(1) A is the origin and B is any other point of the ray AB.

(ii) Arrow on one side indicates that the ray has infinite length on that side.

(iii) If P, Q, R are points on the line 1 such that P-Q-R, then the P Q

ray QP and the ray QR are called opposite rays.

5. Plane :

(i) A plane is represented as shown in the figure.

(ii) A plane is infinite on all sides. A plane has infinite number

of points.

(iii) Two parallel lines or two intersecting lines or three non-collinear points. determine 

a plane. This means one and only one plane can pass through two parallel Lines 

or two intersecting lines or three non-collinear

 (iv) There is exactly one plane passing through a line and a point

not on the line.

6. Angle:

(i) Two non-collinear rays with a common origin form an angle.

(ii) The sign '∠' is used to refer to an angle. Such as ∠ABC or ∠B.

(iii) In the figure, the vertex of the angle is point B. Ray BA and 

ray BC are called the arms of ∠ABC.

(iv) The points on the plane of the angle are divided into three 

regions :

(a) on the angle (Points A, B, C)

(b) its interior portion (Point P) 

(c) its exterior portion (Points M and N).

No point on the plane is common to any two of the three parts.

(v) If the measures of two angles are the same, they are said to be congruent angles.

(vi) ∠ABC ≅ ∠PQR means ∠ABC and ∠PQR are congruent and is read as `Angle ABC is

congruent to angle PQR”.

(vii) If ∠ABC ≅ ∠PQR, then m ∠ABC =m ∠PQR and vice-versa.

(viii) If ∠ABC = ∠PQR and ∠PQR ≅ ∠XYZ, then ∠ABC ≅ ∠XYZ.

7. Types of angles :

(a) Right angle : An angle of measure 90° is called a right angle.

In the figure, ∠CDF is a right angle.

(b) Acute angle : An angle which measures more than 0° but less

than 90° is called an acute angle. In the figure, ∠LMN is an acute angle.

(c) Obtuse angle : An angle which measures more than 90° but

less than 180° is called an obtuse angle. In the figure, ∠TQR is an obtuse angle.

(d) Straight angle : An angle of measure 180° is called a

straight angle. In the figure, ∠AOB is a straight angle.

(e) Zero angle : An angle which measures 0° is called a zero

angle. (In this case, there is no rotation of ray OA.]

(f) Complete angle : An angle of measure 360° is called a

complete angle.

The figure shows a complete angle.

(g) Reflex angle : An angle which measures more than 180° but

less than 360° is called a reflex angle.

In the figure, ∠DEF is a reflex angle.

8. Adjacent angles :

When two angles have a common vertex, a common arm and no  common interior portion, they are said to be the adjacent angles of

each other. In the figure, ∠AOB and ∠BOC are adjacent angles.

9. Linear pair of angles :

(i) When the two sides of adjacent angles, which are not common, form a pair of opposite rays, the adjacent angles are said to be the angles in a linear pair.

 In the figure, ∠AMB and ∠BMC are the angles in a linear pair.

(ii) The sum of the measures of the angles in a linear pair is 180°.

10. Vertically opposite angles :

(i) When the arms of one angle are the opposite rays of the arms of the other angle,

these angles are called the vertically opposite angles of each other.

(ii) In the adjoining figure, ∠AMD and ∠BMC is one pair of the 

opposite angles. ∠AMC and ∠BMD is the other pair.

(iii) The vertically opposite angles are congruent.

(iv) If vertically opposite angles are right angles, the lines formed

by the opposite rays are perpendicular to each other.

11. Supplementary angles :

(i) When the sum of the measures of two angles is 180°, they are called supplementary angles.

(ii) The supplementary angles need not be adjacent angles.

 (iii) Adjacent angles may be supplementary. If the adjacent angles are supplementary,

they form a linear pair.

(iv) The supplementary angle of a right angle is a right angle.

(v) The supplementary angle of an obtuse angle is an acute angle and vice-versa.

(vi) The supplementary angles of the same angle, are all congruent.

12. Complementary angles :

(i) When the sum of the measures of two angles is 90°, they are said to be the complementary angles of each other.

(ii) In a pair of complementary angles, both the angles are acute angles.

(iii) The complementary angles of the same angle, are congruent.

[Note : The difference between the supplementary and the complementary angles of a given angle is 90°. ]

 13. An Angle Bisector :

(i) If point D is in the interior of ∠AMB such that ∠AMD ≅ ∠DMB,

then the ray MD is called the bisector of ∠AMB.

(ii) There is one and only one angle bisector of an angle

14. Angles made by a transversal :

(i) Interior angles : If a line intersects two coplanar lines in

distinct points, two pairs of interior angles are formed.

In the figure, ∠a and ∠b also ∠c and ∠d are interior angles.

(ii) Alternate angles : When two lines are intersected by a

transversal, then two pairs of alternate angles are formed.

e.g. when the lines land mare intersected by the transversal n, then ∠a and ∠b is one pair of alternate angles and

∠c and ∠d is the other pair.

(iii) Corresponding angles : When two lines are intersected

by a transversal, then four pairs of corresponding angles are formed.

In the adjoining figure, line l and line m are intersected by a transversal n.

Pairs of corresponding angles are ∠a, ∠b; ∠c, ∠d; Le, ∠f and ∠g, ∠h.