equidistant
referring to the fact that the distance between two or more points is equal
locus
a set of points whose location is determined by a specific set of conditions
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the segment.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the segment.
Angle Bisector Theorem
If a point is on an angle bisector, it is equidistant from each side of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from both sides of the angle, then the point lies on the bisector of the angle.
circumcenter of a triangle
the point of intersection of the perpendicular bisectors of a triangle
circumscribed circle
a circle that contains a polygon so that it passes through each vertex of the polygon
incenter of a triangle
the point of intersection of the angle bisectors of a triangle
inscribed circle
a circle that is contained within a polygon so that the circle intersects each side of the polygon at exactly one point
Circumcenter Theorem
The circumcenter of a triangle is equidistant from each vertex of the triangle.
Incenter Theorem
The incenter of a triangle is equidistant from each side of a triangle.
altitude of a triangle
a segment that extends from the vertex of a triangle to the opposite side and is perpendicular to the side
centroid of a triangle
the point of intersection of the medians of a triangle
median of a triangle
a segment that extends from a vertex of the triangle to the midpoint of the opposite side
orthocenter of a triangle
the point of intersection of all three altitudes of a triangle
Centroid Theorem
The centroid of a triangle is located 2/3 of the distance between the vertex and the midpoint of the opposite side of the triangle along each median.
Midsegment of a triangle
a segment that extends from the midpoint of one side of a triangle to the midpoint of another side of the triangle.
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to one side of the triangle, and its length is exactly half the length of the side to which it is parallel.
indirect proof
an argument that begins with the assumption that a conclusion is false, then uses logical reasoning to show that the assumption leads to a contradiction
inequality
a statement that compares two expressions that are not equal
Theorem 5.5A
If one side of a triangle is longer than another side, then the measure of the angle opposite the longer side will be greater than the measure of the angle opposite the shorter side.
Theorem 5.5B
If the measure of one angle of a triangle is greater than the measure of another angle, then the side opposite the angle with the greater measure will be longer than the side opposite the angle with the lesser measure.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
SAS Inequality Theorem / Hinge Theorem
If two sides of one triangle are congruent to the corresponding two sides of a second triangle, but the included angles are not congruent, then the third side of the triangle with the larger included angle will be longer.
SSS Inequality Theorem / Converse of the Hinge Theorem
If two sides of one triangle are congruent to the corresponding two sides of a second triangle, but the third side of one triangle is longer than the third side of the second triangle, then the measure of the included angle of the congruent sides of the first triangle will be greater than the measure of the included angle of the congruent sides of the second triangle.
Pythagorean triple
a set of three nonzero positive whole numbers, a, b, and c, such that a2 b2 = c2
Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the measure of its third side, then the triangle is a right triangle.
Pythagorean Inequality Theorem
If a, b, and c represent the three sides of a triangle and if c is the longest side, and if
a2+b2 (more than) c2, then the triangle is acute, but if a2+b2 (less than) c2, then the triangle is obtuse.
45�-45�-90� Triangle Theorem
In a 45�-45�-90� triangle, each leg is congruent and the length of the hypotenuse is times the length of a leg.
30�-60�-90� Triangle Theorem
In a 30�-60�-90� triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg.