Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
How to Use This Page
This page has step-by-step solutions for every exercise in Samacheer Kalvi Class 6 Maths Term 1 Chapter 4: Geometry. Work through each problem on your own first, then check your answer here. Where a calculation is involved, every step of the working is shown — no skips.
Key Concepts (Quick Recap)
Points, Lines, Line Segments, Rays
| Concept | Definition | ||
|---|---|---|---|
| Point | A dot that marks a position. It has no length, width, or thickness. Denoted by a capital letter (e.g. P). | ||
| Line | A straight path that extends endlessly in both directions. It has no endpoints. Denoted as AB with arrows on both ends. | ||
| Line Segment | A part of a line with two endpoints and a fixed length. Denoted as AB with a bar on top. | ||
| Ray | A part of a line with one endpoint (origin) that extends endlessly in one direction. Denoted as AB with an arrow on one end. | ||
| Intersecting Lines | Two lines that cross each other at a point. | ||
| Parallel Lines | Two lines in the same plane that never meet, no matter how far extended. Denoted AB \ | \ | CD. |
Angles
| Type | Measure |
|---|---|
| Acute angle | Greater than 0° and less than 90° |
| Right angle | Exactly 90° |
| Obtuse angle | Greater than 90° and less than 180° |
| Straight angle | Exactly 180° |
| Reflex angle | Greater than 180° and less than 360° |
| Complementary angles | Two angles whose sum is 90° |
| Supplementary angles | Two angles whose sum is 180° |
Triangles
| Classification | Types |
|---|---|
| By sides | Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (no equal sides) |
| By angles | Acute-angled (all angles < 90°), Right-angled (one angle = 90°), Obtuse-angled (one angle > 90°) |
Key Formulas
- Complement of angle x° = (90 − x)°
- Supplement of angle x° = (180 − x)°
- Sum of angles of a triangle = 180°
Exercise 4.1 Solutions — Points, Lines, Line Segments, Rays
(a) All possible lines
A line is named using any two points on it. From the figure, list every line by picking two points on each:
- Line AB
- Line CD
- Line EF
- Line GH
- Line IJ
(b) All possible line segments
A line segment is the portion of a line between two endpoints. List every pair of endpoints:
- Segment AB
- Segment CD
- Segment EF
- Segment GH
- Segment IJ
(c) All possible rays
A ray starts at one endpoint and extends infinitely. For each line, two rays are possible (one from each endpoint):
- Ray AB, Ray BA
- Ray CD, Ray DC
- Ray EF, Ray FE
- Ray GH, Ray HG
- Ray IJ, Ray JI
Parallel lines are lines that never meet:
- CD \|\| EF
- CD \|\| IJ
- EF \|\| IJ
Intersecting lines are lines that cross each other:
- AB and CD intersect at point P
- AB and EF intersect at point Q
- AB and GH intersect at point R
- AB and IJ intersect at point R
- GH and IJ intersect at point R
- The two line segments are QP and QR.
- Point Q is the common endpoint of both segments.
(b) Through two given points:
Through any two distinct points, exactly one straight line can be drawn. This is a fundamental axiom of geometry.
Exercise 4.2 Solutions — Angles
Complement of x° = (90 − x)°
Complement of 20° = 90° − 20° = 70°
Supplement of x° = (180 − x)°
Supplement of 90° = 180° − 90° = 90°
Two angles are complementary if their sum equals 90°.
78° + 12° = 90°
Since the sum is exactly 90°, the two angles are complementary.
∠ABD = ∠ABC − ∠CBD
∠ABD = 90° − 30° = 60°
The angles formed are:
- ∠AOB — angle between ray OA and ray OB
- ∠BOZ — angle between ray OB and ray OZ
- ∠AOZ — angle between ray OA and ray OZ
Let the other angle be x°.
x° + 45° = 90°
x° = 90° − 45° = 45°
Let the other angle be y°.
y° + 110° = 180°
y° = 180° − 110° = 70°
Exercise 4.3 Solutions — Triangles
(b) A triangle with sides 6 cm, 6 cm, 4 cm
Two sides are equal, one is different.
