How to find missing angles in a triangle with one angle

Triangle is a geometrical figure. It is a polygon with three sides (i.e., Three Vertices and Three edges) and sum of it’s internal angles in a triangle is 180o. There are mainly three types of triangles. They are-

1. Equilateral Triangle– All the internal angles in an equilateral triangles are same. i.e., ∠A=∠B=∠C=60o. Where A,B,C are internal angle.
2. Isosceles Triangle– It is a triangle with two sides of equal length and respective angles are also equal. i.e., two interior angles are equal.
3. Scalene Triangle- A scalene triangle have three sides with different interior angles but the summation of those angles is 180o.

How to find the missing Angle in the triangle?

By the Angle sum property, the sum of all the interior angles of a triangle is 180°.

∠a + ∠b + ∠c = 180°

Let’s look into different scenarios of finding the missing angles in a triangle.

Sample Questions

Question 1: What is the missing angle in the given triangle?

Solution:

We know that the sum of interior angles in a triangle is 180o.

So, 70o + 60o + x = 180o

130o + x = 180o

x = 180o – 130o

x = 50o

So the missing angle in the given triangle is 50o.

Question 2: Find the missing angle in the given right-angled triangle?

Solution:

We know that in a right angled triangle one angle should be 90o and other two angles sum should be 90o.

So, 45o + x = 90o

x = 90o – 45o

x = 45o

So the missing angle in the given right angled triangle is 45o.

Question 3: Find the missing angle in the given isosceles triangle?

Solution:

The given triangle is isosceles triangles which indicates that it has two sides of equal length and two equal angles.

So in this case we can find two angles of a triangle when one angle is given.

From the above statements x=y (Equation-1) and

We know that the sum of interior angles in a triangle is 180o.

So, 30o + x + y = 180o

30o + x + x = 180o

30o + 2x = 180o

2x = 180o – 30o

2x = 150o

x = 150o/2

x = 75o

Form equation-1 y = 75o

So the missing angles in the given triangle are 75o, 75o.

In a triangle at any of the angles respective to that side the sum of interior and exterior angle is equal to 180o. Even in some cases, we can find the exterior angle in the problem statement in place of the interior angle. So we can find the interior angle by the formula-

Interior angle = 180o – Exterior angle

Question 4: What is the missing angles in the given triangle?

Solution:

Given

One interior angle = 60o

Other exterior angle = 130o

From exterior angle we can find interior angle by subtraction the exterior angle from 180o

Interior angle = 180o – 130o

                      = 50o

Now we have two angle we can find third angle by subtracting the summation of two angles from 180o

60o + 50o + x = 180o

110o + x = 180o

x = 180o – 110o

x = 70o

The missing angle in this triangle is 70o

Question 5: Find the angles in a given equilateral triangle

Solution:

The given triangle is an equilateral triangle

So not only the length of all sides are equal but also the angles are also equal.

∠x = ∠y = ∠z (Equation-1)

We know that sum of interior angles is 180o

So x + y + z = 180o

x + x + x = 180o   (As all angles are same)

3x = 180o

x = 180o/3

x = 60o

From equation-1 y = 60o and z = 60o

The missing angles in the given equilateral triangle are 60o, 60o, 60o

These are all the possible ways to find missing angles in a triangle.

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Find angle C.

Possible Answers:

C=72

None of these

C=53

C=70

C=65

Explanation:

First, know that all the angles in a triangle add up to 180 degrees.

Each triangle has 3 angles.  Thus, we have the sum of three angles as shown:

where we have angles A, B, and C. In our right triangle, one angle is 25 degree and we'll call that angle A. The other known angle is 90 degrees and we'll call this angle B.  Thus, we have

Simplify and solve for C.

Which of the following can be two angle measures of a right triangle?

Correct answer:

Explanation:

A right triangle cannot have an obtuse angle; this eliminates the choice of 100 and 10.

The acute angles of a right triangle must total 90 degrees. Three choices can be eliminated by this criterion:

The remaining choice is correct:

A right triangle has an angle that is 15 more than twice the other.  What is the smaller angle?

Correct answer:

Explanation:

The sum of the angles in a triangle is 180.  A right triangle has one angle of 90.  Thus, the sum of the other two angles will be 90.

Let =  first angle and = second angle

So the equation to solve becomes or

Thus, the first angle is and the second angle is .

So the smaller angle is

Angle  in the triangle shown below (not to scale) is 35 degrees. What is angle ?

Possible Answers:

 degrees

 degrees

 degrees

 degrees

Correct answer:

 degrees

Explanation:

The interior angles of a triangle always add up to 180 degrees. We are given angle  and since this is indicated to be a right triangle we know angle  is equal to 90 degrees. Thus we know 2 of the 3 and can determine the third angle.

Angle  is equal to 55 degrees.

Which of the following cannot be true of a right triangle?

Possible Answers:

One leg can be longer than the hypotenuse.

A right triangle can have an obtuse angle.

A right triangle can be equilateral.

None of the other statements can be true of a right triangle.

The measures of the angles of a right triangle can total .

Correct answer:

None of the other statements can be true of a right triangle.

Explanation:

All of these statements are false.

A right triangle can be equilateral.

False: An equilateral triangle must have three angles that measure  each.

One leg can be longer than the hypotenuse.

False: Each leg is shorter than the hypotenuse.

A right triangle can have an obtuse angle.

False: Both angles of a right triangle that are not right must be acute.

The measures of the angles of a right triangle can total .

False: The measures of any triangle total .

In triangle , what is the measure of angle ?

Correct answer:

Find the degree measure of the missing angle.

Correct answer:

Explanation:

All the angles in a triangle add up to 180º.

To find the value of the remaining angle, subtract the known angles from 180º:

Therefore, the third angle measures 43º.

The right triangle  has two equal angles, what is each of their measures?

Correct answer:

Explanation:

The internal angles of a triangle always add up to 180 degrees, and it was given that the triangle was right, meaning that one of the angles measures 90 degrees.

This leaves 90 degrees to split evenly between the two remaining angles as was shown in the question.

Therefore, each of the two equal angles has a measure of 45 degrees.

What is the missing angle in this right triangle?

Correct answer:

Explanation:

The angles of a triangle all add up to .

This means that .

Using the fact that 90 is half of 180, we can figure out that the missing angle, x, plus 34 adds to the remaining 90, and we can just subtract

.

Solve for :

Correct answer:

Explanation:

The angles of a triangle add together to 180 degrees. We already know that one of the angles is 90 degrees, so we can subtract 90 from 180: the other 2 angles have to add to 90 degrees.

We can now subtract to get x:

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