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Triangles and their properties •Triangle Angle sum Theorem •External Angle property •Inequalities within a triangle •Triangle inequality theorem •Medians •Altitude •Perpendicular Bisector •Angle Bisector 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 2 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 Exterior Angle Theorem Exterior Angle Theorem 1 and 3 Inequalities Within a Triangle If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal ________________. in the same order P 11 M 8 13 L LP < PM < ML mM < mL < mP Inequalities Within a Triangle If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal ________________. in the same order W 45° 75° 60° J mW < mJ < mK JK < KW < WJ K Inequalities Within a Triangle In a right triangle, the hypotenuse is the side with the greatest measure ________________. W 5 3 X 4 WY > XW WY > XY Y Inequalities Within a Triangle The longest side is BC So, the largest angle is The largest angle is A L So, the longest side is MN 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 Every triangle has 1. 3 medians, 2. 3 angle bisectors and 3. 3 altitudes. 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 are a vertex and a midpoint of the B opposite side Any triangle has three medians. L M A N Let L, M and N be the midpoints of AB, BC and AC respectively. CL, AM and NB are medians of ABC. C The point where all 3 medians intersect Centroid Is the point of concurrency The centroid is 2/3’s of the distance from the vertex to the side. 10 2x 32 5x 16 X The centroid is the center of balance for the triangle. You can balance a triangle on the tip of your pencil if you place the tip on the centroid angle bisector of a triangle a segment that bisects an angle of the triangle and goes to the opposite side. Any triangle has three angle bisectors. B E A In the figure, AF, DB and EC F MD are angle bisectors of ABC. C Note: An angle bisector and a median of a triangle are sometimes different. Let M be the midpoint of AC. The median goes from the vertex to the midpoint of the opposite side. BM is a median BD is a angle bisector of ABC. 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 This makes the Incenter an equidistance from all 3 sides Let AD be a bisector of BAC, M B P lie on AD, PM AB at M, P A D NP AC at N. N C Then P is equidistant from AB and AC. Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. The converse of this theorem is not always true. Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the bisector of the angle. Using the Angle Bisector Theorem • What is the length of RM? Because angle N has been bisected, I know that each point along the bisector is equidistant to the sides Since MR and RP are both perpendicular to each side and touch the bisector, I know they are equal 7x = 2x + 25 5x = 25 x= 5 What is the length of FB? Because angle C has been bisected, I know that each point along the bisector is equidistant to the sides Since BF and FD are both perpendicular to each side and touch the bisector, I know they are equal 6x +3 = 4x + 9 2x +3 = 9 2x = 6 x = 3 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. B C A ACUTE OBTUSE Can a side of a triangle be its altitude? YES! A G C B RIGHT If ABC is a right triangle, identify its altitudes. BG, AB and BC are its altitudes. Orthocenter is where all the altitudes intersect. Orthocenter The orthocenter can be located in the triangle, on the triangle or outside the triangle. Obtuse Right Legs are altitudes 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 equidistance from the endpoints of the segment. A C AB is the perpendicular bisector of CD D B This makes the Circumcenter an equidistance from the 3 vertices Perpendicular Bisector Perpendicular Bisector Using the Perpendicular Bisector Theorem • What is the length of AB? Since BD perpendicular to the side opposite B and bisects AC, I know that BD is a perpendicular bisector. Since BD is a perpendicular bisector, I know that BA and BC are congruent since they are connected to the vertex and the end of the bisected line. AB = 4x AB = 4(5) AB = 20 4x = 6x – 10 –2x = – 10 x=5 BC = 6x – 10 BC = 6(5) – 10 BC = 20 What is the length of QR? Since SQ is perpendicular to the side opposite Q and bisects PR, I know that SQ is a perpendicular bisector. Since SQ is a perpendicular bisector, I know that PQ and QR are congruent since they are connected to the vertex and the end of the bisected line. PQ = 3n – 1 PQ = 3(3) –1 PQ = 8 3n – 1= 5n – 7 – 1= 2n – 7 6 = 2n 3=n QR = 5(n) – 7 QR = 5(3) – 7 QR = 8 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 Midsegment Theorem The midsegment of a triangle is parallel to the third side and is half as long as that side. 1 DE AC 2 B D A E DE AC C 1. Identify the 3 pairs of parallel lines shown above UW TX WY VT YU XV 2a. If LK = 46, what is NM ? MN is half as long as LK 2(MN) = 46 MN = 23 2b. If JK = 5x + 20 and NO = 20, find x NO is half and big as JK 2(20) = 5x +20 40 = 5x + 20 x=4 Example 1 In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU. 16 ft PR = ________ 5 ft TU = ________ Example 2 In the diagram, XZ and ZY are midsegments of triangle LMN. Find MN and ZY. 53 cm MN = ________ 14 cm ZY = ________ Example 3 In the diagram, ED and DF are midsegments of triangle ABC. Find DF and AB. 2 (DF ) = AB 5X+2 2 (3x – 4 ) = 5x + 2 6x – 8 = 5x + 2 3X – 4 x–8= 2 x = 10 x = ________ 10 DF = ________ 26 AB = ________ 52 Perpendicular Bisectors • A point is equidistant from two objects if it is the same distance from each. Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Angle Bisectors • The distance from a point to a line is the length of the perpendicular segment from the point to the line. Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. There are 3 of each of these special segments in a triangle. The 3 segments are concurrent. They intersect at the same point. This point is called the point of concurrency. The points have special names and special properties. Altitude .. Vertex .. 90° .. Orthocenter Angle Bisector.. Angle into 2 equal angles .. Incenter Perpendicular Bisector… 90° .. bisects side .. Circumcenter Median .. Vertex .. Midpoint of side ..Centroid Give the best name for AB A | A | B B Median Altitude A A B B None Angle Bisector A | B | Perpendicular Bisector Survival Training You’re Stranded On A Triangular Shaped Island. The Rescue Ship Can Only Dock On One Side Of The Island But You Don’t Know Which Side. At Which Point Of Concurrency Would You Set Up Camp So You Are An Equal Distance From All 3 Sides? INCENTER What If The Ship Could Only Dock At One Of The Vertices? Would You Change The Location Of Your Camp ? If So, Where? YES CIRCUMCENTER Where would you place a fire hydrant to make it equidistance to the houses and equidistance to the streets? POST Angle bisector for the streets Perpendicular bisector for houses POST