(c) A triangle with sides 3 cm, 4 cm, 5 cm
All three sides are of different lengths.
(b) A triangle with angles 90°, 45°, 45°
One angle is exactly 90°.
(c) A triangle with angles 120°, 30°, 30°
One angle is greater than 90° (120° is obtuse).
Sum of all three angles of a triangle = 180°.
If two angles are 90° each, their sum = 90° + 90° = 180°.
This leaves 0° for the third angle, which is not possible — an angle of a triangle cannot be 0°.
Sum of all three angles of a triangle = 180°.
If all three angles are greater than 60°, the minimum sum would be just above 60° + 60° + 60° = 180°. But since each angle must be strictly greater than 60°, the sum would exceed 180°, which is impossible.
Sum of angles = 180°
90° + 35° + x° = 180°
125° + x° = 180°
x° = 180° − 125° = 55°
Sum of angles = 180°
45° + 65° + x° = 180°
110° + x° = 180°
x° = 180° − 110° = 70°
Classification: All angles (45°, 65°, 70°) are less than 90°, so it is an acute-angled triangle.
Since all three angles are different, all three sides are different — so it is also a scalene triangle.
Since the triangle is isosceles, the other equal angle is also 40°.
40° + 40° + x° = 180°
80° + x° = 180°
x° = 100°
Sum = 70° + 80° + 40° = 190°
Since 190° ≠ 180°, these angles cannot form a triangle.
Additional Questions — Full Solutions
AQ 1: From the figure, name the parallel lines and intersecting lines with their intersection points
Parallel lines:
- CD \|\| EF
- CD \|\| IJ
- EF \|\| IJ
Intersecting lines and their intersection points:
| Intersecting Lines | Point of Intersection |
|---|---|
| AB and CD | P |
| AB and EF | Q |
| AB and GH | R |
| AB and IJ | R |
| GH and IJ | R |
Note: AB is a transversal cutting through the parallel lines CD, EF, and IJ.
AQ 2: Name the line segments QP and QR. Which point is the endpoint of both?
- Line segments: QP and QR
- Common endpoint: Q
Point Q is the vertex from which both segments extend — Q is shared by both QP and QR.
AQ 3: How many lines can pass through one point? Through two points?
Through one point: Infinite lines can pass through a single point. You can rotate a line around that point in any direction.
Through two points: Exactly one straight line can pass through two given points. This is Euclid's first axiom.
AQ 4: What is the minimum and maximum number of points on a line?
- Minimum: 2 points (a line is determined by at least 2 points)
- Maximum: Infinite points (a line extends endlessly and contains infinitely many points)
AQ 5: What is the maximum and minimum number of intersection points when three lines are drawn in a plane?
Minimum (0): When all three lines are parallel, no pair intersects.
Maximum (3): When no two lines are parallel and no three lines are concurrent (passing through the same point), each pair produces one intersection:
- Line 1 ∩ Line 2 → point A
- Line 2 ∩ Line 3 → point B
- Line 1 ∩ Line 3 → point C
Total = 3 distinct points.
AQ 6: Find the complement of 20°
Complement = 90° − 20° = 70°
AQ 7: Find the supplement of 90°
Supplement = 180° − 90° = 90°
The supplement of a right angle is a right angle itself.
AQ 8: Are 78° and 12° complementary?
78° + 12° = 90°
Yes, they are complementary angles because their sum is 90°.
AQ 9: ∠ABC = 90°, ∠CBD = 30°. Find ∠ABD.
∠ABD = ∠ABC − ∠CBD = 90° − 30° = 60°
AQ 10: Name the angles formed by rays OA, OB, OZ from point O
The three angles are:
1. ∠AOB — between ray OA and ray OB 2. ∠BOZ — between ray OB and ray OZ 3. ∠AOZ — between ray OA and ray OZ (the largest angle, if OZ is beyond OB)
AQ 11: What is the alternate name of ∠XYZ?
