Angle Relationships in Triangles Focus on Reasoning. Essential question: What are some theorems about angle measures in triangles? G-CO.3.

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1 Name lass 4- Date ngle Relationships in Triangles Focus on Reasoning Essential question: What are some theorems about angle measures in triangles? G-O..0 Investigate the angle measures of a triangle. Use a straightedge to draw a large triangle on a sheet of paper. ut out the triangle. Tear off the angles of the triangle. D Place the angles together so their sides are adjacent and their vertices meet at a point. Take note of how the angles come together. E Repeat the process by drawing different triangles. e sure you try an acute triangle, a right triangle, and an obtuse triangle. In each case, note how the angles come together. REFLET a. ompare your work with that of other students. What always seems to be true about the three angles of a triangle when they are placed together? b. Make a conjecture: What can you say about the sum of the angle measures in a triangle? c. n equiangular triangle has three congruent angles. What do you think is true about the angles of an equiangular triangle? Why? d. In a right triangle, what is the relationship of the measures of the two acute angles? The relationship you investigated above is known as the Triangle Sum Theorem. hapter 4 4 Lesson

2 The Triangle Sum Theorem The sum of the angle measures in a triangle is 80. m + m + m = 80 The proof of the Triangle Sum Theorem depends upon a postulate known as the Parallel Postulate. The Parallel Postulate Through a point P not on a line l, there is exactly one line parallel to l. P l Prove the Triangle Sum Theorem. The sum of the angle measures in a triangle is 80. Given: Prove: m + m + m = 80 Understand the plan for the proof. Draw a line through that is parallel to. This creates three angles that form a straight angle, so the sum of their measures is 80. Use the fact that alternate interior angles have the same measure to conclude that the sum of the measures of the angles in a triangle is l omplete the proof. REFLET Statements. Draw l through point parallel to... m 4 = m and m 5 = m.. m 4 + m + m 5 = Reasons. ngle ddition Postulate and definition of straight angle a. Give an indirect proof to show why it is not possible for a triangle to have two right angles. hapter 4 4 Lesson

3 corollary to a theorem is a statement that can be proved easily by using the theorem. useful corollary to the Triangle Sum Theorem involves exterior angles of a triangle. When you extend the sides of a polygon, the original angles may be called interior angles and the angles that form linear pairs with the interior angles are the exterior angles. Each exterior angle of a triangle has two remote interior angles. remote interior angle is an interior angle that is not adjacent to the exterior angle. Interior angle Remote interior angles Exterior angle Exterior angles Prove the Exterior ngle Theorem. The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Given: Prove: m 4 = m + m omplete the proof. Statements. and 4 are supplementary.. Reasons 4. m + m 4 = Triangle Sum Theorem 4. m + m 4 = m + m + m m 4 = m + m 5. REFLET a. Explain how you could verify the Exterior ngle Theorem using a method similar to that of the Explore. hapter 4 4 Lesson

4 nother important corollary of the Triangle Sum Theorem is the Quadrilateral Sum Theorem. You will prove the theorem as an exercise. Quadrilateral Sum Theorem The sum of the angle measures in a quadrilateral is 60. PRTIE. omplete the proof that the acute angles of a right triangle are complementary. Given: with a right angle Prove: and are complementary. Statements. is a right angle.. Reasons. m = 90.. m + m + m = Substitution Property of Equality Subtraction Property of Equality 6. and are complementary. 6.. Write a paragraph proof of the Quadrilateral Sum Theorem. Given: Quadrilateral D Prove: m + m + m + m D = 60 (Hint: Draw diagonal and number the angles formed.). 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? D hapter 4 44 Lesson

5 4- Name lass Date Date Name lass Practice dditional Practice 4- ngle Relationships in Triangles LESSON. n area in central North arolina is known as the Research Triangle because of the relatively large number of high-tech companies and research universities located there. Duke University, the University of North arolina at hapel Hill, and North arolina State University are all within this area. The Research Triangle is roughly bounded by the cities of hapel Hill, Durham, and Raleigh. From hapel Hill, the angle between Durham and Raleigh measures From Raleigh, the angle between hapel Hill and Durham measures 4.. Find the angle between hapel Hill and Raleigh from Durham.. The acute angles of right triangle are congruent. Find their measures. The measure of one of the acute angles in a right triangle is given. Find the measure of the other acute angle (90 z) Find each angle measure. 6. m 7. m PRS 8. In ULMN, the measure of an exterior angle at N measures 99. m L = x and m M = x. Find m L, m M, and m LNM. 9. m E and m G 0. m T and m V. In U and UDEF, m = m D and m = m E. Find m F if an exterior angle at measures 07, m = (5x + ), and m = (5x + 5).. The angle measures of a triangle are in the ratio : 4 :. Find the angle measures of the triangle. hapter 4 45 Original content opyright by Holt McDougal. dditions and changes to the original content are the responsibility of the instructor. Lesson Holt McDougal Geometry

6 Problem Solving. The locations of three food stands on a fair s midway are shown. What is the measure of the angle labeled?. large triangular piece of plywood is to be painted to look like a mountain for the spring musical. The angles at the base of the plywood measure 76 and 45. What is the measure of the top angle that represents the mountain peak?. What is the value of? 4. What is the measure of each angle in the banner? 5. What is, the measure of the angle that 6. t takeoff, =. What is, the the pole makes when it first touches measure of the angle the pole makes the ground? with the athlete s body? 7. What is the measure of? D 5 8. What is the measure of? 9. What is the measure of? F 85 H G 90 J 5 68 D 55 hapter 4 46 Lesson