The measures of two angles of a triangle are 66 ∘ and 72 ∘ find the measure of the third angle

We know that every shape in the universe is based on angles. The square is basically four lines connected so that each line makes an angle of 90 degrees with the other line. In this way, a square has four 90 degrees angles on its four sides.

Similarly, a straight line stretched on both sides at 180 degrees. If it turns at any point, it becomes two lines separated by a certain angle. In the same manner, a triangle is basically three lines connected at certain values of angles.

These measures of angles define the type of triangle. Therefore, angles are essential in studying any geometric shape.

In this article, you’ll learn the angles of a triangle and how to find the unknown angles of a triangle when you only know some of the angles. To know the important concepts of triangles, you can consult the previous articles.

What are the Angles of a Triangle?

The angle of a triangle is the space formed between two side lengths of a triangle. A triangle contains interior angles and exterior angles. Interior angles are three angles found inside a triangle. Exterior angles are formed when the sides of a triangle are extended to infinity.

Therefore, exterior angles are formed outside a triangle between one side of a triangle and the extended side. Each exterior angle is adjacent to an interior angle. Adjacent angles are angles with a common vertex and side.

The figure below shows the angle of a triangle. The interior angles are a, b and c, while exterior angles are d, e, and f.

How to Find the Angles of a Triangle?

To find the angles of a triangle, you need to recall the following three properties about triangles:

• Triangle angle sum theorem: This states that the sum of all the three interior angles of a triangle is equal to 180 degrees.

a + b + c = 180º

• Triangle exterior angle theorem: This states that the exterior angle is equal to the sum of two opposite and non-adjacent interior angles.

f = b + a

e = c + b

d = b + c

• Straight line angles. The measure of angles on a straight line is equal to 180º

c + f = 180º

a + d = 180º

e + b = 180º

Let’s work out a few example problems.

Example 1

Calculate the size of the missing angle x in the triangle below.

Solution

By triangle angle sum, theorem, we have,

x + 84º + 43º = 180º

Simplify.

x + 127º = 180º

Subtract 127º on both sides.

x + 127º – 127º = 180º – 127º

x = 53 º

Hence, the size of the missing angle is 53º.

Example 2

Find the size of the interior angles of a triangle that form consecutive positive integers.

Solution

Since, a triangle has three interior angles, then, let the consecutive angles be:

⇒1ST angle = x

⇒ 2ND angle = x + 1

⇒3RD angle = x + 2

But we know that, the sum of the three angles is equal to 180 degrees, therefore,

⇒ x + x + 1 + x + 2 = 180°

⇒ 3x + 3 = 180°

⇒ 3x = 177°

x = 59°

Now, substitute the value of x in the original three equations.

⇒1ST angle = x = 59°

⇒ 2ND angle = x + 1 =59° + 1 = 60°

⇒3RD angle = x + 2 = 59°+ 2 = 61°

So, the triangle’s consecutive interior angles are; 59°, 60°, and 61°.

Example 3

Find the triangle’s interior angles whose angles are given as; 2y°, (3y + 15) ° and (2y + 25) °.

Solution

In triangle, um of interior angles = 180°

2y° + (3y + 15) ° + (2y + 25) ° = 180°

Simplify.

2y + 3y + 2y + 15° + 25° = 180°

7y + 40° = 180°

Subtract 40° on both sides.

7y + 40° – 40° = 180° – 40°

7y = 140°

Divide both sides by 7.

y = 140/7

y = 20°

Substitute,

2y°= 2(20) ° = 40°

(3y + 15) ° = (3 x 20 + 15) ° = 75°

(2y + 25) ° = (2 x 20 + 25) ° = 65°

So, the three interior angles of a triangle are 40°, 75°, and 65°.

 Example 4

Find the value of the missing angles in the diagram below.

Solution

By triangle exterior angle theorem, we have;

(2x + 10) ° = 63° + 87°

Simplify

2x + 10° = 150°

Subtract 10° on both sides.

2x + 10° – 10 = 150° – 10

2x = 140°

Divide both sides by 2 to get;

x = 70°

Now, by substitution;

(2x + 10) ° = 2(70°) + 10 ° = 140 ° + 10 ° = 150 °

Hence, the exterior angle is 150 °

But, straight line angles add up to 180 °. So, we have;

y + 150 ° = 180 °

Subtract 150 ° on both sides.

y + 150 ° – 150 ° = 180 ° – 150 °

y = 30 °

Therefore, the missing angles are 30 ° and 150 °.

Example 5

The interior angles of a triangle are in the ratio 4: 11: 15. Find the angles.

Solution

Let x be the common ratio of the three angles. So, the angles are,

4x, 11x and 15x.

In a triangle, sum of the three angles = 180°

4x + 11x + 15x = 180°

Simplify.

30x = 180°

Divide 30 on both sides.

x =180°/30

x = 6°

Substitute the value of x.

4x = 4(6) ° = 24°

11x = 11(6) ° = 66°

15x = 15(6) ° = 90°

So, the angles of the triangle are 24°, 66°, and 90°.

Example 6

Find the size of angles x and y in the diagram below.

Solution

Exterior angle = sum of two non-adjacent interior angles.

60° + 76° = x

x = 136°

Similarly, sum of interior angles = 180°. Therefore,

60° + 76° + y = 180°

136° + y = 180°

Subtract 136° on both sides.

136° – 136° + y = 180° – 136

y = 44°

Hence, the size of angle x and y is 136° and 44°, respectively.

Example 7

The three angles of a certain triangle are such that the first angle is 20 % less than the second angle, and the third is 20% more than the second angle. Find the size of the three angles.

