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Definitions
- Midpoint: If M is the midpoint of AB, then AM = MB
- Complementary angles: If two angles’ sum is 90,
then they are complementary.
- Supplementary angles: If two angles’ sum is 180, then they are
supplementary.
- Linear pair of angles: If two adjacent angles’ non-common sides
form opposite rays, then they are a linear pair of angles.
- Opposite rays: If two rays that share a common endpoint
form a 180 angle, then those rays are opposite rays.
- Perpendicular lines: If two lines are perpendicular, then they
form four right angles.
- Angle bisector: If a ray contains the vertex of a given
angle and divides the angle into two equal angles, then that ray
is an angle bisector.
- Congruent: If two angles are congruent, then they are
equal.
- Isosceles triangle: A triangle with at least two congruent
sides.
- Median: A line segment from a vertex perpendicular to the opposite side.
- Perpendicular bisector: A line perpendicular to the midpoint of a
segment.
- Altitude: line segment from a vertex perpendicular to the opposite side.
- Parallelogram: a quadrilateral with two pairs of
parallel sides.
- Rectangle: a quadrilateral with four right angles.
- Rhombus: a quadrilateral with four congruent sides.
- Square: a quadrilateral with four right angles and four congruent sides.
- Trapezoid: a quadrilateral with exactly one pair of
parallel sides.
- Isosceles trapezoid: a trapezoid with legs of equal length.
- Median of a trapezoid: a segment connecting the midpoints of the
legs of a trapezoid.
- Statement: If p, then q.
- Inverse: If not p, then not q.
Contrapositive: If not q, then not p.
- Converse: If q, then p.
- Minor arc: the degree measure of a minor arc is
congruent to the central arc.
- Inscribed angle:
An angle whose vertex lies on the circle edge and whose sides
form chords.
- Regular polygon:
A polygon whose sides and angles are congruent.
- Apothem:
distance from the center of a polygon to one of its sides.
- Radius:
distance from the center to the vertex of a polygon.
- Prism:
3-D shape where all alteral sides called the faces are
paralellgram
- Right
prism: faces are perpendicular to the bases
- Oblique
prism: faces
not perpendicular to bases.
- Cylinder:
Polyhedron where the two bases are
congruent circles
- Pyramaids:
A polyhedron containing one base and whose faces are triangles
converging on a common vertex
Theorems
- If there are two lines, then they intersect
in exactly one point.
- If there is one line and a point not on the
line, then they exist in exactly one plane.
- If two lines intersect, then they are
contained in exactly one plane.
- Midpoint theorem: If M is the midpoint of AB, then ĀM=½
ĀB and BM=½AB.
- Angle Bisector theorem: If BX bisects <ABC, then m <ABX=½m<ABC
and m<BC=½ <ABC.
- If vertical angles, then they are
congruent.
- If two angles are a linear pair, then they
are congruent.
- If right angles, then congruent.
- If two lines are perpendicular, then they
form congruent adjacent angles.
- If two lines form congruent angles, then
the lines are perpendicular.
- If the exterior sides of two adjacent acute
angles are perpendicular, then the angles are complementary.
- If two angles are compliments to the same,
or equal, angles, then they are congruent.
- If two angles are supplements to the same,
or equal, angles, then they are congruent.
- If two lines are parallel, then alternate
interior angles are congruent. (Z)
- If alternate interior angles are congruent,
then the lines are parallel.
- If two lines are parallel, then same side
interior angles are supplementary. (U)
- If same side interior angles are
supplementary, then the lines are parallel.
- If a line is perpendicular to one of two
perpendicular lines, then it is perpendicular to the other.
- If two lines are perpendicular to the same
line, then they are parallel to each other.
- If a point is not on a line, then there is
exactly one line through the point parallel to the other line.
- If a point is not on a line, then there is
exactly one line through the point perpendicular to the other
line.
- Triangle sum theorem: If there is a triangle, then the sum of
its angles is 180.
- If a triangle, then the measure of en
exterior angle is equal to the sum of two remote angles.
- If a convex polygon, then the sum of the
measures of the interior angles is (x-2) 180.
