Download Art`s Geometry Notes

Art’s Geometry Notes
http://www.adamford.com/math
September 17, 2010
(This is a work in progress and may have errors and omission.)
Misc..................................................................................................................................... 6
Symbol: ≅ , Congruent................................................................................................. 6
Points Def: Collinear points............................................................................................ 6
Point Postulate: Ruler Postulate...................................................................................... 6
Ray Def: .......................................................................................................................... 6
Ray Def: Opposite rays................................................................................................... 6
Congruence ......................................................................................................................... 7
Congruence: Def ............................................................................................................. 7
Congruence Postulate: Reflexive Property: .................................................................... 7
Congruence Postulate: Symmetric Property: .................................................................. 7
Congruence Postulate: Transitive Property: ................................................................... 7
Congruence Postulate: Substitution Property: ............................................................... 7
Congruence Postulate: Addition Property of Equality: ................................................. 7
Inequality ............................................................................................................................ 8
Inequality Postulate: Addition ........................................................................................ 8
Inequality Postulate: Multiplication by positive ............................................................. 8
Inequality Postulate: Multiplication by negative ............................................................ 8
Inequality Postulate: Transitivity.................................................................................... 8
Inequality Postulate: Subtraction .................................................................................... 8
Logic ................................................................................................................................... 9
Logic def: If-then statements .......................................................................................... 9
Logic def: Converse........................................................................................................ 9
Logic def: Inverse ........................................................................................................... 9
Logic def: Contrapositive ............................................................................................... 9
Segments ........................................................................................................................... 10
Segment Def: ................................................................................................................ 10
Segment Def: Median ................................................................................................... 10
Segment Def: Midpoint................................................................................................. 10
Segment Def: Perpendicular bisector............................................................................ 10
Segment Postulate: Segment Addition Postulate.......................................................... 10
Segment Postulate: Midpoint Postulate ........................................................................ 10
Segment Postulate: Segment Midpoint Postulate ......................................................... 10
Segment Postulate: Congruent Line Segments ............................................................. 10
Segment Postulate: Segment Midpoint Postulate ......................................................... 10
Segment Postulate: Segment Addition Postulate.......................................................... 11
Lines.................................................................................................................................. 12
Line Def: Parallel lines ................................................................................................. 12
Line Def: Perpendicular lines ....................................................................................... 12
Line Def: Skew lines..................................................................................................... 12
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Line Def: Transversal line ............................................................................................ 12
Line Postulate: Minimum number of points ................................................................. 12
Line Postulate: Two point Postulate: ............................................................................ 12
Line Postulate: Line Postulate: ..................................................................................... 12
Line Postulate: Line/Plane Postulate: ........................................................................... 12
Line Postulate: Line Intersection Postulate: ................................................................. 13
Line Postulate: Parallel Line Postulate ......................................................................... 13
Line Postulate: Converse Parallel Line Postulate ......................................................... 13
Line Postulate: Parallel Line Postulate (Alternate)....................................................... 13
Line Postulate: Perpendicular Line Postulate ............................................................... 13
Line Theorem: Parallel Lines Property......................................................................... 13
Line Theorem: Converse with Perpendicular Transversals .......................................... 13
Line Theorem: Parallels are equidistant ....................................................................... 14
Line Theorem: Parallels and congruent transversal segments...................................... 14
Line Theorem: Parallel bisecting one side bisects the other......................................... 14
Line Theorem: Segment joining midpoints is parallel to third side and half as long ... 14
Angles ............................................................................................................................... 15
Angle Def: the angle that intercepts, the segment that subtends .................................. 15
Angle Def: Acute angle ................................................................................................ 15
Angle Def: Obtuse angle............................................................................................... 15
Angle Def: Right angle ................................................................................................. 15
Angls Def: Congruent angles........................................................................................ 15
Angle Def: Straight angle ............................................................................................. 15
Angls Def: Complementary angles............................................................................... 15
Angle Def: Supplementary angles ................................................................................ 16
Angle Def: Vertical angles............................................................................................ 16
