Theorems Flashcards
if two lines intersect, then they intersect in…
exactly one point
through a line and a point not in the line…
there is exactly one plane
if two lines intersect, then… (plane)
exactly one plane contains the lines
what is midpoint theorem
if M is the midpoint of line AB, then AM = 1/2 AB and MB = 1/2 AB
what is angle bisector theorem
if ray BX is the bisector of ∠ABC, then m∠ABX = 1/2 m∠ABC and m∠XBC = 1/2 m∠ABC
what is vertical angle theorem
vertical angles are congruent
if two lines are perpendicular, then they form…
congruent adjacent angles
if two lines form congruent adjacent angles, then the lines are…
perpendicular
f the exterior sides of two adjacent acute angles are perpendicular, then…
the angles are conplementary
what is congruent supplements theorem
fi two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent
what is congruent complements theorem
if two angles are complements of congruent angles (or of the same angle), then the two angles are congruent
if two parallel planes cut by a third plane, then…
the lines of intersection are parallel
what is alt int angle theorem
if two parallel lines are cut by a transversal, then alternate interior angles are congruent
what is same side int angle theorme
if two parallel lines are cut by a transversal, then same-side interior angles are supplementary
what is perpendicular transversal theorem
if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also
what is conv of alt int angle theorem
if two lines are cut by a transversal and alternate interior angles are congruent, then the liens area parallel
what is conv of same side int angle theorem
if two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel
what is conv of perpendicular transversal theorem
IN A PLANE, two lines perpendicular to the same line are parallel
through a point outside a line, there is exactly… (parallel)
one line parallel to the given line
through a point outside a line, there is exactly… (perpendicular)
one line perpendicular to the given line
two lines parallel to a third line are…
parallel to each other
the sum of the measures of the angles of a triangle is…
180
Exterior angle thm
the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles
sum of the measures of the angles of a convex polygon with n sides is…
(n-2)180
the sum of the measures of the exterior angles of any convex polygon
360
isosceles triangle theorem
if tow sides of a triangle are congruent, then the angles opposite those sides are congruent
converse of isosceles triangle theorem?
if two angles of a triangle are congruent, then the sides opposite those angles are congruent
AAS Theorem?
if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
HL theorem?
if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
perpendicular bisector thm?
if a point lies on the perpendicular bisector of a segment, then the point is equidistant form the endpoints of the segment
conv of perpendicular bisector thm?
if a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment
angle bisector thm? (regarding distance)
if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
conv of angle bisector thm (distance)
if a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle
how to prove parallelograms?
if both pairs of opp. sides of a quad are congruent, then the quad is a parallelogram
if one pair of opp. sides of a quad are both congruent and parallel, then the quad is a parallelogram
if both pairs of opp. angles of a quad are congruent, then parallelogram
if diagonals of quad bisect each other, then parallelogram
qualities of a parallelogram?
opp sides congruent
opp angles congruent
diagonals bisect each other
if two lines parallel, then all points on one line are..
equidistant form the other line
congruent transversal thm
if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
a line that contains the midpoint of one side of a triangle and is parallel to another side…
passes through the midpoint of the third side
the segment that joins the midpoints of two sides of a triangle…
is parallel to the third side
is half as long as the third side
diagonals of a rectangle are…
congruent
diagonals of a rhombus are..and…
perpendicular
bisects the angles of the rhombus
sneaky thm?
the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
how to know parallelogram is rectangle?
if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle
how to know parallelogram is rhombus?
fi two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
base angles of an isosceles trapezoid are…
congruent
the median of a trapezoid is…
is parallel to bases
has a length equal to average of the base lengths
exterior angle inequality theorem
the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle
if one side of a triangle is longer than a second side…
then the angle opposite the first side is larger than the angle opposite the second side
if one angle of a triangle is larger than a second angle, then…
the side opposite the first angle is longer than the side opposite the second angle
the triangle inequality?
the sum of the lengths of any two sides of a triangle is greater than the length of the third side
SAS Inequality Theorem
if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle 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.
SSS Inequality Theorem
if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second
SAS Sim. thm
if an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar
SSS Sim Thm
if the sides of two triangles are in proportion, then the triangles are similar
triangle proportionality thm
if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally
triangle angle bisector thm
if a ray bisects an angle of a triangle, then it divides the opp. side into segments proportional to the other two sides
if the altitude is drawn to the hypotenuse of a right triangle, then…
the two triangles formed are similar to the original triangle and to each other
pythagorean thm
in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs
conv of pythag thm
if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle
if the square of the longest sides of a triangle is less than the sum of the squares of the other two sides, then…
triangle is acute
if square of longest side of a triangle is greater than the sum of the squares of the other two sides…
the triangle is obtuse
45-45-90 thm
if a 45-45-90 tirnalge the hypotenuse is sqrt2 times as long as a leg
30-60-90 thm
in a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is sqrt3 times as long as the shorter leg
tangent to circle then the radius…?
if a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency
radius perpendicular to line then?
if a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle
in the same circle or incongruent circles, two minor arcs are congruent if and only if…
their central angles are congruent
in the same circle or in congruent circles… (arcs and chords)
congruent arcs have congruent chords
congruent chords have congruent arcs
a diameter that is perpendicular to a chord
bisects the chord and its arc
in the same circle or in congruent circles (chords and centers)
chords equally distant form the center (or centers) are congruent
congruent chords are equally distant form the center (or centers)
inscribed angles
the measure of an inscribed angle is equal to half the measure of its intercepted arc
the measure of an angle formed by a chord and a tangent is equal to
half the measure of the intercepted arc
measure of angles formed inside and outside circle (all variations)
the measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arc
the measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs
power of a point (all variations)
when two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord
when two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment
when a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segments is equal to the square of the tangent segment