- Exterior angle sum theorem: If a convex polygon, then the sum of the
exterior angles is equal to 360.
- Side Angle Angle congruency theorem: If two angles and a side of a triangle are
congruent to two angle and a side of another triangle, then the
triangles are congruent
- Hypotenuse Leg theorem: If a hypotenuse and a leg of one triangle
are congruent to another hypotenuse and a leg of another
triangle, then the triangles are congruent.
- Two sides of a triangle are congruent, if
and only if, the angles opposite these sides are congruent.
- A point is on a perpendicular bisector, if
and only if, it is equidistant from the endpoints of the line
segment.
- A point is on an angle bisector, if and
only if, it is equidistant from the two sides of the angle.
- If a quadrilateral is a parallelogram, then
the opposite angles are congruent.
- If a quadrilateral is a parallelogram, then
the opposite sides are congruent.
- If a quadrilateral is a parallelogram, then
the diagonals bisect each other.
- If opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
- If the opposite sides of a quadrilateral
are congruent, then the quadrilateral is a parallelogram.
- If the diagonals of a quadrilateral bisect
each other, than the quadrilateral is a parallelogram.
- If one pair of opposite sides of a
quadrilateral is parallel and congruent, then the quadrilateral
is a parallelogram.
- If three parallel lines cut off congruent
segments of one transversal, then the lines cut off congruent
segments of every transversal.
- If a segment connects two sides of a
triangle, then it is parallel and congruent to the third side.
- If a quadrilateral is a rectangle, then the
diagonals are congruent.
- If a
quadrilateral is a rhombus, then the diagonals are
perpendicular.
- If a
quadrilateral is a rhombus, then the diagonals bisect opposite
angles.
- If you have a
right triangle, then the midpoint of the hypotenuse is
equidistant from the three vertices
- If a
parallelogram has one right angle, then it is a rectangle.
- If a
parallelogram had two congruent sides, then it is a rhombus.
- If a trapezoid
is isosceles, then each pair of base angles is congruent.
- If a segment is
a median of a trapezoid, then it is parallel to the bases and
has a length congruent to the 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 then the side opposite the third side.
- If two sides are
added, then the sides are greater than the length of the third
side.
- SSS Inequality: If two sides
of a triangle are equal to two sides of another triangle, but
the third side of the first triangle is less then the third side
of the second triangle, then the opposite angle of the third
side of the first triangle is less then the opposite angle of
the of the third side of the second triangle.
- SAS Inequality: If two sides
of a triangle are equal to two sides of another triangle, and
the included angle of the first triangle is less then the angle
of the second angle, then the third side of the first triangle
is less then the third side of the second side.
- SAS Similarity: If an angle of a triangle is congruent to
an angle of another triangle and the sides including those
angles are in proportion, then the triangles are similar.
- SSS Similarity: If the sides of two triangles are in
proportion, then the triangles are similar.
- If a line
parallel to one side of a triangle intersects the other two
sides, then it divides these sides proportionally.
56a.If three are more parallel lines intersect two or more transversals,
then they will the divide the transversals proportionally.
57.If
a ray bisects an angle of a triangle, then it divides the opposite
side into segments in proportion two the other two sides.
58.If
an altitude is drawn to the hypotenuse of a right triangle, then the
triangles formed are similar to the original triangle and to each
other.
58a.
And the legs are the geometric means of the hypotenuse and
altitude is the geometric leg of the hypotenuse.
59.Pythagorean
theorem:
The sum of the legs equals the square of the hypotenuse, if and only
if, it is a right triangle.
60.In a
45-45-90 right triangle, the hypotenuse is √2 times as long as
a leg.
61.In a
30-60-90 right triangle, the relationship between the lengths is
a-a√3-2a.
62.A line
is perpendicular to a radius of a circle on a point on the circle,
if and only if, the line is a tangent.
63.If
there is a point outside a circle and tangents go through that
circle from the point, then the tangents are congruent.
64.In a
circle or congruent circles, if two central angles are congruent,
then the arcs are equal.
65.In a
circle, if chords are congruent, then intercepted arcs are
congruent.
66.In a
circle, if a diameter is perpendicular to a chord, then it bisects
the chord and its intercepted arcs.
