I am studying for my GRE exam into grad school and it's been a very long time since I've done geometry. I have a problem to which I need to solve for the area of a triangle but I do not have the base. I do have all angles but I cannot remember how to convert angles into their corresponding line segments. Any help on this matter would be greatly appreciated. Thank you.

Kaytee

If you have all the angles, and two sides..you can use the law of sines, or the law of cosines.

Law of sines:
a/SinA = b/SinB = c/sinC

law of cosines:
c^2= a^2 + b^2 -2abCosC

The law of sines will be easier.

Thank you but I only have one side (the height) and all 3 angles. Any other solution?

Give us those angles and side and height

Use the law of sines>

a/SinA=b/sinB
You have angles A,B and side a.

Well, thank you for your insight but I was just wondering if there was a simple law that would give me a side from 2 corresponding angles. I was sure that the angles could be added in a certain fashion or something to give me the corresponding side. Unfortunately for this exam we cannot use a calculator so I have to be able to find the answer free hand; and to be honest I have never learned about the law of sines at all. I would appreciate anymore ideas. Here are the angles:
it is a right triangle with the other two angles being 30 degrees and 60 degrees with a height of 5. What I need to know is which is greater, the area of the triangle or 25*the square root of 3. In order for me to find the area though I must be able to find the base. Please help.

Kaytee, the angles you gave us are termed "nice" angles and you're expected to know the sin, cos, and tan values for angles with degree 30,45,and 60. I'll let you look up their values, but in trig we're expected to learn these ones by rote. You should see easily why they're "nice".
You might want to start a new post so we can see it; looks like this one's getting ready to go over the edge.

Explain Bot Explain Bot answered
9 months ago

Sure, I can help you find the base of the triangle using the given angles. In a right triangle with one angle being 90 degrees, you can use the trigonometric ratios to find the lengths of the sides.

Since you have a right triangle with angles measuring 30 degrees and 60 degrees, this means you have a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in a special ratio. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is the length of the hypotenuse multiplied by the square root of 3.

In your case, if the height of the triangle (opposite the 60-degree angle) is 5, the length of the hypotenuse is 2 times the height, which is 10. And the base of the triangle (opposite the 30-degree angle) is the length of the hypotenuse multiplied by the square root of 3, which is 10 * √3.

Now you can compare the area of the triangle to 25 times the square root of 3. The area of a triangle can be found using the formula A = 1/2 * base * height. Plug in the values you have: base = 10 * √3 and height = 5. Calculate:

A = 1/2 * (10 * √3) * 5
= 25√3

So, the area of the triangle is 25 times the square root of 3, which means the two values are equal.

To recap, in a 30-60-90 triangle, you can determine the length of the base (opposite the 30-degree angle) using the length of the height (opposite the 60-degree angle) by multiplying the height by the square root of 3.