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Sec 4.1 Apply Triangle Sum Properties VOCABULARY Triangle ________________________________________________________________________ Interior angles: angles inside the figure. Ex: angles 4, 6, and 7. Exterior angles: angles created by extending the sides of a figure, lie outside the figure. Ex: angles 1, 2, 3, 5, 8, 9, 10, 11, 12. 1 2 5 6 9 10 4 3 7 8 11 12 Corollary to a theorem ________________________________________________________________________ ________________________________________________________________________ CLASSIFYING TRIANGLES BY SIDES Scalene Triangle Isosceles Triangle Equilateral Triangle ____ congruent sides At least ____ congruent sides ____ congruent sides CLASSIFYING TRIANGLES BY ANGLES Acute Triangle Right Triangle Obtuse Triangle Equiangular Triangle ____ acute angles ____ right angle ____ obtuse angles ____ congruent angles Notice that an equilateral triangle is also isosceles. An equiangular triangle is also acute. Example 1 Classify ∆RST by its sides. Determine if the triangle is a right triangle. Step 1 Use the distance formula RS = __8.1__ (5- -3) 2 (2 - 3 ) 2 x2 x1 2 y2 y1 2 8.1 ST = _________ ( ) to find the side lengths. RT = ________ ) ( 2 2 ) ( ) ( 2 2 ) Step 2 Check for right angles. The slope of RT is __________. The slope of ST is ____________. The product of the slopes is ______, so RT ST and RTS is a ________ angle. Therefore, ∆RST is a ____________ _ triangle. THEOREM 4.1: TRIANGLE SUM THEOREM mA + mB + mC = ___________ The sum of the measures of the interior angles of a triangle is ______. THEOREM 4.2: EXTERIOR ANGLE THEOREM Angles A, B and C are interior angles. A and B are called REMOTE INTERIOR angles because they are far away from the exterior angle at C. m1 = m_ __ + m__ _ The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Example 2 Use the diagram at the right to find the measure of DCB, A and 1 . Solve 3x+16 = 52 + 2x x = 36 so mA=2(36) = 72, m1 = 180-(52+72) = 56. mDCB= 3(36)+16 = 124. To check your answers: The exterior angle = the sum of the 2 remote interior angles. 124 = (52+72) smile! Example 3 Use the diagram at the right to find the measure of DCB. Solution Step 1 Write and solve an equation to find the value of x. 3x 9 = x + 73 x = _____ Exterior Angle Theorem Solve for x. Step 2 Substitute x = ____ in 3x 9 to find mDCB. 3x 9 = 3 ____ 9 = _____. The measure of DCB is _____. COROLLARY TO THE TRIANGLE SUM THEOREM mA + mB = _____ The acute angles of a right triangle are _____________.