Angle Def: Corresponding, Adjacent, Vertical, Alternate, Consecutive, Same-side ... 16
Angle Def: Linear Pair.................................................................................................. 17
Angle Def: Adjacent angles .......................................................................................... 17
Angle Def: Alternate exterior angles ............................................................................ 17
Angle Def: Alternate interior angles............................................................................. 17
Angle Theorem: Vertical Angles Theorem................................................................... 18
Angle Postulate: Linear Pair Postulate: ........................................................................ 18
Angle Postulate: Protractor Postulate ........................................................................... 18
Angle Postulate: Protractor Postulate: .......................................................................... 18
Angle Postulate: Angle Addition Postulate .................................................................. 18
Angle Postulate: Angle Bisector Postulate ................................................................... 18
Angle Postulate: Linear Pair Postulate: ........................................................................ 18
Angle Postulate: Corresponding Angles Postulate ....................................................... 19
Angle Postulate: Converse of Corresponding Angles Postulate.................................. 19
Angle Postulate: Converse of Interior Angles Postulate.............................................. 19
Angle Postulate: Angle Addition Postulate .................................................................. 19
Angle Postulate: Angle Bisector Postulate ................................................................... 19
Angle Postulate: Bisector Postulate .............................................................................. 19
Angle Theorem: Right Angle........................................................................................ 19
Angle Theorem: Supplements of the Same Angle........................................................ 20
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Angle Theorem: Complements of the Same Angle ...................................................... 20
Angle Theorem: Alternate Interior Angles Theorem.................................................... 20
Angle Theorem: Alternate Exterior Angles Theorem .................................................. 20
Angle Theorem: Consecutive Interior Angles Theorem............................................... 20
Angle Theorem: Converse of Alternate Interior Angles Theorem ............................... 20
Angle Theorem: Converse of the Alternate Exterior Angles Theorem ........................ 20
Angle Theorem: Converse of Alternate Interior Angles............................................... 20
Angle Theorem: Converse of the Alternate Exterior Angles........................................ 20
Angle Theorem: Converse of Consecutive Interior Angles.......................................... 20
Polygons............................................................................................................................ 22
Polygon Def: ................................................................................................................. 22
Polygon Def: Convex polygon ..................................................................................... 22
Polygon Def: Types, number of sides........................................................................... 22
Polygon Def: Diagonal ................................................................................................. 22
Polygon Def: Regular ................................................................................................... 22
Polygon Def: Polygonal region..................................................................................... 22
Polygon Theorem: Sum of Interior Angles formula .................................................... 22
Polygon Theorem: Sum of Exterior Angles.................................................................. 23
Polygon Def: Similar polygons..................................................................................... 23
Triangles ........................................................................................................................... 24
Triangle Def:................................................................................................................. 24
Triangle Def: Vertex angle ........................................................................................... 24
Triangle Def: Altitude................................................................................................... 24
Triangle Def: Median.................................................................................................... 24
Triangle Def: Scalene triangle ...................................................................................... 24
Triangle Def: Isosceles triangle .................................................................................... 24
Triangle Def: Equilateral triangle ................................................................................. 24
Triangle Def: Acute triangle ......................................................................................... 24
Triangle Def: Obtuse triangle ....................................................................................... 25
Triangle Def: Right triangle.......................................................................................... 25
Triangle Def: Equiangular triangle .............................................................................. 25
Triangle Def: Interior and exterior angles .................................................................... 25
Triangle Postulate: Side-Side-Side (SSS)..................................................................... 26
Triangle Postulate: Angle-Side-Angle (ASA) .............................................................. 26
Triangle Postulate: Side-Angle-Side (SAS).................................................................. 26
Triangle Postulate: Note: SSA is NOT a postulate. (May fail)..................................... 26
Triangle Theorem: Triangle Sum.................................................................................. 26
Triangle Theorems: Exterior Angles in a Triangle ....................................................... 26
Triangle Theorem: Third Angle.................................................................................... 26
Triangle Theorem: Angle-Angle-Side (AAS) Congruence .......................................... 26
Triangle Theorem: HL Congruence.............................................................................. 26
Triangle Theorem: Base Angles ................................................................................... 27
Triangle Theorem: Converse of Base Angles............................................................... 27
Triangle Theorem: Midsegment Theorem .................................................................... 27
Triangle Theorem: Perpendicular Bisector Theorem: .................................................. 27
Triangle Theorem: Converse of the Perpendicular Bisector Theorem ......................... 27
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Triangle Theorem: Concurrency of Perpendicular Bisectors ....................................... 27
Triangle Theorem: Angle Bisector Theorem................................................................ 27
Triangle Theorem: Converse of the Angle Bisector Theorem...................................... 27
