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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 any side of the triangle is extended beyond its vertex. Visually, if you have a triangle ABCABC and you draw a straight line continuing from side BCBC towards a point DD, the angle ∠ACD\angle ACD formed outside the triangle is the exterior angle at vertex CC.

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The Exterior Angle Property states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. For instance, in triangle ABCABC with exterior angle ∠ACD\angle ACD, the interior opposite angles are ∠CAB\angle CAB (or ∠A\angle A) and ∠ABC\angle ABC (or ∠B\angle B).

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Identifying interior opposite angles is crucial for calculations. If the exterior angle is at vertex CC, the 'adjacent' interior angle is ∠ACB\angle ACB. The 'opposite' interior angles are the ones located at the other two vertices, AA and BB, which do not share the vertex CC.

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An exterior angle and its adjacent interior angle always form a linear pair. This means they lie on a straight line and their measures add up to 180∘180^\circ. Visually, the ray CBCB and the extension CDCD form a straight line, making ∠ACD+∠ACB=180∘\angle ACD + \angle ACB = 180^\circ.

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The exterior angle is always strictly greater than either of its two interior opposite angles individually. Because the exterior angle is the sum of two positive interior angles (e.g., Ext=Int1+Int2Ext = Int_1 + Int_2), it follows that Ext>Int1Ext > Int_1 and Ext>Int2Ext > Int_2.

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Every triangle can have a total of six exterior angles, two at each vertex. At any given vertex, the two exterior angles formed by extending different sides are vertically opposite and therefore equal in measure.

📐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.