Find the measure of each marked angle calculator

Video Transcript

were given two angles that form a right angle because the diagram has a box designating the angle, a right angle is 90 degrees, so we can add the two expressions that are given for these two angles and set that those two extra some of those two expressions equal to 90. So let's do that eight X minus one. The expression for the upper angle. The lower angle expression is five X. When you add them together, they equal 90 degrees. Now we have one equation with one unknown we can solve for X and after we have X weaken. Substitute X back into the expressions for the two angles to give us our angle measure for each angle. So combining like terms to solve this equation for X we have 13 x minus one equals 90. We can add one to both sides of the equation, and we have 13. X equals 91. Divide by 13 both sides and X equals seven degrees. Now seven is an intermediate answer because we need to now substitute seven back into are two expressions for the two angles. So when we do that, we have eight times seven minus one equals 55 degrees is the upper angle. And when we substitute X equal seven into the other expression for our angle, we get 35 degrees. As for the other angle now, we answered the question and we can perform a quick check when we have these two angles together. Does 55 plus 35 equal 90 degrees? Yes, they dio and we've checked our answer.

Video Transcript

were given two angles that form a right angle, and we know that it's a right angle because there's a box in the diagram of the angles in the problem, and we know that right angle is 90 degrees, so 90 degrees has shown with two angles. Here, two angles that form a total of 90 degrees are called complementary angles. Now, to find the value of each angle, we need to add the expressions that are given for each angle. One of the angles is labeled with the expression for X, and the other expression is labeled for. The other angle is labeled with the expression three X plus 13. Now, when you add both of them together, they have to equal 90 degrees because the diagram tells us it's a right angle. So we'll solve this one equation with one unknown for X and after we find X will substitute that value back into each expression to give us the angle measure of each ankle. So, combining like terms to solve this equation, we have seven X plus 13 equals 90. Subtract 13 both sides of the equation, and we have seven X equals 77 divide by seven on both sides of the equation and X equals 11 X is an important intermediate result which we will now substitute into our two expressions for each angle. So the one angle was four x four times 11 for X equals 44 degrees. The other angle expression was three X plus 13. Substitute X equals 11 into that expression, and we have 33 plus 13 is 46 degrees and those are answers for the angle measures of the degrees, we can perform a quick check. We add those two angles together 44 46. Do we get 90 degrees? Yes, and our answer is correct.

Created by Hanna Pamuła, PhD candidate

Reviewed by Bogna Szyk

Last updated: Aug 26, 2022

Triangle angle calculator is a safe bet if you want to know how to find the angle of a triangle. Whether you have three sides of a triangle given, two sides and an angle or just two angles, this tool is a solution to your geometry problems. Below you'll also find the explanation of fundamental laws concerning triangle angles: triangle angle sum theorem, triangle exterior angle theorem, and angle bisector theorem. Read on to understand how the calculator works, and give it a go - finding missing angles in triangles has never been easier!

How to find the angle of a triangle

There are several ways to find the angles in a triangle, depending on what is given:

1. Given three triangle sides

Use the formulas transformed from the law of cosines:

• cos(α) = (b² + c² - a²)/ 2bc,

  so α = arccos [(b² + c² - a²)/(2bc)]

• cos(β) = (a² + c² - b²)/ 2ac,

  so β = arccos [(a² + c² - b²)/(2ac)]

• cos(γ) = (a² + b² - c²)/ 2ab,

  so γ = arccos [(a² + b² - c²)/(2ab)]

1. Given two triangle sides and one angle

If the angle is between the given sides, you can directly use the law of cosines to find the unknown third side, and then use the formulas above to find the missing angles, e.g. given a,b,γ:

• calculate c = √[a² + b² - 2ab * cos(γ)]
• substitute c in α = arccos [(b² + c² - a²)/(2bc)]
• then find β from triangle angle sum theorem: β = 180°- α - γ

If the angle isn't between the given sides, you can use the law of sines. For example, assume that we know a, b, α:

• a / sin(α) = b / sin(β) so β = arcsin[b * sin(α) / a]
• As you know, the sum of angles in a triangle is equal to 180°. From this theorem we can find the missing angle: γ = 180°- α - β

1. Given two angles

That's the easiest option. Simply use the triangle angle sum theorem to find the missing angle:

• α = 180°- β - γ
• β = 180°- α - γ
• γ = 180°- α - β

In all three cases, you can use our triangle angle calculator - you won't be disappointed.