Triangle Theorem: Concurrency of Angle Bisectors Theorem .................................... 27
Triangle Theorem: Angle opposite to longer side is larger .......................................... 27
Triangle Theorem: Side opposite to larger angle is longer........................................... 27
Triangle Theorem: Triangle inequality......................................................................... 28
Triangle Theorem: SAS inequality ............................................................................... 28
Triangle Theorem: SAS inequality ............................................................................... 28
Triangle postulate: AA similarity ................................................................................. 28
Triangle Theorem: SAS similarity................................................................................ 28
Triangle Theorem: SSS similarity ................................................................................ 28
Triangle Theorem: Parallel proportionality .................................................................. 28
Triangle Theorem: Angle bisector ................................................................................ 28
Quadrilaterals.................................................................................................................... 29
Quadrilateral Def: ......................................................................................................... 29
Quadrangle Def:............................................................................................................ 29
Quadrilateral Def: Parallelogram.................................................................................. 29
Quadrilateral Def: Median of a Parallelogram.............................................................. 29
Parallelogram Theorem: Opposite sides are congruent ................................................ 29
Parallelogram Theorem: Opposite angles are congruent .............................................. 29
Parallelogram Theorem: Diagonals .............................................................................. 29
Quadrilateral Theorem: Congruent pairs of opposite sides .......................................... 29
Quadrilateral Theorem: Congruent and parallel opposite sides.................................... 29
Quadrilateral Theorem: Congruent pairs of opposite angles ........................................ 29
Quadrilateral Theorem: Bisecting diagonals ................................................................ 29
Quadrilateral Def: Rectangle ........................................................................................ 29
Rectangle Theorem: Diagonals are congruent.............................................................. 30
Quadrilateral Def: Rhombus ........................................................................................ 30
Rhombus Theorem: Diagonals are perpendicular......................................................... 30
Rhombus Theorem: Diagonals bisect angles................................................................ 30
Quadrilateral Def: Trapezoid ........................................................................................ 30
Trapezoid Theorem: Congruent base angles................................................................. 30
Trapezoid Def: Median ................................................................................................. 30
Trapezoid Theorem: Median is parallel and has average length .................................. 30
Circles ............................................................................................................................... 31
Circle Def: Radius, diameter, circumference................................................................ 31
Circle Def: Congruent circles ....................................................................................... 31
Circle Def: Chord, secant, tangent, point of tangency.................................................. 31
Circle Def: Sector ......................................................................................................... 31
Circle Def: Central angle .............................................................................................. 31
Circle Def: Intercepts and subtends .............................................................................. 31
Circle Postulate: Arc addition....................................................................................... 31
Circle Def: Minor arc.................................................................................................... 31
Circle Def: Major arc .................................................................................................... 32
Circle, Def: Measure of an arc...................................................................................... 32
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Circle Theorem: Tangent to a circle ............................................................................. 32
Circle Theorem: Tangent to a circle converse .............................................................. 32
Circle Theorem: Tangent Segments from a Point......................................................... 32
Circle Def: circumscribed and inscribed polygon ........................................................ 32
Circle Theorem: Three points determine a circle ......................................................... 32
Circle Def: Inscribed angle ........................................................................................... 32
Circle Theorem: Inscribed angle, intercepted arc ......................................................... 32
Solids................................................................................................................................. 33
Solid Def: Parallelpiped................................................................................................ 33
Solid Def: Opposite faces ............................................................................................. 33
Solid Def: Polyhedron................................................................................................... 33
Solid Def: Right cylinder .............................................................................................. 33
Solid Def: Pyramid ....................................................................................................... 33
Solid Def: Right cone.................................................................................................... 33
Solid Def: Right prism .................................................................................................. 33
Solid Def: Slant height.................................................................................................. 33
Solid Def: Small circle.................................................................................................. 33
Solid Def: S.A............................................................................................................... 33
Start 344 ........................................................................................................................ 34
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Misc
Symbol:
≅ , Congruent
Points Def: Collinear points
Points are said to be collinear if they lie along the same line.
Point Postulate: Ruler Postulate
Any two distinct points on a line can be assigned 0 and 1. Once assigned, the
distance between any two points equals the absolute value of the difference
between their coordinates.
Ray Def:
A ray is a part of a line with exactly one endpoint that extends infinitely in one
direction. Rays are named by their endpoint and a point on the ray.
