How to find angles of a triangle with one angle and 2 sides

Q: How do you find the angle of a triangle with two sides and an angle?

SAS is when we know two sides and the angle between them. use The Law of Cosines to calculate the unknown side, then use The Law of Sines to find the smaller of the other two angles, and then use the three angles add to 180° to find the last angle.

Question

When you have two sides of a triangle and the angle between them, otherwise known as SAS (side-angle-side), you can use the law of cosines to solve for the other three parts. Consider the triangle ABC where a is 15, c is 20, and angle B is 124 degrees. The following figure shows what this triangle looks like.

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How do you find the angle of a triangle with two sides and an angle?

A sample triangle that allows for the law of cosines.

Now, to solve for the measure of the missing side and angles:

Find the measure of the missing side by using the law of cosines.
Use the law that solves for side b.
You end up with the value for b2. Take the square root of each side and just use the positive value (because a negative length doesn’t exist).
The length of side b is about 31.
Find the measure of one of the missing angles by using the law of cosines.
Using the law that solves for a, fill in the values that you know.
Solve for cos A by simplifying and moving all the other terms to the left.
Using a scientific calculator to find angle A, you find that A = cos–1(0.916) = 23.652, or about 24 degrees.

You can also switch to the law of sines to solve for this angle. Don’t be afraid to mix and match when solving these triangles.

Find the measure of the last angle.
Determine angle B by adding the other two angle measures together and subtracting that sum from 180.
180 – (124 + 24) = 180 – 148 = 32. Angle B measures 32 degrees.

How about an application that uses this SAS portion of the law of cosines? Consider the situation: A friend wants to build a stadium in the shape of a regular pentagon (five sides, all the same length) that measures 920 feet on each side. How far is the center of the stadium from the corners? The left part of figure shows a picture of the stadium and the segment you’re solving for.

You can divide the pentagon into five isosceles triangles. The base of each triangle is 920 feet, and the two sides are equal, so call them both a. Refer to the right-hand picture in the preceding figure. Use the law of cosines to solve for a, because you can get the angle between those two congruent sides, plus you already know the length of the side opposite that angle.

Determine the measure of the angle at the center of the pentagon.
A circle has a total of 360 degrees. Divide that number by 5, and you find that the angle of each triangle at the center of the pentagon is 72 degrees.
Use the law of cosines with the side measuring 920 feet being the side solved for.
Because the other two sides are the same measure, write them both as a in the equation.
Solve for the value of a.
The distance from the center to a corner is between 782 and 783 feet.
The computations here involve using rounded values. It’s usually best to hold off doing the rounding until you’re ready to report your final answer. In these cases, it didn’t really matter, but you want to be cautious if more accuracy is needed.

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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

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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How do you find the angle of a triangle with two sides and an angle?

"SAS" is when we know two sides and the angle between them. use The Law of Cosines to calculate the unknown side, then use The Law of Sines to find the smaller of the other two angles, and then use the three angles add to 180° to find the last angle.