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Triangle Angle Sum Theorem • The sum of the measures of the angles of a C triangle is 180°. m∠A + m∠B + m∠C = 180 Ex: If m∠A = 30 and m∠B = 70; what is m∠C ? B A m∠A + m∠B + m∠C = 180 30 + 70 + m∠C = 180 100 + m∠C = 180 m∠C = 180 – 100 = 80 1 Exterior Angle Theorem P The measure of an exterior angle of a triangle is equal to sum of its ___________________ remote interior angles In the triangle below, recall that 1, 2, and 3 are interior _______ angles of ΔPQR. 1 2 Q 3 4 R Angle 4 is called an exterior _______ angle of ΔPQR. An exterior angle of a triangle is an angle that forms a linear _________, pair (they add up to 180) with one of the angles of the triangle. In ΔPQR, 4 is an exterior angle because 3 + 4 = 180. Remote interior angles of a triangle are the two angles that do not form ____________________ a linear pair with the exterior angle. In ΔPQR, 1, and 2 are the remote interior angles with respect to 4. Exterior Angle Theorem 1 In the figure, which angle is the exterior angle? 5 which angles are the remote the interior angles? 2 and 3 If 2 = 20 and 3 = 65 , find 5 2 20 65 3 60 85 If 5 = 90 and 3 = 60 , find 2 30 4 90 5 Triangle Inequality Theorem The sum of the measures of any two sides of a triangle is greater than the measure of the third side. _______ b Triangle Inequality Theorem a+b>c a a+c>b c b+c>a Triangle Inequality Theorem Can 16, 10, and 5 be the measures of the sides of a triangle? No! 16 + 10 > 5 16 + 5 > 10 However, 10 + 5 > 16 Medians, Altitudes, Angle Bisectors Perpendicular Bisectors Just to make sure we are clear about what an opposite side is….. B A C Given ABC, identify the opposite side 1. of A. BC 2. of B. AC 3. of C. AB A new term… Point of concurrency • Where 3 or more lines intersect Definition of a Median of a Triangle A median of a triangle is a segment whose endpoints L M A are a vertex and a midpoint of the opposite side. B N C The point where all 3 medians intersect Centroid Is the point of concurrency The centroid is 2/3 the distance from the vertex to the side. 10 2x 32 5x 16 X angle bisector of a triangle a segment that bisects an angle of the triangle and goes to the opposite side. The Incenter is where all 3 Angle bisectors intersect Incenter Is the point of concurency Any point on an angle bisector is equidistance from both sides of the angle Definition of an Altitude of a Triangle A altitude of a triangle is a segment that has one endpoint at a vertex and the other creates a right angle at the opposite side. The altitude is perpendicular to the opposite side while going through the vertex Any triangle has three altitudes. Acute Triangle Orthocenter is where all the altitudes intersect. Orthocenter A Perpendicular bisector of a side does not have to start at a vertex. It will form a 90° angles and bisect the side. Circumcenter Is the point of concurrency Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. A C AB is the perpendicular bisector of CD D B The Midsegment of a Triangle is a segment that connects the midpoints of two sides of the triangle. The midsegment of a triangle is parallel to the third side and is half as long as that side. B D A D and E are midpoints E DE is the midsegment C DE AC 1 DE AC 2 Example 1 In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU. 16 ft PR = ________ 5 ft TU = ________ Give the best name for AB A | A | B B Median Altitude A A B B None Angle Bisector A | B | Perpendicular Bisector