An angle named with three letters can also be written:
- ∠Y (using just the vertex letter)
- ∠ZYX (reversing the order of the outer letters)
Both ∠Y and ∠ZYX refer to the same angle as ∠XYZ.
AQ 12: Two angles share one common point. What are they called?
When two angles share a common vertex and a common arm, they are called adjacent angles.
AQ 13: Find the complement of 60° and the supplement of 60°
Complement of 60°:
90° − 60° = 30°
Supplement of 60°:
180° − 60° = 120°
AQ 14: How many lines can pass through three collinear points?
Since the three points lie on the same straight line, exactly one line passes through all of them.
AQ 15: How many lines can pass through a single point?
Infinite lines can pass through a single point. Every possible direction from that point defines a different line.
AQ 16: Construct a 45° angle using a protractor
Step-by-step construction:
1. Draw the base ray: Draw a ray OA of any convenient length (say 6 cm). Point O is the vertex. 2. Place the protractor: Place the protractor so that its centre (midpoint of the baseline) is exactly on point O and the baseline aligns with ray OA (the 0° mark sits on OA). 3. Mark the angle: Find the 45° mark on the protractor's inner scale (since OA starts from 0°). Place a small dot or mark point B at 45°. 4. Draw the second ray: Remove the protractor. Using a ruler, draw ray OB joining point O to point B. 5. Label and verify: Label the angle as ∠AOB = 45°. Verify visually — 45° is half a right angle, so ray OB should bisect the right-angle quadrant.
Alternative method — bisecting a 90° angle:
1. Construct a 90° angle ∠AOC using a protractor. 2. Find the midpoint between 0° and 90° on the protractor scale — that is 45°. 3. Mark point B at 45° and draw ray OB. 4. ∠AOB = 45° is the required angle.
Common Mistakes
1. Confusing line, line segment, and ray — A line has no endpoints (extends both ways), a line segment has two endpoints, and a ray has one endpoint. Always check endpoints before naming.
2. Writing complement instead of supplement (and vice versa) — Complement = 90° sum, Supplement = 180° sum. Use the mnemonic: "Complement starts with C — think Corner (90°). Supplement starts with S — think Straight (180°)."
3. Forgetting the degree symbol — Always write 90° not 90. An angle measure without the ° symbol is incomplete and costs marks in the exam.
4. Saying "infinite points" when asked about lines through two points — Through one point, infinite lines pass. Through two points, exactly one line passes. Students often mix these up.
5. Naming the angle incorrectly — ∠XYZ means vertex Y is in the middle. Writing ∠XZY would mean a different vertex. The vertex letter is always the middle letter in a three-letter angle name.
6. Assuming all equilateral triangles have angles other than 60° — Every equilateral triangle has all three angles equal to 60°. Always. There is no other possibility.
7. Adding angles that do not share a vertex and calling them complementary or supplementary — Complementary and supplementary angles must share a relationship in their measures. Two random angles that happen to add to 90° or 180° are not automatically complementary or supplementary unless they share the appropriate geometric relationship.
Exam Checklist
Before you walk into the exam, make sure you can tick off every item below:
- [ ] I can define and distinguish between point, line, line segment, and ray
- [ ] I can identify parallel lines (\|\|) and intersecting lines in a figure
- [ ] I can name a line, line segment, or ray using correct notation
- [ ] I can state how many lines pass through 1 point (infinite) and through 2 points (one)
- [ ] I can classify angles as acute, right, obtuse, straight, or reflex
- [ ] I can find the complement of any angle: (90 − angle)°
- [ ] I can find the supplement of any angle: (180 − angle)°
- [ ] I can verify whether two angles are complementary (sum = 90°) or supplementary (sum = 180°)
- [ ] I can name an angle in all three ways (e.g., ∠ABC, ∠B, ∠CBA)
- [ ] I can classify a triangle by sides (equilateral, isosceles, scalene)
- [ ] I can classify a triangle by angles (acute, right, obtuse)
- [ ] I can find a missing angle in a triangle using the 180° rule
- [ ] I can construct an angle of a given measure using a protractor
- [ ] I can explain why a triangle cannot have two right angles or all angles > 60°