67.Two
chords are congruent, if and only if, the two chords are equidistant
from the center.
68.If an
angle is inscribed in a circle, then its measure is half of its
intercepted arc.
69.If a
quadrilateral is inscribed in a circle, then opposite angles are
supplementary.
70.If a
triangle is inscribed inside a semicircle, then it is a right
triangle.
71.If
inscribed angles intercept the same are congruent arcs, then they
are congruent to each other.
72.If two
chords intersect in a circle, then the angles formed are half the
sum of the intercepted arcs.
73.If two
tangents, a tangent and a secant, or two secants to a triangle start
form the same exterior point, then the external angles are half the
difference of intercepted arcs.
74.If two
chards intersect, then the product of two segments of one chord
equals the product of two segments of another chord.
75.If two
secants are drawn to a circle from the same exterior point, then the
product of the exterior secant segment of one secant is equal to the
product of the exterior secant segment and the entire segment of
another secant.
76.If a
secant and a tangent are drawn to a circle from the same exterior
point, then the product of the exterior secant segment and the
entire secant segment is equal to the square of the tangent segment.
77.Area
of a Rectangle= b*h
78.Area
of a Triangle= ½ b*h
79.Area
of a paralellagram= b*h
80.Area
of a rhombus= ½ the product of the diagonals
81.Area
of a regular polygon=
½ apothem * perimeter
82.Lateral
Area of Prism:
p of base times height
83.Total
Area of Prism:
Lateral Area + 2Base
84.Volume
of Prism:
Base * Height
85.Lateral
Area of Cylinder:
Height * 2пr
86.Total
Area of Cylinder: 2пr*H
+ 2пr²
87.Volume
of a Cylinder: h* пr²
88.Lateral
Area of a pyramid:
one half perimeter times slant height
89.Total
Area of a Pyramid:
LA +B
90.Volume
of a Pyramid:
one third base times height
91.Lateral
Area of a cone:
пrl
92.Total
area of a cone:
пr²+пrl
93.Volume
of a cone:
one third пr² times height
94.Surface
Area of a Sphere:
4пr²
95.Volume
of a Sphere:
four thirds пr³
96.If two
solids are similar, then the ratio of the corresponding sides equals
the ratio of the perimeters; the ratio of the area equals the square
of the ratio of corresponding sides, and the ratio of the volume of
the ratio of the corresponding side.
97.
The
Distance formula:
√ (x-x)² + (y-y)²
Postulates
- Ruler Postulate: Two angles have a positive distance.
- Segment Addition Postulate: If B is between A and C then AB + BC
= AC
- Protractor Postulate: All angles have a measurement of 0 –
180.
- Angle Addition Postulate: Two angles that are adjacent can be added
if they are not over 180 and do not overlap.
- Two points make a line.
- Three non-collinear points make a plane.
- If two points are in a plane, then the line
that contains the point is in that plane.
- Two planes intersect in a line.
- If two lines are parallel, then
corresponding angles are congruent. (F)
- If corresponding angles are congruent, then
lines are parallel. (F)
- Side Side Side Congruency Postulate: If three sides of a triangle are congruent
to three sides of another triangle, then the triangles are
congruent.
- Angle Side Angle Congruency Postulate: If two sides and the included angle of a
triangle are congruent to two sides and the included angle of
another triangle, then the triangles are congruent.
- Angle Side Angle Congruency Postulate: If two angles and the included side of a
triangle are congruent to two angles and the included side of
another triangle, then the triangles are congruent.
- AA similarity Postulate: If two or more angles of a triangle are
equal to two angles of another triangle, then the triangles are
congruent.
- Arc addition postulate: If consecutive arcs, then they can be
added.
- The area of a square= side of the square ²
- If two shapes are congruent, then areas are
equal.
- Area addition postulate: two non-overlapping areas can be added.
- Area of circle: пr²
- Circumference of a circle: 2пr
- Ratio of area of a sector: area of sector/ area of circle equals
central angle of a sector/ central angle of a circle
- Ratio of length of a sector: length of a sector/ circumference equals
central angle/ 360
- Geometric probability:
Desired outcome/ total outcome
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