Class 9 — Mathematics (NCERT)
Lines and Angles
Class 9
- ✓By the end of this lesson students will be able to identify and define different pairs of angles such as adjacent angles, linear pairs, and vertically opposite angles.
- ✓By the end of this lesson students will be able to state and apply the properties of angles formed when a transversal intersects two parallel lines.
- ✓By the end of this lesson students will be able to state and apply the Angle Sum Property of a triangle.
- ✓By the end of this lesson students will be able to solve problems involving various angle relationships and properties of parallel lines and triangles.
Key concepts
A line is a breadthless length. A ray is a part of a line with one endpoint. A line segment is a part of a line with two endpoints. An angle is formed when two rays originate from the same endpoint. The rays forming the angle are called the arms of the angle and the common endpoint is called the vertex of the angle.
1. Acute Angle: An angle whose measure is between 0° and 90°.\n2. Right Angle: An angle whose measure is exactly 90°.\n3. Obtuse Angle: An angle whose measure is between 90° and 180°.\n4. Straight Angle: An angle whose measure is exactly 180°.\n5. Reflex Angle: An angle whose measure is between 180° and 360°.
1. Complementary Angles: Two angles whose sum is 90°.\n2. Supplementary Angles: Two angles whose sum is 180°.\n3. Adjacent Angles: Two angles are adjacent if they have a common vertex, a common arm, and their non-common arms are on opposite sides of the common arm.\n4. Linear Pair of Angles: If two adjacent angles form a straight line (their non-common arms are opposite rays), they are called a linear pair of angles. The sum of angles in a linear pair is 180°.\n5. Vertically Opposite Angles: When two lines intersect, the angles opposite to each other at the point of intersection are called vertically opposite angles. Vertically opposite angles are always equal.
A transversal is a line that intersects two or more lines at distinct points. When a transversal intersects two parallel lines, several pairs of angles are formed with specific properties:\n1. Corresponding Angles: Angles in the same relative position at each intersection. If two parallel lines are intersected by a transversal, then each pair of corresponding angles is equal.\n2. Alternate Interior Angles: Angles on opposite sides of the transversal and between the two parallel lines. If two parallel lines are intersected by a transversal, then each pair of alternate interior angles is equal.\n3. Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the two parallel lines. If two parallel lines are intersected by a transversal, then each pair of alternate exterior angles is equal.\n4. Consecutive Interior Angles (or Co-interior Angles / Allied Angles): Angles on the same side of the transversal and between the two parallel lines. If two parallel lines are intersected by a transversal, then each pair of consecutive interior angles is supplementary (their sum is 180°).
The sum of the angles of a triangle is 180°. That is, for a triangle ABC, ∠A + ∠B + ∠C = 180°.\nExterior Angle Property of a Triangle: 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.
Key facts to remember
- 1A linear pair of angles always sums to 180°.
- 2Vertically opposite angles are always equal.
- 3If two parallel lines are intersected by a transversal, then corresponding angles are equal.
- 4If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
- 5If two parallel lines are intersected by a transversal, then consecutive interior angles are supplementary (sum to 180°).
- 6The sum of the angles in any triangle is always 180°.
- 7An exterior angle of a triangle is equal to the sum of its two interior opposite angles.
Worked examples
Example 1
In the given figure, lines PQ and RS intersect at O. If ∠POR : ∠ROQ = 5 : 7, find all the angles.
Answer
∠POR = 75°, ∠ROQ = 105°, ∠POS = 105°, ∠SOQ = 75°.
Remember that angles in a linear pair sum to 180° and vertically opposite angles are equal.
Example 2
In the given figure, if AB || CD, ∠APQ = 50° and ∠PRD = 127°, find x and y.
Answer
x = 50°, y = 77°.
Identify the correct transversal for each pair of angles to apply the properties of parallel lines accurately.
Example 3
In ΔABC, if ∠A = 60°, ∠B = 70°, find ∠C.
Answer
∠C = 50°.
This is a fundamental property of triangles. Always ensure the sum is 180°.
Common mistakes
- ✗Confusing corresponding angles with alternate interior angles or vice-versa.
- ✗Assuming lines are parallel when it is not explicitly stated or proven, leading to incorrect application of angle properties.
- ✗Incorrectly applying the linear pair axiom or vertically opposite angles theorem.
- ✗Making calculation errors when adding or subtracting angles, especially with multiple steps.
- ✗Forgetting that the sum of angles in a triangle is 180° and using other values.
Exam tips
- ★Always draw a clear diagram, even if one is provided, and mark the given angles and lines.
- ★Clearly state the geometric reason for each step in your solution (e.g., 'Linear Pair Axiom', 'Alternate Interior Angles').
- ★Break down complex problems into smaller, manageable steps using one property at a time.
- ★Double-check your calculations, especially when dealing with multiple angles.
- ★Practice identifying different angle pairs quickly and accurately in various configurations.
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