Amy has a master's degree in secondary education and has taught math at a public charter high school.
Watch this video lesson to see why a triangle's angles always add up to 180 degrees. Also, learn how you can use this unique fact about triangles to find an unknown angle in a triangle.
If you cut a rectangular piece of cake in half diagonally, you will end up with two triangles. In this video lesson, we will talk about the angles of a triangle and why they always add up to 180 degrees. We can define a triangle as a flat shape with three straight sides.
The angles of the triangle are the inside corners of a triangle. You can see that there are three of them. No matter what size or kind of triangle you have, you will always have your three angles. Do you see the three angles in all of these triangles? We can label our three angles as a, b, and c. It doesn't matter which one we label which as long as all three are labeled.
Three sides to a triangle
The Formula and Proof
With our angles labeled, we have a formula we can refer to for the angles. It is a + b + c = 180, which tells us that if we add up all of our angles, they will always equal 180.
Why is this? Well, we can prove this with the use of parallel lines. If we extended the line between angles a and b and we drew a line through the tip of angle c that is parallel to the line going through the point of angles a and b, then we will see that all three angles add up to 180 degrees.
If you look at the parallel lines we've drawn and you compare the angles, you will see that angles a and b are the same angles as the angles on either side of angle c. Each side of the triangle becomes the line that intersects the pair of parallel lines, and because of this, the angles that are inside the parallel lines and on opposite sides of the intersecting line are equal.
Another way you can check this is to draw a triangle using a ruler. Then cut it out and then tear each angle off. Then connect the angles. What do they form? Don't they form a straight line at the bottom? This is the same kind of proof as the parallel lines proof. It shows you that all three angles always add up to 180 degrees, or a straight line.
Finding a Missing Angle
Because all triangles add up to 180 degrees, we can use this fact to our advantage. Sometimes we will have a triangle for which we know the measurement of two of the angles. The unknown third angle is easy to find since we know that all of them added up will equal 180. To find the missing angle, we can either use the formula and a bit of algebra or simply subtract the two known angles from 180 to find the unknown third.
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Let's see how this works. We have a triangle here for which we know two of the angles. We see that the two angles measure 49 and 28 degrees, respectively. To find the third, we can plug these values into our formula. Because it doesn't matter how we label our angles, we will just call the 49 degree angle a and the 28 degree angle b. So our formula becomes 49 + 28 + c = 180. To solve for angle c, we need to subtract the 49 and 28 from 180. Doing this, we get 180 - 49 - 28 = 103. So our answer is 103 degrees. We can use this method whenever we need to find an unknown third angle of any triangle we come across as long as the other two are known.
We've learned that triangles are unique shapes with the interesting fact that all of the angles added up will always equal 180 degrees. A triangle, we learned, is defined as a flat shape with three straight sides. Every triangle has three angles, and these three angles will always add up to 180 degrees. We can prove this with the use of parallel lines or by cutting off the angles and lining them up to see that they form a straight line.
The formula for the angles is a + b + c = 180. Unless the problem labels the angles specifically for us, we can label the angles any which way we want, as long as each has been labeled with a, b, or c. To find an unknown third angle of a triangle for which you know the two other angles, you can either use the formula or simply subtract the two known angles from 180.
When you've finished with this video lesson, you should be able to:
Point out and label the angles of triangles
Know that the combined three angles of triangles always equal 180 degrees
Recall the formula for measuring an angle and proving this theory
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