Lines and Angles
Every building, road, and bridge is built using principles of lines and angles. Architects, engineers, and even carpenters use this every day. For us in Class 7, understanding lines and angles is the foundation for all the geometry that follows — triangles, quadrilaterals, and beyond.
Let’s get the concepts sorted, one step at a time.
Types of Angles
An angle is formed when two rays meet at a common point called the vertex.
We measure angles in degrees (°).
Acute Angle
An angle that is less than 90°.
Example: 30°, 45°, 60°, 89° — all acute.
Right Angle
An angle that is exactly 90°.
A corner of a square or a sheet of paper is a right angle.
Obtuse Angle
An angle greater than 90° but less than 180°.
Example: 100°, 120°, 150° — all obtuse.
Straight Angle
An angle that is exactly 180°.
A straight line forms a straight angle. Both rays point in exactly opposite directions.
Reflex Angle
An angle greater than 180° but less than 360°.
We rarely draw these, but the angle on the “other side” of an acute or obtuse angle is a reflex angle.
Complete Angle
An angle of exactly 360° — a full rotation.
Easy way to remember: Acute → small (like “a cute little angle”), Obtuse → big and blunt. Right angle is exactly 90° — the corner of your notebook!
Pairs of Angles
What makes lines and angles interesting is what happens when we look at angles in pairs.
Complementary Angles
Two angles are complementary if their sum is 90°.
40° + 50° = 90° → These two are complementary. 30° + 60° = 90° → Complementary pair.
Note: The two angles don’t need to be next to each other. They just need to add up to 90°.
Finding the complement: Complement of any angle θ = 90° - θ.
Complement of 35° = 90° - 35° = 55°
“Complementary” sounds like “complete” — they complete each other to make 90°.
Supplementary Angles
Two angles are supplementary if their sum is 180°.
110° + 70° = 180° → Supplementary pair. 90° + 90° = 180° → Two right angles are supplementary.
Finding the supplement: Supplement of any angle θ = 180° - θ.
Supplement of 110° = 180° - 110° = 70°
Memory trick: “S” in Supplementary → “S” in Straight angle (180°). “C” in Complementary → “C” in Corner (right angle = 90°).
Adjacent Angles
Two angles are adjacent if:
- They have a common vertex (same corner point).
- They have a common arm (one side is shared).
- Their non-common arms are on opposite sides of the common arm.
Think of a clock at 12:00. The minute and hour hands are along the same line. As the minute hand moves to 3, it forms a 90° angle with the hour hand — this pair of angles (the angle swept and the remaining angle) are adjacent.
Linear Pair of Angles
When two adjacent angles together form a straight angle (180°), they form a linear pair.
If one angle is 70°, the other in the linear pair is 110°.
Key property: Linear pair always adds to 180°.
If two angles form a linear pair: Angle 1 + Angle 2 = 180°
Vertically Opposite Angles
When two straight lines cross each other, they form four angles. The angles that are across from each other (diagonally opposite) are called vertically opposite angles.
When lines AB and CD cross at point O, the angles at O are: ∠AOC, ∠COB, ∠BOD, ∠DOA. ∠AOC and ∠BOD are vertically opposite. ∠COB and ∠DOA are vertically opposite.
Key property: Vertically opposite angles are always equal.
If one angle is 60°, its vertically opposite angle is also 60°. The other two angles are each 120° (since 60° + 120° = 180° as a linear pair).
Vertically opposite angle questions are very common in exams. Remember: they are ALWAYS equal. This is a theorem you must know.
Parallel Lines
Two lines are parallel if they lie in the same plane and never meet, no matter how far they are extended.
We write “line l is parallel to line m” as l ∥ m.
Real-life examples:
- The two rails of a railway track
- Opposite sides of a ruler
- Horizontal lines on a ruled notebook
Transversal
A transversal is a line that cuts across (intersects) two or more lines.
When a transversal cuts two parallel lines, it creates 8 angles — 4 at each intersection point.
