Welcome back, future mathematicians! In this comprehensive guide, we will walk you through the step-by-step solutions for Class 9 Mathematics Chapter 6: Lines and Angles, Exercise 6.2. This exercise focuses on the properties of parallel lines intersected by a transversal, including alternate interior angles, corresponding angles, and co-interior angles. Let's dive in and master these geometry concepts!
Question 1
In the given figure, find the values of x and y and then show that AB || CD.
Given: A transversal line intersects two lines AB and CD. The angle adjacent to x on the straight line is 50°, and the vertically opposite angle to y is 130°.
Solution:
Step 1: Find the value of x
The angles 50° and x lie on a straight line and form a linear pair.
∴ 50° + x = 180° (Linear Pair Axiom)
⇒ x = 180° - 50°
⇒ x = 130°
Step 2: Find the value of y
The angle y and the angle measuring 130° are vertically opposite angles.
∴ y = 130° (Vertically Opposite Angles)
Step 3: Prove AB || CD
From Step 1 and Step 2, we observe that:
x = y = 130°
Since x and y are alternate interior angles and they are equal, by the converse of the alternate interior angles theorem, the lines must be parallel.
∴ AB || CD. (Hence Proved)
Question 2
In the figure, if AB || CD, CD || EF and y : z = 3 : 7, find x.
Solution:
Step 1: Establish relation between lines
Given that AB || CD and CD || EF.
Lines parallel to the same line are parallel to each other.
∴ AB || EF
Step 2: Establish relation between angles
Since AB || EF, then:
x = z ————— (Equation 1) [Alternate interior angles]
Also, since AB || CD, the interior angles on the same side of the transversal are supplementary:
x + y = 180° ——— (Equation 2) [Co-interior angles]
Step 3: Solve using ratio
Substitute Equation 1 in Equation 2:
z + y = 180°
Given ratio, y : z = 3 : 7. Let y = 3k and z = 7k.
⇒ 3k + 7k = 180°
⇒ 10k = 180°
⇒ k = 18°
Therefore, z = 7 × 18° = 126°.
Since x = z, we have:
x = 126°
Question 3
In the figure, if AB || CD, EF ⊥ CD and ∠GED = 126°, find ∠AGE, ∠GEF and ∠FGE.
Solution:
Step 1: Find ∠AGE
Since AB || CD and GE is the transversal line:
∠AGE = ∠GED (Alternate interior angles)
∴ ∠AGE = 126°
Step 2: Find ∠GEF
We are given that ∠GED = 126°.
From the diagram, ∠GED = ∠GEF + ∠FED.
Since EF ⊥ CD, ∠FED = 90°.
⇒ 126° = ∠GEF + 90°
⇒ ∠GEF = 126° - 90°
⇒ ∠GEF = 36°
Step 3: Find ∠FGE
∠AGE and ∠FGE form a linear pair on the straight line AB.
⇒ ∠AGE + ∠FGE = 180°
⇒ 126° + ∠FGE = 180°
⇒ ∠FGE = 180° - 126°
⇒ ∠FGE = 54°
ЁЯТб Aptitude Corner (NTSE / Olympiad Special)
The M-Rule (Zig-Zag Angle Concept):
When solving competitive exams like NTSE, look out for the "M" shaped geometry problems. If line L1 is parallel to line L2, and a sharp bend creates an inward vertex:
Sum of left-facing sharp angles = Sum of right-facing sharp angles
For example, if you have a reflex angle shaped like >
between two parallel lines, you can easily find the middle angle by drawing a parallel auxiliary line. This saves valuable time during exams!
Great job! You have successfully solved the key problems of Exercise 6.2. Regular practice of these parallel line properties will help you score maximum marks in your MP Board Term exams and talent search tests. If you have any questions, drop a comment below!
Academic Discussion & Doubts