These comprehensive RBSE Class 9 Maths Notes Chapter 6 Lines and Angles will give a brief overview of all the concepts.
RBSE Class 9 Maths Chapter 6 Notes Lines and Angles
1. Angle : An angle is formed when two rays originate from the same endpoint. The rays making an angle are called the arms of the angle and the endpoint is called the vertex of the angle. The angle formed by the rays OA and OB as shown in the figure is denoted by ∠AOB or ∠BOA. Here, the common point O is known as vertex of angle and OA and OB are the arms. Sometimes for convenience, an angle is also denoted by numbers or words.
2. Measurement of Angle: Every angle has a measure. The unit of angle measurement is called a “degree”. If we divide a right angle into 90 equal parts then, each part is called a degree.
Since one right angle is divided into 90 equal parts, we can say 1 right angle = 90 degree or 90°.
Similarly, every degree is divided into 60 equal parts and each such part is known as Kala (1 minute).
If one minute is divided into 60 equal parts, then each part is known as Vikla (1 second).
Thus, 1° = 60 Kala (minute) = 60′
1 Kala (minute) = 60 Vikala (second), i.e., 1′ = 60″
In symbolic form, one degree, one minute and one second can be expressed as 1° and 1″.
A protractor is used to measure angles. It has markings from 0° to 180° scale.
3. Acute Angle : An angle whose measure is less than 90° is called an acute angle.
4. Right Angle : An angle whose measure is 90° is called a right angle.
5. Obtuse Angle : An angle whose measure is more than 90° but less than 180° called an obtuse angle.
6. Straight Angle : An angle whose measure is 180° is called a straight angle.
7. Reflex Angle : An angle whose measure is more than 180° but less than 360c called reflex angle.
8. Adjacent Angles: Two angles are adjacent, if they have a common vertex, a common arm and their non-common arms are on different sides of the common arm.
9. Supplementary Angles : Two angles whose Bum is 180° are called supplementary angles.
10. Complementary Angles : Two angles whose sum is 90° are called complementary angles.
11. Vertically Opposite Angles : Two angles are said to form a pair of vertically opposite angles, if their arms form two pairs of opposite rays. In the given figure, lines AB and CD intersect each other at point O and make angles
∠1, ∠2, ∠3 and ∠4. Here ∠1 and ∠3 are vertically opposite angles while ∠2 and ∠4 are another pair of vertically opposite angles.
12. Angles around a point: If a number of rays start from a point and angles are formed, then angles obtained in this form are known as angles about or around a point. In the fig. ∠1, ∠2, ∠3, ∠4, ∠5 and ∠6 are various angles formed around the point O. Also, sum of angles around a point = 360° i.e., ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°
13. Linear Pair angles : If the non-common arms, of two adjacent angles form a fine, then these angles are known as linear pair of angles (their sum is always equal to 180°). Two adjacent angles are linear pair angles only if they are supplementary to each other.
14. Theorem 6.1: If two lines intersect each other, then the vertically opposite angles are equal to each other.
In the figure, ∠COB =∠AOD and ∠AOC = ∠BOD
15. Sub-theorem : Half of vertically opposite angles are in a line.
16. Angles made by a transversal with two or more lines : If a group of two or more fines are intersected by a line at different points, then it is known as transversal. In the given diagram EF is the transversal.
Here, in the figure, three fines AB, CD and EF are making a total of eight angles.
There are four angles ∠1, ∠2, ∠3 and ∠4 at the point G and ∠5, ∠6, ∠7 and ∠8 at the point H. They are named according to their position.
17. Corresponding Angles : The following type of angles are known as corresponding angles-
(i) ∠1and ∠5
(ii) ∠2 and ∠6
(iii) ∠4 and ∠8
(iv) ∠3 and ∠7
It is clear that, corresponding angles fie on the same side of the transversal and their positions are also similar.
18. Alternate Angles : The following type of angles are known as alternate angles-
(i) ∠3 and ∠5 (ii) ∠2 and ∠8
It is clear that, alternate angles fie on opposite sides of the transversal.
19. Interior angles : The following type of angles are known as interior angles.
(i) ∠2 and ∠5 (ii) ∠3 and ∠8
Interior angles lie on the same side of the transversal.
20. Exterior Angles: In the figure, ∠1, ∠4, ∠6 and ∠7 are known as exterior angles. Generally there is no relationship between the various types of angles mentioned above, but if a transversal cuts two or more parallel lines, then:
(i) corresponding angles are equal.
(ii) alternate angles are equal.
(iii) interior angles are supplementary.
The converses of the above statements are also true.
21. Theorem 6.2 : If a transversal intersects two or more parallel lines, then alternate interior angles are equal to each other.
22. Theorem 6.3 (Converse of Theorem 6.2): If a transversal intersects two lines and alternate interior angles are equal then the lines are parallel to each other.
23.. Theorem 6.4: If a transversal intersects two parallel lines, then the sum of interior angles is equal to two right angles or 180°.
24. Theorem 6.5 (Converse of theorem 6.4) : If a transversal intersects two lines and the sum of interior angles on the same side of the transversal is 180° two right angles, then both lines are parallel.
25. Sub-theorem: Perpendiculars drawn to the same line are parallel to each other.
26. If a ray stands on a line then the sum of two adjacent angles so formed is 180° and conversely if the sum of two adjacent angles is 180°, then the ray stands on a line. The above two theorems together may be re-stated as :
“Two adjacent angles are a linear pair if they are supplementary.” This is known as linear pair property.
27. If a transversal intersects two parallel lines, then:
(i) each pair of corresponding angles is equal.
(ii) each pair of alternate interior angles is equal. %
(iii) each pair of interior angles on the same side of the transversal are supplementary.
28. If a transversal intersects two lines such that:
(i) a pair of corresponding angles is equal, or
(ii) a pair of alternate interior angles is equal, or .
(iii) a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel to each other.
29. Lines which are parallel to the same line are parallel to each other.
30. The sum of three angles of a triangle is 180°.
31. If a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angles.