Triangle Geometry Notes for SSC Exams

Triangle geometry is the most important part of geometry in SSC CGL, and four to six questions come from geometry in each Tier 1 paper. Most of them are on the centres of a triangle, angles and similarity. These triangle geometry notes for SSC give you all the standard results in one place with solved examples and practice questions.

Basic Properties

• The sum of the three angles of a triangle is 180 degrees.
• The exterior angle is equal to the sum of the two opposite interior angles.
• The sum of any two sides is greater than the third side.
• The sum of the exterior angles (one at each vertex) is 360 degrees.
• In a right triangle, hypotenuse squared = base squared + height squared (Pythagoras theorem).
• The side opposite the larger angle is larger.

Centres of a Triangle

For a right triangle, the circumcentre is the mid-point of the hypotenuse and the orthocentre is at the right-angle vertex. For an equilateral triangle, all four centres coincide.

Similarity and Congruence

• Congruent triangles have the same shape and size (SSS, SAS, ASA, RHS).
• Similar triangles have the same shape (AAA, SAS, SSS in proportion).
• In similar triangles, the ratio of areas = (ratio of sides) squared.
• Basic proportionality theorem: a line parallel to one side divides the other two sides in the same ratio.
• Mid-point theorem: the line joining the mid-points of two sides is parallel to the third side and half of it.

Useful Formulas

• Area of a triangle = r x s (r is the inradius, s the semi-perimeter).
• Inradius of a right triangle = (base + height - hypotenuse) / 2.
• Circumradius of a right triangle = hypotenuse / 2.
• Circumradius of a triangle = abc / (4 x Area).
• Apollonius theorem: AB squared + AC squared = 2(AD squared + BD squared), where AD is the median.

Solved Examples

Example 1: The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles.
Solution: Sum of the parts = 9. Angles = 180 x 2/9, 180 x 3/9, 180 x 4/9 = 40, 60 and 80 degrees.

Example 2: An exterior angle of a triangle is 110 degrees and one of the opposite interior angles is 50 degrees. Find the other.
Solution: Other angle = 110 - 50 = 60 degrees.

Example 3: The legs of a right triangle are 6 cm and 8 cm. Find the hypotenuse.
Solution: Hypotenuse = square root of (36 + 64) = 10 cm.

Example 4: O is the incentre of triangle ABC and angle A = 60 degrees. Find angle BOC.
Solution: Angle BOC = 90 + A/2 = 90 + 30 = 120 degrees.

Example 5: O is the circumcentre of triangle ABC and angle A = 50 degrees. Find angle BOC.
Solution: Angle BOC = 2A = 100 degrees.

Example 6: The median of a triangle is 15 cm long. How far is the centroid from the vertex?
Solution: The centroid divides the median 2 : 1 from the vertex. Distance = 15 x 2/3 = 10 cm.

Quick Tips to Solve Faster

• Memorise the three angle results: incentre 90 + A/2, circumcentre 2A, orthocentre 180 - A.
• Draw a figure for every question and mark the given angles.
• Remember the Pythagorean triplets to avoid square roots.
• The centroid always divides a median in the ratio 2 : 1 from the vertex.
• Use similarity to find unknown sides when a line is parallel to a side.

Common Mistakes to Avoid

• Mixing up the angle results of the incentre and the orthocentre.
• Taking the centroid to divide the median 1 : 2 from the vertex.
• Applying Pythagoras to a triangle that is not a right triangle.
• Using the sum of angle bisectors in place of medians.
• Forgetting that the area ratio is the square of the side ratio.

Quick Revision Points

• Sum of angles = 180 degrees and exterior angle = sum of opposite interior angles.
• Incentre: 90 + A/2. Circumcentre: 2A. Orthocentre: 180 - A.
• Centroid divides the median 2 : 1.
• Similar triangles: areas are in the ratio of the squares of the sides.
• Right triangle: circumradius = hypotenuse / 2.

Practice Questions With Answers

1. Two angles of a triangle are 45 degrees and 65 degrees. Find the third angle.
   Answer: 70 degrees (180 - 110).
2. O is the orthocentre of triangle ABC and angle A = 70 degrees. Find angle BOC.
   Answer: 110 degrees (180 - 70).
3. A leg of a right triangle is 5 cm and the hypotenuse is 13 cm. Find the other leg.
   Answer: 12 cm (Triplet 5-12-13).
4. Find each angle of an equilateral triangle.
   Answer: 60 degrees (180 / 3).
5. AD is a median of triangle ABC and G is the centroid. If AD = 18 cm, find AG.
   Answer: 12 cm (18 x 2/3).
6. What is the sum of the three exterior angles of a triangle (one at each vertex)?
   Answer: 360 degrees (Standard result).
7. The sides of two similar triangles are in the ratio 3 : 5. Find the ratio of their areas.
   Answer: 9 : 25 (3 squared : 5 squared).
8. The legs of a right triangle are 7 cm and 24 cm. Find the hypotenuse.
   Answer: 25 cm (Triplet 7-24-25).
9. O is the incentre of triangle ABC and angle A = 80 degrees. Find angle BOC.
   Answer: 130 degrees (90 + 40).
10. In triangle ABC, D and E are the mid-points of AB and AC. If BC = 14 cm, find DE.
    Answer: 7 cm (Mid-point theorem).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching practice papers to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Circle Geometry Notes
• Mensuration 2D Notes
• Trigonometry Notes
• Heights and Distances Notes
• SSC Mensuration Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

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Frequently Asked Questions

What is the sum of the angles of a triangle?

The sum of the three angles of a triangle is 180 degrees.

What is the exterior angle property?

An exterior angle of a triangle equals the sum of its two opposite interior angles.

What is the centroid of a triangle?

The centroid is the point where the three medians meet, and it divides each median in the ratio 2 : 1.

What is the incentre of a triangle?

The incentre is the point where the three angle bisectors meet. It is the centre of the inscribed circle.

What is the angle at the circumcentre?

The angle subtended by a side at the circumcentre is twice the opposite angle of the triangle.

What is the orthocentre?

The orthocentre is the point where the three altitudes of a triangle meet.

How many geometry questions come in SSC CGL Tier 1?

Four to six questions come from geometry, and many of them are based on triangles and circles.

How should I practise triangle geometry?

Learn the results, draw a diagram for every question and solve the examples here.