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Transcript
Journal Chapter 5 Daniela Lopez L. Perpendicular Bisector: a line that cuts a segment into two equal parts and forms a right angle. Perpendicular Bisector Theorem: The Perpendicular bisectors of the sides of a triangle are concurrent (meet)) at a point that is equidistant from the vertices of the triangle. The point where they meet is called a circumcenter. Converse: If a point is equidistant from the endpoints of a segment then that point is found on the perpendicular bisector of the segment. PERPENDICULAR BISECTORS: PERPENDICULAR BISECTOR THEOREM: Angle bisector: Line/ray that divides an angle into two congruent sides. Angle Bisector Theorem: If a point is on the bisector of an angle then it is equidistant from the triangles sides. Converse: If a point on the inside of a triangle is equidistant from both sides then it is the angle bisector. Concurrent: when three or more lines intersect they are called concurrent. The point of concurrency is the exact place where they meet. The point where three perpendicular bisector of a triangle meet is called the circumcenter. Concurrency of perpendicular bisectors theorem: perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. A triangle’s angle bisectors are also concurrent, the point where they meet is called the incenter of the triangle. Incenter theorem: the incenter of the triangle is equally distant from the three sides. Median: segment that has endpoints at one vertex and the midpoint of the side opposite to it. Centroid: point where medians of a triangle concur. O CONCURRENCY OF MEDIANS OF A TRIANGLE THEOREM: 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. Point where altitudes meet is called the orthocenter. Altitude of a triangle: perpendicular segment from a vertex to the line on the opposite side. Can be on the outside or inside of a triangle. Concurrency of altitudes of a triangle theorem: Lines containing the altitudes are concurrent. Midsegment: Segment that joins the midpoints of two sides of a triangle, when the three midsegments of a triangle join, they form the midsegment triangle. Triangle midsegment theorem: midsegment of a triangle is parallel to a side of the triangle and half the length of that side. _____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples. O An exterior angle of a triangle is greater than either of the non-adjacent interior angles <BAD is the exterior angle <BAD is >(greater than) <BCA <BAD is > (greater than) <CBA _____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples. O For three lines to form a triangle the sum of two of them must be greater than the measure of the remaining one. Writing Indirect Proofs 1. Identify what must be proven. 2. Assume the opposite (negation) of the conjecture. 3. Use direct reasoning to show that negation is contradicted. 4. Since negation has been proven wrong the original conjecture from step one must be true. Given: m<P>m<R Prove: QR>QP Assume QR is less than or equal to QP. (negation) 2. If QR is less than QP then m<P is less than <m<R because the larger angle is opposite to the longer side. This means QR is greater than or equal to QP. 1. _____(0-10 pts.) Describe the hinge theorem and its converse. Give at least 3 examples. O if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle en.wikipedia.org/wiki/Hinge_theorem _____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples. A2 + B2 > C2 the triangle is acute A2 + B2 = C2 the triangle is right A2 + B2 < C2 the triangle is obtuse