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The Sum of the Angles in a Triangle
Introduction: The sum of the angles in a triangle is 180º. We show why that is true and give several examples.

The Lesson:
In the diagram shown above, we see triangle ABC and the dotted line through A that is parallel to BC. Since angle A, from the triangle, and angles 1 and 2 form a straight (dotted) line, the sum of their measures is 180º. We also know that since the dotted line is parallel to BC that by alternate interior angles.

This gives us:
Substituting for angles 1 and 2 we have
This result is true for all triangles:
Summarizing:
  • Theorem: The sum of the three angles in a triangle is always 180º.
Let's Practice:
  1. If two of the three angles in a triangle have measure 34º and 72º, what is the measure of the third angle?
We have 34º + 72º = 106º. Therefore the third angle has a measure of
180º - 106º = 74º
  1. In a triangle ABC, we have . What is the value of x and what are the measures of angles A, B, and C?
We must have
Collecting like terms and solving gives us:
This gives us:




Notice the following: 35.92º + 57.3º + 86.76º = 179.98º

The reason that this does not add up to exactly 180º is because we rounded off in calculating the value of x.

Example
Example
In a triangle XYZ, .
What is the value of x and what is the measure of angle Z?
What is your answer?
 



M Ransom

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