Chapter 5-6 Flashcards
if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
perpendicular bisector theorem
if a bisector is also perpendicular to the segment it is called this
perpendicular bisector
if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment
converse of the perpendicular bisector theorem
when three or more lines intersect at a common point, the lines are called this
concurrent lines
the point where concurrent lined intersect
point of concurrency
the point of concurrenc of the perpendicular bisectors
circumcenter
the perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle
circumcenter theorem
if a point on the bisector of an angle, then it is equidistant from the sis of the angle
angle bisector theorem
if a point n the interior of an angle is equidistant from the sides of the angle, then it is on h bisector of the angle
converse of the angle bisector theorem
the angle bisectors of a triangle are concurrent, and their point of concurrency is called this
incenter
a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side
median
the point of concurrency of the medians of a triangle and is always inside the triangle
centroid
the medians of triangle intersect at a point called this that is 2/3 of the distance from each vertex to the midpoint of the opposite side
centroid theorem
a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side
altitude
the lines containing the altitude of a triangle are concurrent, intersecting at a point called this
orthocenter
for an real numbers a and b, a>b if and only if there is a positive number c such that a=b+c
definition of inequality
ab
comparison property of inequality
- if a<b>b and b>c, then a>c</b>
transitive property of inequality
- if a>b, then a+c>b+c
2. if a<b+c
addition property of inequality
- if a>b, then a-c>b-c
2. if a<b-c
subtraction property of inequality
the measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles
exterior angle inequality
if one side of a triangle is longer than another side, the the angle opposite the longer side has a greater measure than the angle opposite the shorter side
-same with angles of a triangle
angle-side relationships in triangles
assuming that a conclusion was false and then showing that this assumption led to a contradiction
indirect reasoning
a proof in which you temporarily assume that what you are trying to prove is false
indirect proof or proof by contradiction
the sum of the lengths of any two sides of a triangle must be greater than the length of the third side
triangle inequality theorem
if two sides of a 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 triangle, then the third side of the first triangle is longer than the third side of he second triangle
hinge theorem
if two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the second triangle, then the included angle measure of the first triangle is greater than the included angle measure in the second triangle
converse of the hinge theorem
a segment tht connects any two nonconsecutive vertices
diagonal
the sum of the interior angle measures of an n-sided convex polygon is (n-2)*180
polygon interior angles sum
the sum of the exterior angle measures of a convex polygon, one angle at each vertex, is 360
polygon exterior angles sum
a quadrilateral with both pairs of opposite sides parallel
parallelogram
- if a quadrilateral is a parallelogram, then its diagonals bisect each other
- if a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles
diagonals of parallelograms
a parallelogram with four right angles
rectangle
if a parallelogram is a rectangle, then its diagonals are congruent
if diagonals of a parallelogram are congruent, then the parallelogram is a rectangle
diagonals of a rectangle
a parallelogram with all four sides congruent
rhombus
if a parallelogram is a rhombus, then its diagonals are perpendicular
if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles
diagonals of a rhombus
a parallelogram with four congruent sides and four right angles
square
a quadrilateral with exactly one pair of parallel sides
trapezoid
the parallel sides of a trapezoid
bases
the nonparallel sides in a trapezoid
legs
angles formed by the base and one of the legs
base angles
if the legs of a trapezoid are congruent, then it is this
isosceles trapezoid
- if a trapezoid is isosceles, then each pair of base angles is congruent
- if a trapezoid has one pair of congruent base angles then it is an isosceles trapezoid
- a trapezoid is isosceles if and only if its diagonals are congruent
isosceles trapezoid
the segment that connects the midpoints of the legs of the trapezoid
mid-segment of a trapezoid
the mid-segment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases
trapezoid mid-segment theorem
a quadrilateral with exactly two pairs of consecutive congruent sides
kite
- if a quadrilateral is a kite, then its diagonals are perpendicular
- if a quadrilateral is a kite then exactly one pair of opposite angles is congruent
kites
