Triangle Sum Theorem – Explanation & Examples (2024)

Triangle Sum Theorem – Explanation & Examples (1)We know that different triangles have different angles and side lengths, but one thing is fixed — that each triangle is composed of three interior angles and three sides that can be of the same length or different lengths.

For instance, a right triangle has one angle that is exactly 90 degrees and two acute angles.

Isosceles triangles have two equal angles and two equal side lengths. Equilateral triangles have the same angles and same side lengths. Scalene triangles have different angles and different side lengths.

Even though all of these triangles differ in angles or side lengths, they all follow the same rules and properties.

In this article, you’ll learn about:

  • The Triangle Sum Theorem,
  • Interior angles of a triangle, and
  • How to use the Triangle Sum Theorem to find the interior angles of a triangle?

What is the Interior Angle of a Triangle?

In geometry, the interior angles of a triangle are the angles that are formed inside a triangle.

Interior angles have the following properties:

  • The sum of interior angles is 180 degrees (Triangle Angle Sum Theorem).
  • All interior angles of a triangle are more than 0° but less than 180°.
  • The bisectors of all three interior angles intersect inside a triangle at a point called the in-center, which is the center of the in-circle of the triangle.
  • The sum of each interior angle and exterior angle is equal to 180° (straight line).

What is the Triangle Angle Sum Theorem?

One common property about triangles is that all three interior angles add up to 180 degrees. This now brings us to an important theorem in geometry known as Triangle Angle Sum Theorem.

According to the Triangle Angle Sum Theorem, the sum of the three interior angles in a triangle is always 180°.

We can this as:

∠a + ∠b + ∠c = 180°

Triangle Sum Theorem – Explanation & Examples (2)

How to Find the Interior Angles of a Triangle?

When two interior angles of a triangle are known, it is possible to determine the third angle using the Triangle Angle Sum Theorem. To find the third unknown angle of a triangle, subtract the sum of the two known angles from 180 degrees.

Let’s take a look at a few example problems:

Example 1

Triangle ABC is such that, ∠A = 38° and ∠B = 134°. Calculate ∠C.

Solution

By Triangle Angle Sum Theorem, we have;

∠A + ∠B + ∠C = 180°

⇒ 38° + 134° + ∠Z = 180°

⇒ 172° + ∠C = 180°

Subtract both sides by 172°

⇒ 172° –172° + ∠C = 180° – 172°

Therefore, ∠C = 8°

Example 2

Find the missing angles x in the triangle shown below.

Triangle Sum Theorem – Explanation & Examples (3)

Solution

By Triangle Angle Sum Theorem (Sum of interior angles = 180°)

⇒ x + x + 18°= 180°

Simplify by combining like terms.

⇒ 2x +18°= 180°

Subtract both sides by 18°

⇒ 2x + 18° – 18° = 180° – 18°

⇒ 2x = 162°

Divide both sides by 2

⇒ 2x/2 = 162°/2

x = 81°

Example 3

Find the missing angles inside the triangle below.

Triangle Sum Theorem – Explanation & Examples (4)

Solution

This is an isosceles right triangle; therefore, one angle is 90°

⇒ x + x + 90°= 180°

⇒ 2x + 90°= 180°

Subtract both sides by 90°

⇒ 2x + 90°- 90°= 180° – 90°

⇒ 2x =90°

⇒ 2x/2 = 90°/2

x = 45°

Example 4

Find the angles of a triangle whose second angle exceeds the first angle by 15° and the third angle is 66° more than the second angle.

Solution

Let;

1ST angle = x°

2ND angle=(x + 15) °

3RD angle = (x + 15 + 66) °

By Triangle Angle Sum Theorem,

x° + (x + 15) ° + (x + 15 + 66) ° = 180°

Collect the like terms.

⇒ 3x + 81° = 180°

⇒ 3x = 180° – 81°

⇒ 3x= 99

x =33°

Now substitute x = 33° into the three equations.

1ST angle = x° = 33°

2ND angle=(x + 15) ° = 33° + 15° = 48°

3RD angle = (x + 15 + 66) ° = 33° + 15° + 66° = 81°

Therefore, the three angles of a triangle are 33°, 48° and 81°.

Example 5

Find the missing interior angles of the following diagram.

Triangle Sum Theorem – Explanation & Examples (5)

Solution

Angle y ° and (2x + 10) ° are supplementary angles (sum is 180°)

Therefore,

⇒ y ° + (2x + 10) ° = 180°

⇒ y + 2x = 170°……………… (i)

Also, by Triangle Angle Sum Theorem,

⇒ x + y + 65° = 180°

⇒ x + y = 115° ………………… (ii)

Solve the two simultaneous equations by substitution

⇒ y = 170° – 2x

⇒ x + 170° – 2x = 115°

⇒ -x = 115° -170°

x = 55°

But, y = 170° – 2x

= 170° – 2(55) °

⇒ 170° – 110°

y = 60°

Hence, the missing angles are 60° and 55°

Example 6

Calculate the value of x for a triangle whose angles are; x°, (x + 20) ° and (2x + 40) °.