Let’s call the two parallel lines l and m, and the transversal t.
At the intersection with line l: angles 1, 2, 3, 4. At the intersection with line m: angles 5, 6, 7, 8.
These 8 angles fall into special pairs with special relationships.
Angles Formed by a Transversal
Corresponding Angles
Corresponding angles are in the same position at each intersection — both in the top-left, or both in the bottom-right, etc.
Pairs: (∠1, ∠5), (∠2, ∠6), (∠3, ∠7), (∠4, ∠8)
Property: When the lines are parallel, corresponding angles are equal.
If ∠1 = 65°, then ∠5 = 65°.
Corresponding angles make an “F” shape (or a reverse F). If you can see the letter F in the diagram, those are corresponding angles.
Alternate Interior Angles
These are angles that are:
- On alternate (opposite) sides of the transversal.
- Between (interior to) the two parallel lines.
Pairs: (∠3, ∠6) and (∠4, ∠5).
Property: When the lines are parallel, alternate interior angles are equal.
If ∠3 = 75°, then ∠6 = 75°.
Alternate interior angles make a “Z” shape (or reverse Z). Look for the letter Z in the figure.
Co-interior Angles (Also Called Allied or Same-Side Interior Angles)
These are angles that are:
- On the same side of the transversal.
- Between the two parallel lines.
Pairs: (∠3, ∠5) and (∠4, ∠6).
Property: When the lines are parallel, co-interior angles are supplementary (they add up to 180°).
If ∠3 = 110°, then ∠5 = 70°.
Co-interior angles make a “C” shape (or reverse C). They are “C”o-interior — and they add up to 180°.
When l ∥ m and t is a transversal:
- Corresponding angles = equal
- Alternate interior angles = equal
- Co-interior angles = supplementary (add to 180°)
Using These Properties to Find Missing Angles
Example: Two parallel lines are cut by a transversal. One angle is 65°. Find the angles marked as corresponding, alternate interior, and co-interior.
- Corresponding angle = 65° (equal)
- Alternate interior angle = 65° (equal)
- Co-interior angle = 180° - 65° = 115° (supplementary)
Example: A transversal cuts two parallel lines. One angle is (3x + 10)° and its alternate interior angle is 70°. Find x.
Since alternate interior angles are equal:
3x + 10 = 70 3x = 60 x = 20
Summary Table
| Angle Pair | Condition | Property |
|---|---|---|
| Complementary | Any two angles | Sum = 90° |
| Supplementary | Any two angles | Sum = 180° |
| Linear pair | Adjacent, on a straight line | Sum = 180° |
| Vertically opposite | When two lines cross | Always equal |
| Corresponding (parallel lines) | Same position at each intersection | Equal |
| Alternate interior (parallel lines) | Opposite sides, between lines | Equal |
| Co-interior (parallel lines) | Same side, between lines | Sum = 180° |
5 Common Mistakes to Avoid
Mistake 1: Confusing complementary and supplementary
Complementary = 90°. Supplementary = 180°. Many students mix these up. Memory trick: “C” comes before “S” in the alphabet, and 90 comes before 180.
Mistake 2: Assuming vertically opposite angles are supplementary
Vertically opposite angles are EQUAL, not supplementary. They add to 180° only if they are right angles (both 90°). In general, they are just equal to each other.
Mistake 3: Applying parallel-line angle properties to non-parallel lines
Corresponding, alternate interior, and co-interior rules ONLY work when the two lines are parallel. If the lines are not parallel, these rules don’t apply.
Mistake 4: Confusing alternate interior with alternate exterior angles
Interior angles are between the two parallel lines. Exterior angles are outside. Many students mix these up. In the NCERT Class 7 syllabus, the focus is on alternate interior angles.
Mistake 5: Saying “vertically opposite” when they mean “adjacent”
Adjacent angles share a side. Vertically opposite angles are directly across the intersection. They are not the same thing at all.
Practice Questions
Question 1: Find the complement of 62°.