Sum of angles in a triangle - Triangle angle sum theorem

The theorem states that interior angles of a triangle add to 180°:

α + β + γ = 180°

How do we know that? Look at the picture: the angles denoted with the same Greek letters are congruent because they are alternate interior angles. Sum of three angles α, β, γ is equal to 180°, as they form a straight line. But hey, these are three interior angles in a triangle! That's why α + β + γ = 180°.

Exterior angles of a triangle - Triangle exterior angle theorem

An exterior angle of a triangle is equal to the sum of the opposite interior angles.

• Every triangle has six exterior angles (two at each vertex are equal in measure).
• The exterior angles, taken one at each vertex, always sum up to 360°.
• An exterior angle is supplementary to its adjacent triangle interior angle.

Angle bisector of a triangle - Angle bisector theorem

Angle bisector theorem states that:

An angle bisector of a triangle angle divides the opposite side into two segments that are proportional to the other two triangle sides.

Or, in other words:

The ratio of the BD length to the DC length is equal to the ratio of the length of side AB to the length of side AC:

|BD|/|DC|= |AB|/|AC|

Finding missing angles in triangles - example

OK, so let's practice what we just read. Assume we want to find the missing angles in our triangle. How to do that?

1. Find out which formulas you need to use. In our example, we have two sides and one angle given. Choose angle and 2 sides option.
2. Type in the given values. For example, we know that a = 9 in, b = 14 in and α = 30°. If you want to calculate it manually, use law of sines:

• a / sin(α) = b / sin(β), so

  β = arcsin[b * sin(α) / a] =

  arcsin[14 in * sin(30°) / 9 in] =

  arcsin[7/9] = 51.06°

• From the theorem about sum of angles in a triangle, we calculate that γ = 180°- α - β = 180°- 30° - 51.06° = 98.94°

1. The triangle angle calculator finds the missing angles in triangle. They are equal to the ones we calculated manually: β = 51.06°, γ = 98.94°; additionally, the tool determined the last side length: c = 17.78 in.

FAQ

How do I find angles in a triangle?

To determine the missing angle(s) in a triangle, you can call upon the following math theorems:

• The fact that the sum of angles is a triangle is always 180°;
• The law of cosines; and
• The law of sines.

Which set of angles can form a triangle?

Every set of three angles that add up to 180° can form a triangle. This is the only restriction when it comes to building a triangle from a given set of angles.

Why can't a triangle have more than one obtuse angle?

This is because the sum of angles in a triangle is always equal to 180°, while an obtuse angle has more than 90° degrees. If you had two or more obtuse angles, their sum would exceed 180° and so they couldn't form a triangle. For the same reason, a triangle can't have more than one right angle!

How do I find angles of the 3 4 5 triangle?

Let's denote a = 5, b = 4, c = 3.

1. Write down the law of cosines 5² = 3² + 4² - 2×3×4×cos(α). Rearrange it to find α, which is α = arccos(0) = 90°.
2. You can repeat the above calculation to get the other two angles.
3. Alternatively, as we know we have a right triangle, we have b/a = sin β and c/a = sin γ.
4. Either way, we obtain β ≈ 53.13° and γ ≈ 36.87.
5. We quickly verify that the sum of angles we got equals 180°, as expected.

Hanna Pamuła, PhD candidate

30 60 90 triangle45 45 90 triangleArea of a right triangle… 15 more

What is marked angle?