Triangles and Properties
~14 min read
- What: Classification of triangles, the angle-sum property, the exterior angle theorem, Pythagoras, congruence (SSS, SAS, ASA, RHS), similarity, centroid/incentre/circumcentre/orthocentre, and triangle inequality.
- Why it matters: CDS papers from 2007 to 2023 average 4–6 questions per sitting — one of the largest geometry heads.
- Key fact: Pythagoras: in a right triangle with legs \(a, b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\). Memorise the common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41.
Triangles is the largest geometry chapter in CDS Maths. The good news: most questions test 5–7 core properties (angle sum, exterior angle, Pythagoras, congruence, similarity), and those rules are tight. Master them and you collect 4–6 marks per paper.
This page is built from CDS Previous Year Questions across 2007–2023, plus NCERT Class 10 Triangles and Class 8 Baudhayana-Pythagoras Theorem. Pair with Lines and Angles, Area and Perimeter, and Trigonometry.
What This Topic Covers
CDS scope: (1) classification — by sides (scalene, isoceles, equilateral) and angles (acute, right, obtuse); (2) angle-sum and exterior angle; (3) Pythagoras theorem and triples; (4) congruence — SSS, SAS, ASA, AAS, RHS; (5) similarity — AA, SSS, SAS proportional; (6) special centres — centroid, incentre, circumcentre, orthocentre; (7) median, altitude, perpendicular bisector, angle bisector; (8) triangle inequality — sum of any two sides > third side.
Why This Topic Matters
- 4–6 CDS questions per paper.
- Pythagoras alone is 1–2 marks per paper.
- Underpins Height and Distance and Quadrilateral and Polygon.
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2007-II | 3 | Pythagoras, angle sum |
| 2009-II | 4 | Similarity, congruence |
| 2010-I/II | 4 | Centroid, exterior angle |
| 2011-I/II | 4 | Pythagoras, isoceles |
| 2012-I/II | 5 | Mixed |
| 2013-I/II | 5 | Similarity ratio, area |
| 2014-I/II | 4 | Congruence, special centres |
| 2015-I/II | 5 | Pythagoras, similarity |
| 2016-I/II | 5 | Mixed |
| 2017-I/II | 5 | Centroid, exterior |
| 2018-I/II | 4 | Pythagoras triples |
| 2019-II | 3 | Mixed |
| 2020-I/II | 4 | Similarity, area ratio |
| 2021-I/II | 4 | Special centres, Pythagoras |
| 2022-I / 2023-I | 4 | Mixed |
For two similar triangles, the ratio of areas = (ratio of corresponding sides)². If sides are in ratio 2:3, areas are in ratio 4:9. Volumes (in 3D similarity) would be 8:27. CDS tests this every other sitting.
Core Concepts
Angle-Sum and Exterior Angle
Exterior angle = sum of the two non-adjacent interior angles.
Each exterior angle of any polygon = 360° / \(n\) (for a regular \(n\)-gon).
Pythagoras Theorem
Congruence Criteria
SSS: three sides equal. SAS: two sides and included angle. ASA: two angles and included side. AAS: two angles and a non-included side. RHS: right angle, hypotenuse, one side (for right triangles).
SSA (two sides and a non-included angle) does NOT prove congruence. CDS sometimes plants this — only the listed five criteria work.
Similarity
SSS: three sides in proportion.
SAS: two sides in proportion with included angle equal.
Ratio of areas = (ratio of sides)².
Special Centres
Centroid: intersection of medians; divides each median in 2:1 ratio from vertex. Incentre: intersection of angle bisectors; centre of inscribed circle. Circumcentre: intersection of perpendicular bisectors; centre of circumscribed circle. Orthocentre: intersection of altitudes.
Triangle Inequality
Equivalently: the sum of any two sides must exceed the third side. Used to check if three lengths can form a triangle.
Worked Examples
Example 1 — Pythagoras (2007-II)
Q: Find the hypotenuse of a right triangle with legs 9 and 12.
- \(c^2 = 81 + 144 = 225 \implies c = 15\). (3-4-5 triple scaled by 3: 9-12-15.)
Example 2 — Angle Sum (2010-I)
Q: In triangle \(ABC\), \(\angle A = 60°\) and \(\angle B = 70°\). Find \(\angle C\).
- \(\angle C = 180° - 60° - 70° = 50°\).
Example 3 — Similarity Area Ratio (2013-II)
Q: Two similar triangles have sides in ratio 3:5. Find the ratio of their areas.
- Area ratio = \((3)^2 : (5)^2 = 9 : 25\).
Example 4 — Triangle Inequality (2014-I)
Q: Can sides 4, 5, 10 form a triangle?
- Check: \(4 + 5 = 9 < 10\). Inequality fails.
- Answer: no, these cannot form a triangle.
