Which theorem states that the measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle?
(Line and Angle Relationships Unit Test)
conexxese
The theorem that states the measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle is known as the Exterior Angle Theorem.
To prove this theorem, you can use the fact that the sum of the angles in a triangle is always 180 degrees.
Here are the steps to prove the Exterior Angle Theorem:
1. Draw a triangle and label its three interior angles A, B, and C.
2. Choose one angle, let's say angle A, as the exterior angle.
3. Draw a line that extends from one side of the triangle (let's call it side BC) at angle B, creating angle D on the exterior of the triangle.
4. Label the two remote interior angles of the exterior angle A as angles B and C.
5. Now, consider the sum of the interior angles of the triangle. We know that angle A + angle B + angle C = 180 degrees.
6. Next, notice that angle B + angle D = 180 degrees since the line that forms angle D is a straight line, making the two angles supplementary.
7. Since angle B = angle B (the interior angle is congruent to itself), we can substitute it into the equation from step 6. This gives us angle B + angle B + angle C = 180 degrees.
8. Combining like terms, we have 2 angle B + angle C = 180 degrees.
9. Rearrange the equation to isolate angle C: angle C = 180 degrees - 2 angle B.
10. Now, notice that angle C is the sum of angles B and D (angle D is the exterior angle). So, we can rewrite it as angle C = angle B + angle D.
11. Substitute the values of angle C and angle D derived from the previous steps into this equation: 180 degrees - 2 angle B = angle B + angle D.
12. Simplify the equation: 180 degrees - 2 angle B = 180 degrees. By subtracting 180 degrees from both sides, we get -2 angle B = 0.
13. Divide both sides by -2 to solve for angle B: angle B = 0/(-2) = 0 degrees.
14. Since angle B = 0 degrees, we can substitute it into angle C = 180 degrees - 2 angle B, which becomes angle C = 180 degrees - 2(0) = 180 degrees.
15. Therefore, angles B and C are each 180 degrees, which confirms that the measure of the exterior angle is equal to the sum of the measures of the two remote interior angles.
So, the Exterior Angle Theorem states that for any triangle, the measure of the exterior angle is equal to the sum of the measures of the two remote interior angles.
It should be Dr WHAT. What are your choices?
I'd say it is the one which states
the measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle