17. Geometry: lines, angles and triangles
Free study material · concepts, shortcuts & solved questions
Angles on a straight line sum to 180°; angles around a point sum to 360°. Vertically opposite angles are equal. Triangle angles sum to 180°, and an exterior angle equals the sum of the two opposite interior angles. Pythagoras applies to right triangles: hypotenuse² = base² + perpendicular².
Method before formula For similarity, corresponding sides are proportional and areas are in the square of the side ratio. Congruence requires exact matching conditions such as SSS, SAS, ASA/AAS or RHS. |
Formula and decision table
Fact | Use |
3–4–5 triangle | Right triangle; area with legs 3 and 4 is 6 |
Equilateral triangle | Each angle 60° |
Similar triangles | Same shape; scale factor controls sides and area |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
A triangle’s interior angles sum to: | Basic triangle angle sum is 180°. | B. 180° |
A 3–4 right triangle has hypotenuse: | 3²+4²=25. | A. 5 |
If side ratio of similar triangles is 2:3, area ratio: | Areas scale as the square. | C. 4:9 |
What an examiner is testing
The options around Geometry: lines, angles and triangles usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: A triangle’s interior angles sum to: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 180°, because Basic triangle angle sum is 180°.
Example 2: A 3–4 right triangle has hypotenuse: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 5, because 3²+4²=25.
Example 3: If side ratio of similar triangles is 2:3, area ratio: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 4:9, because Areas scale as the square.
Chapter practice
1. A triangle’s interior angles sum to:
(A) 90° (B) 180° (C) 270° (D) 360°
Answer: B. 180° | Explanation: Basic triangle angle sum is 180°.
2. A 3–4 right triangle has hypotenuse:
(A) 5 (B) 6 (C) 7 (D) 12
Answer: A. 5 | Explanation: 3²+4²=25.
3. If side ratio of similar triangles is 2:3, area ratio:
(A) 2:3 (B) 3:2 (C) 4:9 (D) 8:27
Answer: C. 4:9 | Explanation: Areas scale as the square.
SSC CGL speed and trap clinic
For Geometry: lines, angles and triangles, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Geometry: lines, angles and triangles? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Geometry: lines, angles and triangles question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |