Chapter 10 - Properties of Triangles Revision Notes - Class 7 Mathematics

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📌 Key Points

A triangle has 3 sides, 3 angles, and 3 vertices.
The sum of all interior angles of a triangle is always 180°.
Equilateral triangle: all sides equal, all angles 60°.
Isosceles triangle: 2 sides equal, 2 angles equal (base angles).
Scalene triangle: all sides different, all angles different.
Acute triangle: all angles < 90°.
Right triangle: one angle = 90°, other two angles are complementary.
Obtuse triangle: one angle > 90°.
Exterior angle = sum of two remote interior angles.
Exterior angle is always greater than either remote interior angle.
Triangle inequality: Sum of any two sides > third side.
In isosceles triangle, angles opposite equal sides are equal.
Median: line from vertex to midpoint of opposite side.
All 3 medians meet at centroid.
Altitude: perpendicular from vertex to opposite side.

📘 Important Definitions

Triangle

A polygon with three sides, three angles, and three vertices.

Equilateral Triangle

Triangle with all three sides equal. Each angle measures 60°.

Isosceles Triangle

Triangle with two sides equal. The angles opposite the equal sides are also equal (base angles).

Scalene Triangle

Triangle with all three sides of different lengths and all angles different.

Right Triangle

Triangle with one angle equal to 90° (right angle). The side opposite the right angle is hypotenuse.

Exterior Angle

Angle formed when one side of triangle is extended beyond a vertex. Equals sum of two remote interior angles.

Median

Line segment from a vertex to the midpoint of the opposite side.

Altitude

Perpendicular line segment from a vertex to the opposite side or its extension.

🔢 Formulas & Laws

Angle Sum Property

∠A + ∠B + ∠C = 180°

True for any triangle

Exterior Angle

Exterior angle = sum of two remote interior angles

If ∠ACD is exterior, then ∠ACD = ∠A + ∠B

Triangle Inequality

a + b > c, b + c > a, c + a > b

All three conditions must be satisfied

Equilateral Triangle Angles

Each angle = 60°

Since all angles equal and sum is 180°

Right Triangle

∠A + ∠B = 90° (for acute angles)

The two acute angles are complementary

⚠️ Common Mistakes

✗ Wrong: Thinking angle sum can be different in different triangles.

✓ Correct: Angle sum is ALWAYS 180° for any triangle.

✗ Wrong: Confusing median and altitude.

✓ Correct: Median goes to midpoint. Altitude is perpendicular. In equilateral triangle only, they're same.

✗ Wrong: Forgetting to check all three conditions of triangle inequality.

✓ Correct: Must verify: a+b>c AND b+c>a AND c+a>b

✗ Wrong: In isosceles triangle, thinking all angles are equal.

✓ Correct: Only BASE angles are equal. Vertex angle can be different.

✗ Wrong: Exterior angle equals one of the remote interior angles.

✓ Correct: Exterior angle equals SUM of two remote interior angles, not equal to either.

✗ Wrong: In right triangle, thinking exterior angle at right angle vertex has special property.

✓ Correct: Exterior angle rule applies at ALL vertices, including right angle.

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Finding missing angles using angle sum property

1m★★★

Classifying triangles by sides and angles

1m★★★

Using exterior angle property to find angles

2m★★★

Checking if three given lengths form valid triangle

2m★★

Properties of isosceles triangle (equal sides, equal angles)

2m★★

Solving equations with angles in ratio form

3m★★

🎯 Last-Minute Recall