Example 5 — Exterior Angle (2017-I)
Q: An exterior angle of a triangle is 110°, and one interior opposite angle is 40°. Find the other interior opposite angle.
- Exterior = sum of two non-adjacent interiors: \(110° = 40° + x \implies x = 70°\).
Example 6 — Centroid (2010-II)
Q: The centroid of a triangle divides each median in what ratio?
- 2:1 from vertex. So vertex side is twice the side from base.
Example 7 — Pythagorean Triple Recognition (2018-II)
Q: A ladder 13 m long leans against a wall, base 5 m from wall. How high does it reach?
- 5-12-13 triple recognised → height = 12 m. Or compute: \(\sqrt{169 - 25} = \sqrt{144} = 12\).
How CDS Tests This Topic
Six recurring archetypes: (1) Pythagoras with one or two legs given, (2) angle sum (\(180°\)) given two angles, (3) similarity ratio of sides and areas, (4) triangle inequality test, (5) exterior angle theorem, (6) centroid 2:1 ratio, or other special-centre identity.
Exam Shortcuts (Pro-Tips)
Shortcut 1 — Pythagorean Triples
Memorise: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41. Recognise any scaling (6-8-10, 9-12-15, etc.). Solves 80% of right-triangle problems instantly.
Shortcut 2 — Area Ratio = Side² Ratio
For similar triangles. If linear ratio is \(k\), area ratio is \(k^2\), volume ratio (for 3D analogues) is \(k^3\).
Shortcut 3 — Exterior Angle = Sum of Opposites
Exterior angle of a triangle = sum of the two non-adjacent interior angles. Skip the angle-sum approach.
Shortcut 4 — Centroid 2:1
The centroid divides each median into 2 parts: 2 (from vertex) : 1 (from midpoint of opposite side). Useful in coordinate geometry too.
Shortcut 5 — Equilateral Triangle Identities
Side \(a\): height \(\tfrac{\sqrt{3}}{2}a\); area \(\tfrac{\sqrt{3}}{4}a^2\); inradius \(\tfrac{a}{2\sqrt{3}}\); circumradius \(\tfrac{a}{\sqrt{3}}\); all centres coincide.
Common Question Patterns
Pattern 1 — Pythagoras Application
Right triangle with two sides given; find the third. Recognise common triples for speed.
Pattern 2 — Angle Sum
Two angles given; find third. Or use exterior angle = sum of remote interiors.
Pattern 3 — Similarity
Two similar triangles with sides in known ratio. Find unknown side, area ratio, etc.
Pattern 4 — Triangle Inequality
Three lengths given; can they form a triangle? Sum of any two > third.
Pattern 5 — Special Centres
Centroid 2:1, incentre angle bisector, circumcentre perpendicular bisector. Identify which centre is being asked.
Preparation Strategy
Week 1. Memorise Pythagorean triples and standard identities (equilateral). Drill 20 Pythagoras problems and 10 angle-sum problems.
Week 2. Similarity, congruence criteria, and special centres. Practice area ratio of similar triangles. Cross-train with Area and Perimeter and Lines and Angles.
Mock testing. Use CDS mock tests. Most slip-ups: missing the Pythagorean triple recognition, or confusing area-ratio with side-ratio (\(k\) vs \(k^2\)).
Drill Triangles and Properties
CDS mocks with Pythagoras, similarity, triangle inequality, and special-centre problems. Six archetypes — five identities — reflex.
Start Free Mock TestFrequently Asked Questions
What are the five congruence criteria?
SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), and RHS (right angle + hypotenuse + one side, for right triangles). SSA is NOT a valid criterion.
What is the centroid and where is it?
The centroid is the intersection of the three medians (lines from each vertex to the midpoint of the opposite side). It divides each median in a 2:1 ratio (2 from vertex, 1 from base midpoint).
How do I check if three lengths form a triangle?
Apply the triangle inequality: the sum of any two sides must exceed the third. Equivalently, the third side must lie between the difference and the sum of the other two: \(|a - b| < c < a + b\).
For similar triangles, how do sides, areas and perimeters relate?
If linear ratio is \(k\): sides are in \(k\), perimeters in \(k\), medians in \(k\), altitudes in \(k\), but areas are in \(k^2\). Memorise this hierarchy.
What is the exterior angle theorem?
The exterior angle of a triangle equals the sum of the two non-adjacent interior angles. For example, if interior angles are 50° and 60°, the exterior at the third vertex is 110°. (Or interior at third vertex is 70°, exterior is its supplement, 110°.)
Which Pythagorean triples should I memorise?
3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29. Also recognise any multiple (6-8-10, 9-12-15, 10-24-26, etc.). These triples cover most CDS right-triangle problems.
Which CDS Maths topics connect to Triangles?
Lines and Angles — angle relationships. Area — Heron and basic formula. Trigonometry — right-triangle ratios. Height and Distance — application.