Depends on what is given. SOHCAHTOA where O=opposite side, H=hypotenuse, A=opposite sides of a triangle in relation to the angle you are seeking.C=cosine, S=sine, T=tangent. So it depends on what is given and what is sought to be any more specific!
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To find an angle without using a calculator, you will need to use a trig identity. Determine which angle it is and use its corresponding trig identity.
A corresponding angle is related to a primary angle. Subtract the primary angle measure from 180 degrees, to obtain the corresponding angle measure.
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If the angle is a lone, random angle, I believe you would need a protractor to determine the precise size of the angle (in "degrees"). However, you could, in this case, roughly guess as to whether the angle is acute, obtuse, or right (if the little rectangle is shown in the angle). Of course, if an angle is in a position where one can determine its measure using known postulates or theorems, finding the size of this angle becomes much easier. For example, if you know the measure of one angle and you must determine the measure of another angle, but these two angles are vertical angles, or are corresponding angles (by the corresponding angles postulate), you can indeed determine the measure of this angle without a protractor. Additionally, another example is that if you knew a pair of angles were either supplementary angles, complementary angles, or a linear pair, and you were given the measure of one of these angles, you could determine the measure of the other angle without a protractor. Therefore, it depends on the angle you're looking at.
To find the measure of an angle, you need to know the size of the entire angle and the other angles within the angle. Then, you subtract the smaller, known angles from the entire, large angle and you should get the measure of the missing angle.