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Advanced Geometry LT 3.1 – Triangle Sum and Exterior Angle Theorems Exploring the Triangle Sum Theorem Activity: 1. Each member of your group should draw a different triangle on the half sheets of paper provided. Make sure you have at least one acute, obtuse and right triangle in you group. 2. Label the angles in your triangle with 1 – 3. 3. Use the tracing paper to trace the 3 angles of the triangle as 3 adjacent angles. (Trace , then trace adjacent to it, then . They should all share the same vertex.) 4. Note what is formed once all three angles are traced side by side. Reflect: 1. Compare your tracing paper with your group. What always seems to be true about the three angles of a triangle when they are placed together? Does it matter what type of triangle it is? 2. Make a conjecture: What can you say about the sum of the angle measures in all triangles? 3. An equiangular triangle has three congruent angles. What do you think is true about the angles in an equiangular triangle? Why? 4. In a right triangle, what is the relationship of the measures of the two acute angles? Why? The Triangle Sum Theorem The sum of the angle measures in a triangle is _______. In order to prove the Triangle Sum Theorem we need the following postulate: The Parallel Postulate Through a point P not on a line l, there is exactly one line parallel to l. Proving the Triangle Sum Theorem Given: Prove: Statements Reasons 1. 1. Given 2. Draw line l through point B parallel to (label the 2 new angles 4 and 5) 2. Parallel Postulate 3. __________________________ 3. _____________________________ 4. __________________________ 4. Angle Addition Postulate and definition of a straight angle 5. __________________________ 5. _____________________________ Reflect: 5. Use the Triangle Sum Theorem to explain why it is not possible for a triangle to have two right angles. Exploring the Exterior Angle Theorem When you extend the sides of a polygon, the original angles are called interior angles and the angles that form linear pairs with the interior angles are called the exterior angles. Each exterior angle of a triangle has two remote interior angles. A re mote interior angle is an interior angle that is not adjacent to the exterior angle. For example: Original Triangle Triangle With One Side Extended Activity: 1. Extend one side of the triangle you drew earlier. 2. Label the exterior angle as 3. Determine which angles are the remote interior angles to 4. Use your tracing paper to trace the remote interior angles adjacent to each other Reflect: 6. What do you notice about the two remote interior angles when they are placed together ? Is it true for all of your group members’ triangles? Proving the Exterior Angle Theorem Given: Prove: Statements Reasons 1. 1. Given 2. 2. _______________________ 3. __________________________ 3. Triangle Sum Theorem 4. 4. _______________________ 5. 5. _______________________ Reflect: 7. If two angles of one triangle are congruent to two angles of another triangle, must the third angles of the triangles also be congruent? Why or why not? 8. According to the definition of an exterior angle, one of the sides of the triangle mus t be extended in order to see it. How many ways can this be done for any vertex? How many exterior angles is it possible to draw for one triangle?