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After solving for in each of the diagrams in problem 2-28, Jerome thinks he sees a pattern. He notices that the measure of an exterior angle, the angle formed by extending a side of a triangle, is related to two of the angles of the triangle.

  1. Do you see a pattern? Study the results of problem 2-28 to help.

    Pick examples of angles that could realistically be placed in for and (remote interior angles) in the triangle above.
    For example, and calculate the value for (the exterior angle) using the work you did in problem 8-6.
    Is there a connection between and (remote interior angles) and (exterior angle)?

    If you picked other angles for and (remote interior angles) would your conjecture still work?

  1. In the example below, angles and are called remote interior angles of the given exterior angle, because they are not adjacent to the exterior angle. Write a conjecture about the relationship between the measures of the remote interior and exterior angles of a triangle.
    Triangle with bottom side extending to the left, triangle angles labeled as follows, top is a, bottom right is b, bottom left is c, linear angle with bottom left angle, labeled x, & exterior angle, top & right bottom angles also labeled remote interior angles.

    The measure of the exterior angle is equal to the sum of the remote interior angles.

  2. Prove that the conjecture you wrote for part (b) is true for all triangles. Your proof can be written in any form, as long as it is convincing and provides reasons for all statements.

  3. Now that you have proved the relationship between an exterior angle and the corresponding remote interior angles in a triangle, add this theorem to your Theorem Graphic Organizer.