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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)