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LESSON 3.4 TRIANGLE-SUM THEOREM During this lesson, you will: Classify triangles and find the measures of their angles Use exterior angles of triangles Classifying Triangles Triangles can be classified by the lengths of their sides and the measures of their angles. VOCABULARY Isosceles Scalene Equilateral 5 12 13 All 3 sides ≡ No sides ≡ At least 2 sides ≡ VOCABULARY Obtuse Right Equiangular Draw and label an acute triangle (Ex. 89,45, 46). 60 60 ≡ angles; each 60º 120 60 Exactly one 90 º angle Exactly one angle > 90 º Classify each triangle below based upon the lengths of its sides: EQUILATERAL SCALENE ISOSCELES Classify each triangle based upon its angle measures: ACUTE OBTUSE RIGHT EQUIANGULAR Check for Understanding: Use your protractor and ruler to draw and label the following. If not possible, write not possible. isosceles right triangle isosceles acute triangle isosceles obtuse triangle Alert! An equilateral triangle is always equiangular. An isosceles triangle may be right, obtuse, or acute. A scalene triangle may be right, obtuse, or acute. Alert! An equiangular triangle is always equilateral. A right triangle may be isosceles or scalene. An obtuse triangle may be isosceles or scalene. INVESTIGATION Write a, b, and c in the interiors of the three angles of your triangle. Geometry Carefully cut out your triangle if it was not pre-cut for you. Tear off the three angles. Mrs. McConaughy Arrange the three angles so that their vertices meet at a fixed point to show their sum. 10 INVESTIGATIVE RESULTS: THEOREM Triangle Sum Conjecture: The sum of the measures of the interior angles of any triangle is 180º ▲ Sum = 180. B A mA + m B + m C = 180 C Ex. 1 Z Find m Z X 48º mX + mY + mZ = 180 48 + 67 + mZ = 180 115 + mZ = 180 mZ = 65 67º Y Ex. 2 Find the values of a, b and c. C EX. 2 cº 70º Find c aº c + 70 = 90 A c = 20º D Find a. Use Triangle Sum a + mADC + c = 180 a + 90 + 20 = 180 a + 110 = 180 a = 70º bº B Find b. Use Triangle Sum b = 20º Use your triangle sum conjecture to calculate the measure of the third angle in each triangle below. Third Angle Conjecture: If two angles of one triangle are equal in measure to two angles of another triangle, then the remaining angles _____________________. Geometry Mrs. McConaughy 14 Exterior Angle Theorem Remote interior angles A B 2 Exterior angle 1 3 x C m x = m 1 + m 2 The measure of an exterior angle in any triangle equals the sum of the two remote interior angles. Examples: x Justify your answer. x = ____ x + y = _____ 30 x y 135 x = ____ x x x 90 45 144 x = ____