Complement = 90° - 62° = 28°
Check: 62° + 28° = 90° ✓
Question 2: Two angles are supplementary. One is 45°. Find the other.
Supplement = 180° - 45° = 135°
Check: 45° + 135° = 180° ✓
Question 3: Two lines intersect. One of the angles formed is 130°. Find all four angles.
Angle 1 = 130° (given) Angle 2 = 180° - 130° = 50° (linear pair with Angle 1) Angle 3 = 130° (vertically opposite to Angle 1) Angle 4 = 50° (vertically opposite to Angle 2)
The four angles are 130°, 50°, 130°, 50°.
Question 4: Two parallel lines are cut by a transversal. One angle is 75°. Find the co-interior angle on the same side.
Co-interior angles are supplementary (add to 180°). Co-interior angle = 180° - 75° = 105°
Question 5: An angle is 10° more than twice its complement. Find the angle.
Let the angle = x. Its complement = 90° - x.
Given: x = 2(90° - x) + 10° x = 180° - 2x + 10° x + 2x = 190° 3x = 190° x = 190° ÷ 3 ≈ 63.3°
Wait — let’s recheck. x = 2(90 - x) + 10: x = 180 - 2x + 10 = 190 - 2x 3x = 190 x ≈ 63.3°
The angle is approximately 63.3° and its complement is about 26.7°. Check: 2(26.7) + 10 = 53.4 + 10 = 63.4° ≈ 63.3° ✓
Question 6: Two parallel lines are cut by a transversal. An alternate interior angle is (2x + 15)° and the corresponding angle at the first intersection is 65°. Find x.
The angle at the first intersection = 65°. Since alternate interior angles are equal (when lines are parallel), the alternate interior angle at the second intersection = 65°.
But wait — the corresponding angle to this alternate interior angle is also 65° (corresponding angles are equal).
We need to think more carefully. The alternate interior angle of 65° is: 2x + 15 = 65° 2x = 50° x = 25°
Question 7: In a linear pair, the angles are in the ratio 2:3. Find both angles.
Let the angles be 2k and 3k. Linear pair sum = 180°: 2k + 3k = 180° 5k = 180° k = 36°
Angle 1 = 2 × 36° = 72° Angle 2 = 3 × 36° = 108°
Check: 72° + 108° = 180° ✓
Question 8: Can two angles be both complementary and supplementary? Explain.
No. Complementary means their sum is 90°. Supplementary means their sum is 180°. These are different sums, so the same pair cannot satisfy both conditions simultaneously.
The only exception would be if we somehow had 90° = 180°, which is impossible.
Answer: No, two angles cannot be both complementary and supplementary at the same time.
Frequently Asked Questions
Q1: Are all right angles equal?
Yes! Every right angle is exactly 90°. No matter where or how they are formed, all right angles are equal.
Q2: Can an angle be both acute and complementary?
An angle can be acute (less than 90°). Two acute angles can form a complementary pair — for example, 40° and 50°. However, not all pairs of complementary angles are acute — a right angle (90°) has no complement, since 90° + anything > 90°.
Q3: If two lines are cut by a transversal and the corresponding angles are equal, does that mean the lines are parallel?
Yes! This works both ways. If corresponding angles are equal, the lines are parallel. If alternate interior angles are equal, the lines are parallel. These are actually used to prove lines are parallel when we don’t know for sure.
Q4: What’s the difference between a line, a ray, and a line segment?
A line goes forever in both directions. A ray starts at a point and goes forever in one direction. A line segment has two endpoints — it starts and stops. Angles are formed by rays (or line segments meeting at a vertex).
Q5: How many angles does a transversal create when it cuts two lines?
8 angles total — 4 at each intersection point. These 8 angles have all the special relationships (corresponding, alternate interior, co-interior, vertically opposite) we studied.
Q6: In what real-life situation do we use alternate interior angles?
Road engineers use alternate interior angles when designing parallel roads with a crossing road. The angle at one crossing equals the angle at the other. Tiling patterns, window grills, and staircase railings also use these principles.