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

Grade 7CBSE

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

🔑Concepts

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An exterior angle of a triangle is formed when a side of the triangle is extended beyond its vertex. For every triangle, there are six possible exterior angles (two at each vertex).

Triangle ABC with side BC extended to point D, showing exterior angle ACD.
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The measure of an exterior angle of a triangle is equal to the sum of its two interior opposite angles. This is known as the Exterior Angle Property of a triangle.

Diagram showing that the exterior angle equals the sum of interior opposite angles x and y.
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An exterior angle and its adjacent interior angle form a linear pair, meaning their sum is always 180∘180^\circ.

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The exterior angle is always greater than either of its interior opposite angles.

📐Formulae

Exterior Angle=Sum of Interior Opposite Angles\text{Exterior Angle} = \text{Sum of Interior Opposite Angles}

∠ACD=∠BAC+∠ABC\angle ACD = \angle BAC + \angle ABC

Exterior Angle+Adjacent Interior Angle=180∘\text{Exterior Angle} + \text{Adjacent Interior Angle} = 180^\circ

💡Examples

Problem 1:

In triangle ABCABC, side BCBC is produced to DD. If interior angles ∠A=45∘\angle A = 45^\circ and ∠B=65∘\angle B = 65^\circ, find the measure of the exterior angle ∠ACD\angle ACD.

Solution:

  1. Identify the interior opposite angles relative to ∠ACD\angle ACD, which are ∠A\angle A and ∠B\angle B.
  2. Apply the Exterior Angle Property: ∠ACD=∠A+∠B\angle ACD = \angle A + \angle B.
  3. Substitute the given values: ∠ACD=45∘+65∘\angle ACD = 45^\circ + 65^\circ.
  4. Calculate the sum: ∠ACD=110∘\angle ACD = 110^\circ.

Explanation:

According to the property, an exterior angle equals the sum of its two interior opposite angles. By adding the two given interior angles, we find the exterior angle.

Problem 2:

An exterior angle of a triangle measures 120∘120^\circ. If one of the interior opposite angles is 55∘55^\circ, find the measure of the other interior opposite angle.

Solution:

  1. Let the unknown interior opposite angle be xx.
  2. Use the formula: Exterior Angle=Interior Opposite Angle 1+Interior Opposite Angle 2\text{Exterior Angle} = \text{Interior Opposite Angle 1} + \text{Interior Opposite Angle 2}.
  3. Set up the equation: 120∘=55∘+x120^\circ = 55^\circ + x.
  4. Solve for xx: x=120∘−55∘x = 120^\circ - 55^\circ.
  5. x=65∘x = 65^\circ.

Explanation:

We use the Exterior Angle Property in reverse. Since the sum of the interior opposite angles must equal the exterior angle, we subtract the known interior angle from the exterior angle to find the missing one.

Problem 3:

In the given figure, find the value of the unknown interior angle xx if the exterior angle is 110∘110^\circ and one interior opposite angle is 30∘30^\circ.

Triangle with an exterior angle of 110 degrees and interior opposite angles of 30 degrees and x.

Solution:

  1. According to the Exterior Angle Property: Exterior Angle=Sum of interior opposite angles\text{Exterior Angle} = \text{Sum of interior opposite angles}
  2. Substitute the given values: 110∘=30∘+x110^\circ = 30^\circ + x
  3. Solve for xx: x=110∘−30∘x = 110^\circ - 30^\circ x=80∘x = 80^\circ

Explanation:

The exterior angle of 110∘110^\circ must equal the sum of the non-adjacent interior angles (xx and 30∘30^\circ). By subtracting the known interior angle from the exterior angle, we find xx.

Problem 4:

Find the value of yy in the following triangle where the interior opposite angles are 2y2y and 3y3y, and the exterior angle is 125∘125^\circ.

Triangle with interior opposite angles labeled 2y and 3y, and an exterior angle of 125 degrees.

Solution:

  1. Using the Exterior Angle Property: 2y+3y=125∘2y + 3y = 125^\circ
  2. Combine like terms: 5y=125∘5y = 125^\circ
  3. Divide by 5: y=125∘5y = \frac{125^\circ}{5} y=25∘y = 25^\circ

Explanation:

Since the sum of the interior opposite angles equals the exterior angle, we set up an algebraic equation 5y=125∘5y = 125^\circ to solve for the variable yy.