Exercise 4.3 — Parallel Lines

Lines and Angles – Transversal and Parallel Lines

When a line crosses two other lines at two distinct points, it is called a transversal. A transversal cutting across two lines creates exactly 8 angles — four at each intersection point. Understanding how these angles relate to one another is one of the most important skills in Class 9 Geometry, and forms the backbone of Exercise 4.3 in Chapter 4 of the CBSE, Telangana, and Andhra Pradesh Mathematics syllabus.

Types of Angle Pairs Formed by a Transversal

When a transversal meets two lines, the 8 angles it forms can be grouped into several important pairs. Knowing these pairs by name — and their properties — is essential for solving every question in this exercise.

• Corresponding Angles — Pairs (∠1, ∠5), (∠2, ∠6), (∠3, ∠7), (∠4, ∠8). They lie on the same side of the transversal, one at each intersection. When lines are parallel, corresponding angles are equal.
• Alternate Interior Angles — Pairs (∠3, ∠5) and (∠4, ∠6). These lie between the two lines but on opposite sides of the transversal. When lines are parallel, they are equal.
• Alternate Exterior Angles — Pairs (∠1, ∠7) and (∠2, ∠8). These lie outside the two lines on opposite sides. Equal when lines are parallel.
• Co-interior (Same-side Interior) Angles — Pairs (∠4, ∠5) and (∠3, ∠6). These lie between the lines on the same side of the transversal. When lines are parallel, they are supplementary (add up to 180°).
• Co-exterior (Same-side Exterior) Angles — Pairs (∠1, ∠8) and (∠2, ∠7). Lie outside both lines on the same side. Supplementary when lines are parallel.

Key Properties When Lines Are Parallel

All the properties above flip into powerful tools once you know two lines are parallel. These are the theorems you will use again and again in Exercise 4.3:

• Each pair of corresponding angles is equal: ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, ∠4 = ∠8
• Each pair of alternate interior angles is equal: ∠3 = ∠5 and ∠4 = ∠6
• Each pair of alternate exterior angles is equal: ∠1 = ∠7 and ∠2 = ∠8
• Each pair of co-interior angles is supplementary: ∠3 + ∠6 = 180° and ∠4 + ∠5 = 180°
• Each pair of co-exterior angles is supplementary: ∠1 + ∠8 = 180° and ∠2 + ∠7 = 180°
• Lines parallel to the same line are parallel to each other (transitive property of parallel lines).

Worked Example – Finding Unknown Angles (Q2)

Given AB ∥ CD and CD ∥ EF with the ratio y : z = 3 : 7, find x. Since AB ∥ EF (lines parallel to the same line are parallel), we use two key properties. First, x and y are co-interior angles between AB and EF, so x + y = 180°. Second, x and z are alternate interior angles, so x = z. Substituting gives y + z = 180°. Splitting 180° in the ratio 3:7 gives y = 54° and z = 126°. Since x = z, we get x = 126°.

Worked Example – Auxiliary Line Technique (Q4 & Q19)

Several questions in this exercise involve a point that does not lie on either parallel line. The standard strategy is to draw an auxiliary line through that point, parallel to both given lines. This splits the unknown angle into two parts, each of which can be found using co-interior or alternate interior angle properties. For example, in Q4 with PQ ∥ ST and ∠PQR = 110°, ∠RST = 130°, drawing line l through R parallel to both gives a = 70° (co-interior with PQ) and c = 50° (co-interior with ST). Since a + b + c = 180° (angles on a straight line), b = 60°, so ∠QRS = 60°.

Common Mistakes to Avoid

• Confusing co-interior with alternate angles — Co-interior angles are on the same side of the transversal and add to 180°. Alternate angles are on opposite sides and are equal. Mixing these up is the most frequent error in board exams.
• Forgetting the auxiliary line — When a vertex lies between two parallel lines (not on either one), you must draw a helper line before applying any property. Trying to apply properties directly leads to wrong answers.
• Not using the transitive property — If AB ∥ CD and CD ∥ EF, then AB ∥ EF. Many students miss this shortcut and struggle to connect the given information.
• Ratio problems — When a ratio like y:z = 3:7 is given and y + z = 180°, always find the total parts first (3 + 7 = 10), then multiply each fraction by 180°.
• Sign errors in algebraic questions — In problems like Q15(ii) where corresponding angles give 2x + 15 = 3x − 20, carefully rearrange: move all x terms to one side and constants to the other.

What This Exercise Prepares You For

Mastery of transversal and parallel line properties here directly supports the triangle chapter, where the angle sum property is proved using a parallel line drawn through a vertex. It also connects to Exercise 4.1 on basic angle terms and Exercise 4.2 on pairs of angles. For Class 10 students, these properties reappear in similar triangles and coordinate geometry when proving lines are parallel using slopes. In both CBSE and Telangana/Andhra Pradesh board exams, proof-based and calculation-based questions from Lines and Angles regularly carry 3–5 marks.