Class 9 Maths
Lines and Angles
Parallel Lines & a Transversal
A line a line which intersects two or more lines at distinct points is called a transversal. Line l intersects lines m and n at points P and Q respectively. Therefore, line l is a transversal for lines m and n. Observe that four angles are formed at each of the points P and Q. ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are called exterior angles, while ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are called interior angles
Corresponding angles: (i) ∠ 1 and ∠ 5 (ii) ∠ 2 and ∠ 6 (iii) ∠ 4 and ∠ 8 (iv) ∠ 3 and ∠ 7
Alternate interior angles: (i) ∠ 4 and ∠ 6 (ii) ∠ 3 and ∠ 5
Alternate exterior angles: (i) ∠ 1 and ∠ 7 (ii) ∠ 2 and ∠ 8
Interior angles on the same side of the transversal: (i) ∠ 4 and ∠ 5 (ii) ∠ 3 and ∠ 6
Interior angles on the same side of the transversal are also referred to as consecutive interior angles or allied angles or co-interior angles.
Now, let us find out the relation between the angles in these pairs when line m is parallel to line n.
Measure any pair of corresponding angles and find out the relation between them. You may find that: ∠ 1 = ∠ 5, ∠ 2 = ∠ 6, ∠ 4 = ∠ 8 and ∠ 3 = ∠ 7. From this, you may conclude the following axiom.
Axiom 3: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. This Axiom is also referred to as the corresponding angles axiom
Let’s check if the converse is also true.
Converse Statement: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel.
It can be verified by observations/construction that the converse statement is true. Thus we have one more Axiom.
Axiom 4: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.
This Axiom is also referred to as the corresponding angles axiom.
Lets now use this the corresponding angles axiom, to prove relationship between the alternate interior angles, ∠ 4 and ∠ 5 , ∠ 3 and ∠ 6 when a transversal intersects two parallel lines.
Theorem 2: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
Since this is a theorem, let’s try to prove it.
To prove: ∠ 4 = ∠ 5, given l is parallel to m.
Proof:
∠ 4 = ∠ 7 (Corresponding angle Axiom) – (i)
∠5 = ∠ 7 (Vertical Opposite angle Theorem) – (ii)
Using I & ii , we get ∠ 4 = ∠ 5
Converse of theorem 2 is also true. It is stated in Theorem 3.
Theorem3: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
Since this is a theorem, let’s try to prove it.
To prove: n is parallel to m, given ∠ 4 = ∠ 5
Proof:
∠ 4 = ∠ 5 (Given) – (i)
∠5 = ∠ 7 (Vertical Opposite angle Theorem) – (ii)
Using I & ii , we get ∠ 4 = ∠ 7, but these are corresponding angles. So as per Corresponding angle axion, line n is parallel to line m.
Similarly we have theorems for interior angle on same side of Transversal.
Theorem 4: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
Theorem 5: If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
Class 9 Maths Lines and Angles NCERT Chapter 6 Free Notes for Best Revision
Revision of Class 9 Maths Lines and Angles is a crucial aspect of effective learning. Revision plays a vital role in the learning process and is especially important before exams. Here are some key points you can consider emphasizing in your content:
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- Spacing Effect: Spacing out revision sessions over time, rather than cramming all at once, has been shown to improve long-term retention. Create a revision schedule leading up to the exams to allow for spaced practice.
- Regular Revision over Cramming: Regular and consistent revision throughout the academic year is very important. Waiting until the last moment to cram everything can be overwhelming and less effective than spaced-out revision.
- Self-Assessment: Assess your understanding periodically through quizzes or self-tests. This helps you to gauge your progress and identify areas that need further attention. Refer Class 9 Lines and Angles Online Tests.
- Balanced Approach: Remind students to strike a balance between revision and other activities. Adequate rest, exercise, and relaxation are essential for optimal learning and performance.
- Seeking Help: If you face difficulties during the revision process, Refer the videos of Class 9 Maths Lines and Angles. Clearing doubts promptly is crucial for a better grasp of the subject matter. You can always ask your doubts on Lines and Angles Class 9 Maths NCERT Chapter 6. “Ask a Question” section of LearnoHub.com
By highlighting the benefits and strategies of effective revision, you can approach your studies more mindfully and achieve better results in your exams. Best of luck bachhon!
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Remember, studying with full concentration is a skill that takes time and practice to develop. If you find your mind wandering during study sessions, gently bring your focus back to the task at hand and be patient with yourself. With consistent effort, you can improve your ability to concentrate and make the most of your study time.
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