Solution

Sum of interior angles = 180°

x° + (x + 20) ° + (2x + 40) ° = 180°

Simplify.

x + x + 2x + 20° + 40° = 180°

4x + 60° = 180°

Subtract 60 from both sides.

4x + 60° – 60°= 180° – 60°

4x = 120°

Now divide both sides by 4.

4x/4 = 120°/4

x = 30°

Therefore, the angles of the triangle are 30°, 50°, and 100°.

Example 7

Find the missing angles in the diagram below.

Triangle Sum Theorem – Explanation & Examples (6)

Solution

Triangle ADB and BDC are isosceles triangles.

∠ DBC = ∠DCB = 50°

∠ BAD = ∠ DBA = x°

Therefore,

50° + 50° + ∠BDC = 180°

∠BDC = 180° – 100°

∠BDC = 80°

But, z° + 80° = 180° (Angles on a straight line)

Hence, z = 100°

In triangle ADB:

z° + x + x = 180°

100° + 2x = 180°

2x = 180° – 100°

2x = 80°

x = 40°

Triangle Sum Theorem – Explanation & Examples (2024)

FAQs

Triangle Sum Theorem – Explanation & Examples? ›

Final answer:

What is the answer to the triangle sum theorem? ›

Answer: The sum of the three angles of a triangle is always 180 degrees. To find the measure of the third angle, find the sum of the other two angles and subtract that sum from 180.

What is the triangle sum theorem 9th grade? ›

The sum of the interior angles in a triangle is supplementary. In other words, the sum of the measure of the interior angles of a triangle equals 180°. So, the formula of the triangle sum theorem can be written as, for a triangle ABC, we have ∠A + ∠B + ∠C = 180°.

What is the triangle theorem explanation? ›

The Triangle Inequality Theorem states:
  1. The difference between any two sides will be less than the third since the sum of any two sides is bigger than the third.
  2. The sum of any two sides is always greater than the sum of the 3rd side.
  3. The longest side in a triangle is the side opposite a greater angle.

What can you conclude about the triangle sum theorem? ›

One thing is for certain for all types of triangles: the measures of the three interior angles add up to 180 degrees. The Triangle Angle Sum Theorem states that the sum of all three interior angles of any triangle will always be 180 degrees.

What is the formula for the sum of a triangle? ›

For Δ A B C , the formula for the angle sum property of a triangle is ∠ A + ∠ B + ∠ C = 180 ∘ . What Is the angle sum theorem for any polygon? According to the angle sum theorem for any polygon, the sum of all interior angles is equal to ( n − 2 ) × 180 ∘ , where n is the total number of sides of the polygon.

Can a triangle have two right angles? ›

A triangle can't have two right angles. If a triangle has two right angles, then the sum of its three angles will be more than the two right angles i.e. more than 180∘ which is impossible​.

How do you explain the theorem? ›

A theorem can be defined as a statement that can be proved to be true based on known and proved facts; all theorems contain a math rule and at least one proof. The Pythagorean theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides of the triangle.

What is the theorem of triangle grade 9? ›

Theorem 1: The sum of all the three interior angles of a triangle is 180 degrees. Theorem 2: The base angles of an isosceles triangle are congruent. The angles opposite to equal sides of an isosceles triangle are also equal in measure. Where ∠B and ∠C are the base angles.

What is the formula of triangle explanation? ›

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h. Hence, to find the area of a tri-sided polygon, we have to know the base (b) and height (h) of it.

What is the sum theorem for triangular numbers? ›

However, in general, a triangular number looks like the following sum. 1 + 2 + 3 + ⋯ + ( n − 2 ) + ( n − 1 ) + n Adding the first and last terms gives n + 1 adding the second and second-to-last terms gives n + 1 , and so on.

What is the Pythagorean theorem simple sum? ›

All the Pythagoras theorem triangles follow the Pythagoras theorem which says that the square of the hypotenuse is equal to the sum of the two sides of the right-angled triangle. This can be expressed as c2 = a2 + b2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle.

What is the corollary to the triangle sum theorem? ›

A useful corollary to the Triangle Sum Theorem involves exterior angles of a triangle. When you extend the sides of a polygon, the original angles may be called interior angles and the angles that form linear pairs with the interior angles are the exterior angles.

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