krit.club logo

The Triangle and its Properties - Angle Sum Property of a Triangle

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

The Angle Sum Property states that the sum of all internal angles in any triangle is always exactly 180∘180^{\circ}.

A triangle ABC showing the relationship between its three interior angles.
•

In an isosceles triangle, the angles opposite to the equal sides are equal. If we know the vertex angle, we can find the base angles using the angle sum property.

An isosceles triangle with equal base angles marked as x.
•

In an equilateral triangle, all three angles are equal. Since the sum is 180∘180^{\circ}, each angle must be 60∘60^{\circ} because 3×60∘=180∘3 \times 60^{\circ} = 180^{\circ}.

An equilateral triangle with all interior angles labeled as 60 degrees.
•

The angle sum property helps in finding an unknown angle when two angles are given, or when a relationship between the angles is known (like a ratio).

📐Formulae

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}

Third Angle=180∘−(Sum of the other two angles)\text{Third Angle} = 180^{\circ} - (\text{Sum of the other two angles})

In an Equilateral Triangle: x+x+x=180∘  ⟹  3x=180∘  ⟹  x=60∘\text{In an Equilateral Triangle: } x + x + x = 180^{\circ} \implies 3x = 180^{\circ} \implies x = 60^{\circ}

In a Right-Angled Triangle: 90∘+∠1+∠2=180∘\text{In a Right-Angled Triangle: } 90^{\circ} + \angle 1 + \angle 2 = 180^{\circ}

💡Examples

Problem 1:

In △XYZ\triangle XYZ, the measure of ∠X=55∘\angle X = 55^{\circ} and the measure of ∠Y=65∘\angle Y = 65^{\circ}. Find the measure of ∠Z\angle Z.

Solution:

  1. According to the Angle Sum Property, ∠X+∠Y+∠Z=180∘\angle X + \angle Y + \angle Z = 180^{\circ}.
  2. Substitute the given values into the equation: 55∘+65∘+∠Z=180∘55^{\circ} + 65^{\circ} + \angle Z = 180^{\circ}.
  3. Calculate the sum of the known angles: 120∘+∠Z=180∘120^{\circ} + \angle Z = 180^{\circ}.
  4. Subtract 120∘120^{\circ} from both sides to find the unknown angle: ∠Z=180∘−120∘\angle Z = 180^{\circ} - 120^{\circ}.
  5. Final result: ∠Z=60∘\angle Z = 60^{\circ}.

Explanation:

To find a missing angle in a triangle, we subtract the sum of the two known angles from 180∘180^{\circ}.

Problem 2:

One of the acute angles of a right-angled triangle is 42∘42^{\circ}. Find the measure of the other acute angle.

Solution:

  1. In a right-angled triangle, one angle is always 90∘90^{\circ}. Let the three angles be 90∘90^{\circ}, 42∘42^{\circ}, and xx.
  2. Use the Angle Sum Property: 90∘+42∘+x=180∘90^{\circ} + 42^{\circ} + x = 180^{\circ}.
  3. Simplify the equation: 132∘+x=180∘132^{\circ} + x = 180^{\circ}.
  4. Solve for xx: x=180∘−132∘x = 180^{\circ} - 132^{\circ}.
  5. Final result: x=48∘x = 48^{\circ}.

Explanation:

Since one angle is fixed at 90∘90^{\circ}, the other two angles must add up to 90∘90^{\circ}. We can simply subtract the given acute angle from 90∘90^{\circ} to find the answer: 90∘−42∘=48∘90^{\circ} - 42^{\circ} = 48^{\circ}.

Problem 3:

The angles of a triangle are in the ratio 2:3:52:3:5. Find the measure of each angle of the triangle.

A right-angled triangle with angles marked in the ratio 2x, 3x, and 5x.

Solution:

Let the angles be 2x2x, 3x3x, and 5x5x. By Angle Sum Property: 2x+3x+5x=180∘2x + 3x + 5x = 180^{\circ} 10x=180∘10x = 180^{\circ} x=180∘10=18=18∘x = \frac{180^{\circ}}{10} = 18 = 18^{\circ} Now, find each angle: Angle 1 = 2×18∘=36∘2 \times 18^{\circ} = 36^{\circ} Angle 2 = 3×18∘=54∘3 \times 18^{\circ} = 54^{\circ} Angle 3 = 5×18∘=90∘5 \times 18^{\circ} = 90^{\circ}

Explanation:

We use the ratio to represent angles as multiples of a common variable xx. Their sum is equated to 180∘180^{\circ} to solve for xx.

Problem 4:

In △PQR\triangle PQR, ∠P=2∠Q\angle P = 2\angle Q and ∠R=3∠Q\angle R = 3\angle Q. Find all three angles.

Triangle PQR with angles expressed in terms of x.

Solution:

Let ∠Q=x\angle Q = x. Then, ∠P=2x\angle P = 2x and ∠R=3x\angle R = 3x. By the Angle Sum Property: ∠P+∠Q+∠R=180∘\angle P + \angle Q + \angle R = 180^{\circ} 2x+x+3x=180∘2x + x + 3x = 180^{\circ} 6x=180∘6x = 180^{\circ} x=30∘x = 30^{\circ} So, ∠Q=30∘\angle Q = 30^{\circ} ∠P=2×30∘=60∘\angle P = 2 \times 30^{\circ} = 60^{\circ} ∠R=3×30∘=90∘\angle R = 3 \times 30^{\circ} = 90^{\circ}

Explanation:

By expressing all angles in terms of ∠Q\angle Q, we create a linear equation based on the angle sum property.

Angle Sum Property of a Triangle Class 7 Notes & Examples