What is the relation between the lengths of the sides of a triangle vs. the angles?

Question

If you had the lengths of a triangles sides, how could you figure out the angle measurements? For example, triangle ABC has the following measurements:
AB=5cm
BC=4cm
AC=3cm

Please answer this question with precise step-by-step instructions on how to do it.

Thanks

Answer 1

This triangle is a rather special one, and people who are familiar with geometry or trigonometry tend to know that the triangle with sides 3, 4 and 5 is right-angled.
This is because 3^2 + 4^2 = 9 + 16 = 25, and 25 = 5^2.

If you call the triangle ABC, and let the sides be AB = 3cm, BC = 4cm, and CA = 5cm, then the right angle is at B, because that is opposite the longest side.

The ratio CB / BA is the ratio of the side opposite A to the side adjacent to A. From the mnemonic SOH CAH TOA, the tangent of the angle A is CB / BA = 4 / 3.
Angle A is therefore arctan(4/3) = 53.1 degrees approx.

Now you know angle A (53.1 deg) and angle B (90 deg), you can add these together, getting 143.1 deg. You know that the angles of a triangle add up to 180 deg, and so you can subtract 143.1 from 180, getting angle C = 36.9 deg.

If you had three sides of a triangle and did not know it to be right angled, you would need a more advanced formula. One such formula would be the cosine rule.

Answer 2

The measurements that you give are part of a rather famous right-angle triangle. 3,4,5 make up the right, bottom, and hypothenuse of a right triangle. The angle at vertex C is 90 degrees. The angle at A is inv-sin(4/5) and the angle at B is inv-sin(3/5).

Another famous triangle has lengths 1,2,sqrt(3). That has angles 30,60, and 90 degrees.

To solve for the generic triangle when you have just the lengths of each side, you can use the law of cosine:

c^2 = a^2 + b^2 - 2*a*b*cosine(C)

Plug in a, b, and c, then solve for angle C

You can apply the same equation for other two angles. Use A+B+C = 180 degrees as verification.

Answer 3

Your triangle is a right triangle as the sides are 3, 4, and 5. This may be proved with the Pythagorean theorem if you wish.

The previous answerer gave you the correct formulas to solve for the angles in a right triangle. One angle you already know, the angle opposite the side of 5 cm (the hypotenuse) is 90 degrees.
.

Answer 4

use law of cosines ..... or.... c^2=a^2+b^2 - 2ab(cosC)

side opposite each vertex is named with the lower case of that letter. c=5cm, a=4cm, b=3cm....

By the way, this is actually a right angled triangle because

5^2 = 3^2 + 4^2

so angle c is 90 degrees

now use sinB = opp / hyp
sinB=3/5=0.6
use a scientific calculator and do 0.6 second f sin... and you'll get the angle..36.9degrees

which leaves angle A at 53.1 degrees because the sum of the angles has to be 180 degrees.

Answer 5

The angles of a triangle are in the same proportion as the length of its sides. For example: (1) The sum of the measures of any pair of sides in a triangle will always be greater than the measure of the remaining side. [s + m > l] (2) The longest side of a triangle will always be opposite the greatest angle of the triangle, and the shortest side will always be opposite the smallest angle. [Given ΔABC with m∠A > m∠B > m∠C, we know that BC > AC > AB.]

Answer 6

1. Draw Triangle
2- Use protractor to measure angles
3- Take measurements of angles and apply them to the following definitions to find the type of triangle it is.
Acute - all angles measure less than 90º
Right - one angle measures exactly 90º.
Obtuse - one angle measures more than 90º.
Equiangular - all angles measure the same. (60º) This is the same as the equilateral triangle.

Answer 7

Trigonometry
eg
sin x=opp/hyp
cos x=adj/hyp
tan x=opp/adj