Angle Sum Property

Angle Sum Property
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Max came across a problem of geometry dealing with the angle sum property. He wanted to find the value of \(\angle x\) in the triangle shown below, when the value of \(\angle BAC=50^{\circ}\) and \(\angle ABC=90^{\circ}\).

Triangle ABC, Find the value of x

His friend, Sam, helped him to solve this by telling him about the angle sum property.

This property states that the sum of all the angles of a triangle is always \(180^{\circ}\). 

In the given \(\triangle ABC\), \(\angle BAC + \angle CBA + \angle ACB = 180^{\circ}\) 

So, \(\angle ACB+90^{\circ}+50^{\circ}=180^{\circ}\)

\(\therefore\) \(\angle ACB=\angle x=40^{\circ}\).

Lesson PLan 


What Is Angle Sum Property? 

Definition 

The angle sum property states that the sum of the angles of a triangle is equal to \(180^{\circ}\).

Whether a triangle is an acute, obtuse, or a right triangle, the sum of the angles will always be \(180^{\circ}\).

This can be represented by the formula:

In a \(\triangle ABC\), \(\angle B + \angle A + \angle C = 180^{\circ}\) 

This triangle angle sum theorem is useful for finding the measure of an unknown angle when the values of the other two angles are known.

Angle sum property of a triangle


What Is the Exterior Angle of a Triangle?

The exterior angle of a triangle is formed if any side of a triangle is extended.

Exterior and Interior Angle

In the above image of \(\triangle ABC\), the interior angles are \(a, b, c\) and the exterior angles are \(d, e, f\).

According to the Exterior Angle Property of a triangle, the exterior angle is equal to the sum of the two remote interior angles.

\(i.e.\), in this case,

\(\angle d = \angle b + \angle a\)

\(\angle e = \angle c + \angle a\)

\(\angle f = \angle b + \angle c\)


Proof of the Angle Sum Property

The Steps for Proving the Angle Sum Property of a Triangle

Step 1:
Draw a line \(PQ\) that passes through the vertex A and is parallel to side \(BC\) of the \(\triangle ABC\) as shown below. 

Step 1 Image

Step 2 :
We know that the sum of the angles on a straight line is equal to 180°.

In other words, \(\angle PAB + \angle BAC + \angle CAQ = 180^{\circ}\)

Equation 1: \(\angle PAB + \angle BAC + \angle CAQ = 180^{\circ}\)

Step 2 Image

Step 3:

Now, since line \(PQ\) is parallel to \(BC\)

\(\angle PAB\) = \(\angle ABC\) and \(\angle CAQ\) = \(\angle ACB\).

(Interior alternate angles)

Equation 2: \(\angle PAB\) = \(\angle ABC\)

Equation 3: \(\angle CAQ\) = \(\angle ACB\)

Step 3 Image

Step 4:
Substitute \(\angle PAB\) and \(\angle CAQ\) with \(\angle ABC\) and \(\angle ACB\) respectively, in Equation 1 as shown below.

Equation 1: \(\angle PAB + \angle BAC + \angle CAQ = 180^{\circ}\)

Thus we get,  \(\angle ABC + \angle BAC + \angle ACB = 180^{\circ}\)

Hence proved, in \(\triangle ABC\), \(\angle B + \angle A + \angle C = 180^{\circ}\).

Thus, the sum of all the angles of a triangle is equal to 180°.

 
Challenge your math skills

Challenging Questions

  1. Try to find the angle sum properties of a rectangle, a  hexagon and a pentagon.

Solved Examples

Example 1


 

 

Sam needs to find the measure of the third angle of a triangle ABC in which \(\angle ABC\) = 45° and \(\angle ACB\) = 55°. Can you help him?

Example 1 Image

Solution

We know that \(\angle ABC =45^{\circ}\) and \(\angle ACB =55^{\circ}\)

Using the Angle Sum Property of the triangle,

\(\angle A+\angle B+\angle C=180^{\circ}\).

\(\angle A+45^{\circ}+55^{\circ}=180^{\circ}\).

\(\angle A+100^{\circ}=180^{\circ}\).

\(\angle A=180^{\circ}-100^{\circ}\).

\(\angle A=80^{\circ}\).

So, the third angle: \(\angle A=80^{\circ}\).
Example 2

 

 

If the angles of a triangle are in the ratio \(3:4:5\), determine the value of the three angles.

Solution

Let the angles be \(3x,\,4x,\) and \(5x\)

According to the Angle Sum Property of the triangle,

\(3x+4x+5x=180^{\circ}\).

\(12x=180^{\circ}\).

\(x=15^{\circ}\)

\(\therefore \) The three angles will be:

\(3x=3\times 15=45^{\circ}\) 

\(4x=4\times 15=60^{\circ}\) 

\(5x=5\times 15=75^{\circ}\) 

So, the three angles are \(45^{\circ},\,60^{\circ},\,  75^{\circ}\).

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

 

 

 

 
important notes to remember
Important Notes
  • The sum of the angles of a triangle is always \(180^{\circ}\).

  • If any side of a triangle is extended, then the exterior angle thus formed is equal to the sum of the two interior opposite angles.

  •  A triangle cannot have more than one right angle.

  • In a right-angled triangle, the sum of two acute angles is equal to 90°.

Let's Summarize

We hope you enjoyed learning about angle sum property with the simulations and practice questions. Now, you will be able to easily solve problems on Angle Sum property definition, Angle Sum property steps, and Angle Sum property examples.

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FAQs on Angle Sum Property

1. How do you solve for exterior angles?

The exterior angle of a polygon is the angle formed between any side of a polygon and a line which is extended from the next side. It should be noted that the corresponding interior and exterior angles are supplementary and the exterior angles of a regular polygon are equal in measure.

2. What is angle sum property of a polygon?

We know that the sum of interior angles in a triangle sum up to 180°. In order to find the sum of interior angles of a polygon, we should multiply the number of triangles in the polygon by 180°. Thus, the sum of interior angles of a polygon can be calculated with the formula: S = ( n − 2) × 180°. It should also be noted that the sum of exterior angles of a polygon is 360°

3. What is the sum of all angles in a triangle?

The sum of the angles of any triangle is equal to 180°

4. What is the angle sum property of a hexagon?

Each interior angle of a regular hexagon is 120°. Since the sum of these angles will always be 360°, each exterior angle would be 60° (360° ÷ 6 = 60°).

5. What is the angle sum property of a quadrilateral?

According to the angle sum property of a quadrilateral, the sum of all the four interior angles is 360°

6. What is the exterior angle property?

The exterior angle theorem says that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

7. What is the formula of angle sum property?

The formula for the angle sum property is S = ( n − 2) × 180°

8. How do you find the sum of an angle in a triangle?

The sum of the angles of a triangle is always 180°

 

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