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Transcript

WHAT IS A PERPENDICULAR BISECTOR? A perpendicular bisector is a line that divides a segment into two equal segments in a direction perpendicular to that same segment. This kind of bisector makes up four 90˚ angles, such as in this picture: THEY CAN ALSO BE FOUND IN TRIANGLES… PERPENDICULAR BISECTOR THEOREM The Perpendicular Bisector Theorem states that any given point on the perpendicular bisector is equidistant from the endpoints of the bisected segment. Converse: : If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of that segment. ANGLE BISECTOR THEOREM Angle Bisector: An angle bisector is a ray that divides an angle into two congruent angles. The Theorem: If a point’s on the angle bisector, then it’s equidistant to the sides of the angle. Converse: If a point is equidistant to the sides of an angle, then it’s on its bisector. CONCURRENCY OF PERPENDICULAR BISECTORS OF A TRIANGLE THEOREM The point of Concurrence is where 3 or more lines intersect. Theorem: The perpendicular bisectors of a triangle meet at a point that’s equidistant to all the vertices. CIRCUMCENTER The point in which all the perpendicular bisectors meet. In an acute triangle: the point of concurrency is on the inside of the triangle. In an obtuse triangle: the point of concurrency is on the outside. In a right triangle: it’s on the midpoint of the hypotenuse. CONCURRENCY OF ANGLE BISECTORS THEOREM The angle bisectors of a triangle meet at a point that is equidistant to all the sides of the triangle. INCENTER The point in which all angle bisectors in a triangle meet. CONCURRENCY OF MEDIANS OF A TRIANGLE THEOREM Median: A segment in which one of its endpoints is on the vertex of one of the angles of a triangle and the other endpoint on the midpoint of the opposite side. The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. CENTROID The point in which all the medians meet. ALTITUDE & ORTHOCENTER Segment from the vertex of an angle in a triangle that goes perpendicular to its opposite side. ORTHOCENTER: where these 3 meet. MIDSEGMENT THEOREM MIDSEGMENT: Segment that goes from the midpoint of one side of a triangle to the midpoint of another side. THEOREM: A midsegment that connects two sides of a triangle is both half as long and parallel to the third side. EXTERIOR ANGLE INEQUALITY The exterior angle of a triangle is bigger than any of the non adjacent interior angles. TRIANGLE INEQUALITY The longest side of a triangle has to be greater than the sum of the two shorter sides so the sides can for a triangle. HOW TO WRITE AN INDIRECT PROOF 1. Write the false of the statement. 2. The false is given. 3. Keep on solving as a normal proof until you come to a contradiction. HINGE THEOREM If two triangles have two congruent sides, and the included angle of the first triangle is greater than the included angle of the second triangle, then the third side of the 1st triangle is greater that the one on the second triangle. 30-60-90 Hypotenuse is twice as long as the short leg (Or short leg is half as long as hypotenuse) Long leg is measure of short leg times √3/ Short leg is lenght of long leg divided by √3. 45-45-90 Both legs are congruent. Hypotenuse is length of leg times √2. Legs: Length of the hypotenuse divided by √2. _____(0-10 pts.) Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3 examples of each. _____(0-10 pts.) Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples of each. _____(0-10 pts.) Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle theorem. Explain what a circumcenter is. Give at least 3 examples of each. _____(0-10 pts.) Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3 examples of each. _____(0-10 pts.) Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle theorem. Give at least 3 examples of each. _____(0-10 pts.) Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. Give at least 3 examples. _____(0-10 pts.) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples. _____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples. _____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples. _____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples. _____(0-10 pts.) Describe how to write an indirect proof. Give at least 3 examples. _____(0-10 pts.) Describe the hinge theorem and its converse. Give at least 3 examples. _____(0-10 pts.) Describe the special relationships in the special right triangles (30-60-90 and 45-45-90). Give at least 3 examples of each. _____(0-5 pts.) Neatness and originality bonus. ______Total points earned (120 possible)