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Lesson 75: Sum of Angles in Triangles and Quadrilaterals
In this quick and easy lesson you'll learn some basic facts about the angles in triangles and quadrilaterals (four-sided figures).
Recall Lesson 74 on angles. In every triangle, there are of course three angles. The sum of the three angles in any triangle is always 180°.
This is just a law of math. You can see this for yourself by drawing any type of triangle, and then measuring the angles using a protractor. If you draw a new triangle with one of the angles bigger, the other two will have to be smaller. Try this for yourself on paper. Make one angle really wide, like a big obtuse angle. The other two angles will be small acute angles. With this knowledge, you can easily figure out the measurement of an angle in a triangle, if you know the measurements of the other two angles.
For example, a triangle has a 90° angle and a 30 degree angle. What is the measurement of the other angle? We add 90 + 30 to get 120, and then subtract that from 180 to see that the missing angle is 60°. What about a triangle that has a very small 15° angle, and a slightly bigger 20° angle. What is the missing angle? Using the method above, we can see that it is a wide obtuse angle of 145°.
Look
at the diagram at left. We can deduce three different
formulas. First, c = 180 - (a+b). If we know the sum of
angles a and b, we can then subtract that sum from 180, to determine
the measure of angle c. Next, we can see that angles c and d
lie along a straight angle, which we know measures 180 degrees.
Therefore c = 180 - d.
If you look at the two formulas above that give us the value of c, we can see that (a+b) and d are in the same positions. That must mean that (a+b) and d are equal. We can therefore say that the measure of an exterior angle of a triangle (e.g., d) is equal to the sum of the measures of the two remote interior angles of the triangle (e.g., a and b) .
Four-sided figures are called quadrilaterals. All quadrilaterals have four angles. The sum of the four angles in any quadrilateral is always 360°. For example, a trapezoid has angles of 127°, 93°, and 29°. What is the measurement of the fourth angle? We can add the three known angles, and subtract the result from 360 to get our answer. The sum is 249°, which means that the unknown angle is 111°.
Later you will have much more practice with this topic. For now, just memorize the sum of the measures of the angles in triangle and quadrilaterals.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.