UP Board Class 10 Maths: Properties of Triangles

Properties of Triangles for UP Board Class 10

Ever wondered why a triangle can hold up a roof, a bridge, or even a pizza slice?

💡 In Simple Words: A triangle is a shape with three straight sides. Its angles and lines inside follow a few reliable rules that never change. Knowing those rules makes solving geometry problems feel like following a recipe.

Sum of Angles in a Triangle

The interior angles of any triangle always add up to 180°. Imagine a straight line as a piece of rope stretched tight – that line measures 180°. When you cut the rope into three pieces that meet at a point, you get a triangle, and the three pieces still total the same length.

Example: If two angles are 65° and 70°, the third is 180°‑65°‑70° = 45°.

Exterior Angle Theorem

An exterior angle sits outside the triangle, formed by extending one side. Its measure equals the sum of the two opposite interior angles.

Example: In a triangle where the interior angles are 50° and 60°, the exterior angle next to the third side is 50°+60° = 110°.

Triangle Inequality

The length of any side of a triangle must be less than the sum of the other two sides, but greater than their difference. Think of three sticks trying to form a closed loop – if one stick is too long, the loop can’t close.

Example: Sides 5 cm, 7 cm, and 13 cm cannot form a triangle because 5+7 = 12 which is less than 13.

Median, Altitude and Angle Bisector

Median: A line from a vertex to the midpoint of the opposite side. It splits that side into two equal parts.

Altitude: A perpendicular (right‑angled) line from a vertex to the line containing the opposite side. It measures the ‘height’ of the triangle.

Angle Bisector: A line that cuts an interior angle into two equal angles.

These three lines have handy relationships. For instance, the three medians intersect at a single point called the centroid, which is always two‑thirds of the way from each vertex.

Quick Revision Table

Worked Example: Finding an Unknown Angle

In ΔABC, ∠A = 40°, and the exterior angle at B is 130°. What is ∠C?

1. Exterior angle at B = 130° = ∠A + ∠C (by exterior angle theorem).
2. Plug in ∠A: 130° = 40° + ∠C.
3. ∠C = 130° – 40° = 90°.

So the triangle is a right‑angled triangle at C.

Worked Example: Using the Triangle Inequality

Can sides 8 cm, 15 cm, and 22 cm form a triangle?

• Check 8 + 15 > 22 → 23 > 22 ✅
• Check 8 + 22 > 15 → 30 > 15 ✅
• Check 15 + 22 > 8 → 37 > 8 ✅

All three conditions hold, so the sides do form a triangle.

How to Remember the Key Rules

• 180° rule = straight line.
• Exterior = two opposite interior angles.
• Inequality = “short + short > long”.
• Median meets at centroid (2/3 from each vertex).
• Altitude = height, always a right angle.
• Angle bisector splits side in the ratio of adjacent sides.

📝 Likely Exam Questions

1. In ΔPQR, ∠P = 55° and ∠Q = 65°. Find ∠R.
   Answer: 180° – 55° – 65° = 60°.
2. Show that a triangle with sides 9 cm, 12 cm, and 20 cm cannot exist.
   Answer: 9 + 12 = 21 > 20, but 9 + 20 = 29 > 12 and 12 + 20 = 32 > 9, so actually it can exist. (Trick: students often mis‑apply; correct answer: it can exist.)
3. In ΔXYZ, the exterior angle at Y is 150° and ∠X = 40°. Find ∠Z.
   Answer: Exterior = ∠X + ∠Z → 150° = 40° + ∠Z → ∠Z = 110°.
4. Prove that the three medians of a triangle intersect at a single point.
   Answer: By coordinate geometry or using vector addition, each median can be expressed as a line joining a vertex to the midpoint of the opposite side. Solving the three equations shows they meet at ( (x1+x2+x3)/3 , (y1+y2+y3)/3 ), the centroid.
5. Find the length of the altitude to side BC in a right‑angled triangle with legs 6 cm and 8 cm.
   Answer: Area = ½·6·8 = 24 cm². Altitude h to hypotenuse (10 cm) satisfies ½·10·h = 24 → h = 4.8 cm.