Lines and Angles Exercise 6.2 - CBSE Class 9 NCERT Solutions

Understanding Parallel Lines and Transversals

When a straight line (called a transversal) intersects two other straight lines, it creates eight distinct angles. If those two lines are parallel, special geometric relationships unfold. First, corresponding angles—which lie in the same relative position at each intersection—are equal. Second, alternate interior angles (the 'Z' angles formed on opposite sides of the transversal but inside the parallel lines) are also equal. Third, consecutive interior angles (also known as co-interior or allied angles, located on the same side of the transversal inside the parallel lines) are supplementary, meaning their sum is exactly 180 degrees. To tackle class 9 maths lines and angles ex 6 2, you must be able to recognize these visual patterns instantly. Always look for 'F' shapes for corresponding angles, 'Z' shapes for alternate interior angles, and 'U' or 'C' shapes for co-interior angles.

Essential Theorems and Axioms for Exercise 6.2

Axiom 6.3 (Corresponding Angles Axiom)

If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

Theorem 6.2 (Alternate Interior Angles)

If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Theorem 6.4 (Consecutive Interior Angles)

If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary (adds up to 180°).

Theorem 6.6 (Lines Parallel to the Same Line)

Lines which are parallel to the same line are parallel to each other.

Step-by-Step Approach to Solve Ex 6.2 Problems

1. Identify Given Information and Parallel Lines — Carefully read the question. Look for notations like AB || CD or CD || EF. Mark these parallel lines on your diagram.
2. Locate the Transversal Line — Find the straight line cutting across the parallel lines. This line determines which angle pairs (alternate, corresponding, or co-interior) relate to each other.
3. Establish Algebraic Relations — Express the unknown angles using variables (like x or y). Set up equations using theorems (e.g., equate alternate interior angles or sum consecutive interior angles to 180°).
4. Solve the Equations and Verify — Solve the linear equations to find the variable. Substitute the value back to find all required angles, checking if linear pairs on straight lines still add up to 180°.

Score Boosters and Common Pitfalls

1. Don't Assume Parallelism: Never assume two lines are parallel just because they look parallel in the figure. You must only use parallel properties if the question explicitly states $AB \parallel CD$, or if you prove them parallel using converse theorems first.

1. State Your Reasons: In CBSE board exams, writing down the mathematical reason in brackets is mandatory! Writing just 'x = 50' gets zero marks. Write 'x = 50° (Alternate interior angles, AB || CD)' to secure full points.

1. Use Linear Pairs Correctly: Remember that angles on a straight line always add up to 180°. Use this to bridge gaps between interior and exterior angles.

Practice Questions with Solutions

• Q: In the given figure, three lines AB, CD, and EF are parallel (AB || CD and CD || EF). A transversal cuts them. If the ratio of interior angles y and z (where y is on CD and z is on EF on opposite sides of the transversal) is y : z = 3 : 7, and x is the interior angle on AB alternate to z, find x. A: Step 1: Write down the given relations. We are given AB || CD and CD || EF. By Theorem 6.6, lines parallel to the same line are parallel to each other, so AB || EF. Step 2: Relate x, y, and z. Since AB || EF, the alternate interior angles are equal. Therefore, x = z. Step 3: Relate x and y. Since AB || CD, the consecutive interior angles on the same side of the transversal are supplementary. Therefore, x + y = 180°. Since x = z, we can substitute z for x to get z + y = 180°. Step 4: Use the ratio y : z = 3 : 7. Let y = 3k and z = 7k. Then, 3k + 7k = 180° => 10k = 180° => k = 18°. Step 5: Calculate x. Since x = z = 7k, we have x = 7 * 18° = 126°. Final answer: x = 126°.
• Q: In a figure, AB || CD, EF is perpendicular to CD and angle GED = 126°. Find angle AGE, angle GEF, and angle FGE. A: Step 1: Understand the given components. We have AB || CD, EF ⊥ CD (which means ∠FED = 90°), and ∠GED = 126°. Step 2: Find ∠AGE. Since AB || CD and GE is a transversal line, the alternate interior angles are equal. Hence, ∠AGE = ∠GED = 126°. Step 3: Find ∠GEF. We know that ∠GED is made of two adjacent angles: ∠GEF and ∠FED. Therefore, ∠GED = ∠GEF + ∠FED. Substituting the values: 126° = ∠GEF + 90° => ∠GEF = 126° - 90° = 36°. Step 4: Find ∠FGE. The angles ∠AGE and ∠FGE lie on the straight line AB, forming a linear pair. Therefore, ∠AGE + ∠FGE = 180° => 126° + ∠FGE = 180° => ∠FGE = 180° - 126° = 54°. Final answer: ∠AGE = 126°, ∠GEF = 36°, and ∠FGE = 54°.
• Q: In a geometry problem, PQ || ST, angle PQR = 110° and angle RST = 130°. Find angle QRS. A: Step 1: Draw a construction line to help solve the problem. Draw a line EF parallel to ST passing through point R. Since PQ || ST and EF || ST, it follows that PQ || EF. Step 2: Use co-interior angles properties. Since PQ || EF and QR is a transversal, the consecutive interior angles sum to 180°. Thus, ∠PQR + ∠QRX = 180° (where X is on the line EF to the left of R). 110° + ∠QRX = 180° => ∠QRX = 70°. Step 3: Similarly, since ST || EF and SR is a transversal, the consecutive interior angles on the right sum to 180°. Thus, ∠RST + ∠SRY = 180° (where Y is on the line EF to the right of R). 130° + ∠SRY = 180° => ∠SRY = 50°. Step 4: Find ∠QRS. Since XRY is a straight line, the angles on it must sum to 180°. ∠QRX + ∠QRS + ∠SRY = 180° => 70° + ∠QRS + 50° = 180° => 120° + ∠QRS = 180° => ∠QRS = 60°. Final answer: ∠QRS = 60°.
• Q: In a figure, a transversal intersects lines AB and CD. The angles given are 50° adjacent to x on line AB (forming a linear pair), and y is vertically opposite to an angle of 130° on line CD. Find x and y, and show that AB || CD. A: Step 1: Find the value of x. The angles 50° and x lie on a straight line AB forming a linear pair. Therefore, 50° + x = 180° => x = 180° - 50° = 130°. Step 2: Find the value of y. The angle y and the angle measuring 130° are vertically opposite angles. Since vertically opposite angles are equal, y = 130°. Step 3: Prove AB || CD. From Steps 1 and 2, we see that x = 130° and y = 130°. Thus, x = y. Since x and y are alternate interior angles for lines AB and CD with respect to the transversal, and they are equal, the lines must be parallel. Final answer: x = 130°, y = 130°, and AB || CD because alternate interior angles are equal.

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