Presentation on theme: "Nina Dorion 9-5. Perpendicular bisector: line that cuts a segment into two equal parts which measure 90. Converse: if a point is equidistant from the."— Presentation transcript:
Nina Dorion 9-5
Perpendicular bisector: line that cuts a segment into two equal parts which measure 90. Converse: if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Examples: 1) AB is the perpendicular bisector of CD 2) FE is perpendicular bisector of IH 3) MN is perpendicular bisector of lk a b cd f e hi M N LK
Angle bisector: divides an angle into two equal parts The angle bisector in this example is AB which bisects
Three or more lines are concurrent if they intersect a single point. Concurrency of Perpendicular Bisectors of a Triangle Theorem: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle Circumcenter: the center of a circle that circumscribes a triangle.
Concurrency of Angle Bisectors of a Triangle Theorem: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The incenter of a triangle is equidistant from all sides of the triangle.
A median is a line joining the vertex to the midpoint of the opposite side. median The centroid is the point of congruency of the three medians of the triangle. Acute triangle right triangle obtuse triangle 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
Altitude of a triangle is where the lines of a triangle intersect The orthocenter is the intersection of three altitudes of a triangle. Concurrency of Altitudes of a Triangle Theorem: The lines containing the altitudes of a triangle are concurrent
a midsegment is parallel to the third part of a triangle and is half the length of the third side midsegment Midsegment Theorem : The segment joining the midpoints of two sides of a triangle is parallel to the third side, and is half the length of the third side 5 10
The relationship between the sides and the angles are that, the opposite side of the shortest angle will be the shortest side, and the opposite side of the largest angle will be the largest side. In this example we can see that segment BC is the smallest because the measure of angle A is the smallest A B C shortest longest shortest
Exterior angle inequality: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
Triangle inequality theorem: Any side of a triangle is always shorter than the sum of the other two sides. C B A a + b > c b + c > a a + c > b Could a triangle have sides of 4, 8, 2? And can you make a triangle with 5,6,7?
Indirect proof: to write an indirect proof you have to assume that the given is true, then stop when the given contradicts itself. Examples: Statement reason
Statement Assume angle A ans
Hinge theorem: 2 triangles have 2 sides that are congruent, but the 3 sides is not congruent, then the triangle with the larger included angle has the longer side. F E H G 82 EF < GF By the hinge theorem