Solution

Let the second angle be x

First angle = x – 20x/100 = x – 0.2x

Third angle = x + 20x/100 = x + 0.2x

Sum of the three angles = 180 degrees.

x + x – 0. 2x + x + 0.2x = 180°

Simplify.

3x = 180°

x = 60°

Therefore,

2nd second angle = 60°

1st angle =48°

3rd angle = 72°

So, the three angles of a triangle are 60°, 48°, and 72°.

Example 8

Calculate the size of angle p, q, r and s in the diagram below.

Solution

exterior angle = sum of the two non-adjacent interior angles.

140° = p + r …………. (i)

This is an isosceles triangle, so,

q = r

Angles on a straight line = 180°

140° + q = 180°

subtract 140 from both sides to get.

q = 40°

But q = r, so r is also 40°

r + s = 180° (linear angles)

40° + s =180°

s = 140°

Sum of interior angles = 180°

p + q + r = 180°

p + 40° + 40° = 180°

p = 180° – 80°

p = 100°

Finding the third angle of a triangle when you know the measurements of the other two angles is easy. All you've got to do is subtract the other angle measurements from 180° to get the measurement of the third angle. However, there are a few other ways to find the measurement of the third angle of a triangle, depending on the problem you're working with. If you want to know how to find that elusive third angle of a triangle, see Step 1 to get started.

1. 1

   Add up the two known angle measurements. All you have to know is that all of the angles in a triangle always add up to 180°. This is true 100% of the time. So, if you know two of the three measurements of the triangle, then you're only missing one piece of the puzzle. The first thing you can do is add up the angle measurements you know. In this example, the two angle measurements you know are 80° and 65°. Add them up (80° + 65°) to get 145°.[1] X Research source Go to source

2. 2

   Subtract this number from 180°. The angles in a triangle add up to 180°. Therefore, the remaining angle must make the sum up the angles up to 180°. In this example, 180° - 145° = 35°.

3. 3

   Write down your answer. You now know that the third angle measures 35°. If you're doubting yourself, just check your work. The three angles should add up to 180° for the triangle to exist. 80° + 65° + 35° = 180°. You're all done.[2] X Research source Go to source

1. 1

   Write down the problem. Sometimes, instead being lucky enough to know the measurements of two of the angles of a triangle, you'll only be given a few variables, or some variables and an angle measurement. Let's say you're working with this problem: Find the measurements of angle "x" of the triangle whose measurements are "x," "2x," and 24. First, just write it down.[3] X Research source Go to source

2. 2

   Add up all of the measurements. It's the same principle that you would follow if you did know the measurements of the two angles. Simply add up the measurements of the angles, combining the variables. So, x + 2x + 24° = 3x + 24°.[4] X Research source Go to source

3. 3

   Subtract the measurements from 180°. Now, subtract these measurements from 180° to get closer to solving the problem. Make sure you set the equation equal to 0. Here's what it would look like:

   • 180° - (3x + 24°) = 0
   • 180° - 3x - 24° = 0
   • 156° - 3x = 0

4. 4

   Solve for x. Now, just put the variables on one side of the equation and the numbers on the other side. You'll get 156° = 3x. Now, divide both sides of the equation by 3 to get x = 52°. This means that the measurement of the third angle of the triangle is 52°. The other angle, 2x, is 2 x 52°, or 104°.[5] X Research source Go to source

5. 5

   Check your work. If you want to make sure that this is a valid triangle, just add up the three angle measurements to make sure that they add up to 180°. That's 52° + 104° + 24° = 180°. You're all done.

1. 1

   Find the third angle of an isosceles triangle. Isosceles triangles have two equal sides and two equal angles. The equal sides are marked by one hash mark on each of them, indicating that the angles across from each side are equal. If you know the angle measurement of one equal angle of an isosceles triangle, then you'll know the measurement of the other equal angle. Here's how to find it:[6] X Research source Go to source

   • If one of the equal angles is 40°, then you'll know that the other angle is also 40°. You can find the third side, if needed, by subtracting 40° + 40° (which is 80°) from 180°. 180° - 80° = 100°, which is the measurement of the remaining angle.

2. 2

   Find the third angle of an equilateral triangle. An equilateral triangle has all equal sides and all equal angles. It will typically be marked by two hash marks in the middle of each of its sides. This means that the angle measurement of any angle in an equilateral triangle is 60°. Check your work. 60° + 60° + 60° = 180°.[7] X Research source Go to source

3. 3

   Find the third angle of a right triangle. Let's say you know you have a right triangle, with one of the other angles being 30°. If it's a right triangle, then you know that one of the angles measures exactly 90°. The same principles apply. All you have to do is add up the measurements of the sides you know (30° + 90° = 120°) and subtract that number from 180°. So, 180° - 120° = 60°. The measurement of that third angle is 60°.[8] X Research source Go to source

• Question

  In a right angle triangle, if one of the other two angles is 35 degrees, find the remaining angle.

  Take the 90, add it to 35. This gives you 125 degrees. Triangles can only ever add up to 180, thus take the difference of 125 and 180 (180-125). This will give you the third remaining angle, which in this case is 55.

• Question

  If one angle of a right triangle is 50 degrees, what will be the measurement of the third angle?

  A right angle triangle always consists of one 90 degree angle, and every triangle must equal 180 degrees. Here is the work for this problem: 90 degrees (representing the right angle) + 50 degrees equals 140 degrees. 180 minus 140 equals 40. Therefore, the remaining angle would be 40 degrees.

• Question

  What if there is only one number?

  Substitute a letter, and work it out like an algebraic equation that you have to solve.

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 24 people, some anonymous, worked to edit and improve it over time. This article has been viewed 238,376 times.

Co-authors: 24

Updated: April 30, 2022

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Categories: Geometry

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