Ray Def: Opposite rays
Two rays with a common endpoint that form a line
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Congruence
Congruence: Def
Congruent geometric objects have the same size and shape.
Two segments are congruent if they have the same length.
We may use tic marks to show that two segments are congruent.
Or to show congruent angles.
Congruence Postulate: Reflexive Property:
of Segments: MN ≅ MN
∠P ≅ ∠P
of Angles:
Congruence Postulate: Symmetric Property:
of Segments: If MN ≅ PQ , then PQ ≅ MN
of Angles:
If ∠P ≅ ∠Q , then ∠Q ≅ ∠P
Congruence Postulate: Transitive Property:
of Segments: If MN ≅ PQ and PQ ≅ ST then MN ≅ ST
of Angles;
If ∠P ≅ ∠Q , and ∠Q ≅ ∠R then ∠P ≅ ∠R
Congruence Postulate: Substitution Property:
If a = b; then b can be put in place of a anywhere.
Example: Given that a = 9 and that a − c = 5. Then 9 − c = 5.
Congruence Postulate: Addition Property of Equality:
If a = b, then a + c = b + c.
You can add the same number to both sides of an equation
Example: If m∠A = 55o , then m∠A + 30o = 55o + 30o •
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Inequality
Inequality Postulate: Addition
If a > b and c ≥ d , then a + c > b + d
Inequality Postulate: Multiplication by positive
a b
If a > b and c > 0 , then ac > bc and >
c c
Inequality Postulate: Multiplication by negative
a b
If a > b and c < 0 , then ac < bc and <
c c
Inequality Postulate: Transitivity
If a > b and b > c , then a > c
Inequality Postulate: Subtraction
If a = b + c and c > 0 , then a > b
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Logic
Logic def: If-then statements
An if-then statement has two parts: the hypothesis and the conclusion.
The form of an if-then statement is “if (hypothesis), then (conclusion)”
Example: If you live in Ohio, then you live in the U.S.
The hypothesis is: you live in Ohio
The conclusion is: you live in the U.S.
If-then statements are symbolized as “if p, then q” (though “if h, then c” would be
easier to remember because h for hypothesis and c for conclusion.)
Logic def: Converse
The converse of an if-then statement is: “if q, then p”.
So if the original if-then statement is: “If you live in Ohio, then you live in the
U.S.” then its converse is: “If you live in the U.S., then you live in Ohio.” The
converse of a true if-then statement may not be true.
Logic def: Inverse
The inverse of an if-then statement is: “if not p, then not q”.
So if the original if-then statement is: “If you live in Ohio, then you live in the
U.S.” then its inverse is: “If you do not live in Ohio, then you do not live in the
U.S.” The inverse of a true if-then statement may not be true.
Logic def: Contrapositive
The contrapositive of an if-then statement is: “if not q, then not p”.
So if the original if-then statement is: “If you live in Ohio, then you live in the
U.S.” then its contrapositive is: “If you do not live in the U.S., then you do not
live in Ohio” The contrapositive of a true if-then statement is always true, too.
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Segments
Segment Def:
- aka line segment; the set of points consisting of two distinct points and all
inbetween them; written
Segment Def: Median
The segment connecting the vertex of an angle in a triangle to the midpoint of the
side opposite it.
Segment Def: Midpoint
The point M of
where AM = AB
Segment Def: Perpendicular bisector
The bisector of a segment perpendicular to it.
Segment Postulate: Segment Addition Postulate
The measure (length) of any line segment can be found by adding the measures of
the smaller segments that comprise it.
Segment Postulate: Midpoint Postulate
See Segment Midpoint Postulate
Segment Postulate: Segment Midpoint Postulate
Any line segment will have exactly one midpoint — no more, and no less.
Segment Postulate: Congruent Line Segments
Two segments are congruent if they have the same length.
Segment Postulate: Segment Midpoint Postulate
Any line segment will have exactly one midpoint — no more, and no less.
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Segment Postulate: Segment Addition Postulate
The measure (length) of any line segment can be found by adding the measures of
the smaller segments that comprise it.
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Lines
Line Def: Parallel lines
Parallel lines are coplanar lines that do not intersect.
Line Def: Perpendicular lines
Two segments, rays, or lines that form a 90 degree angle
Line Def: Skew lines
Non-coplanar lines that don't intersect
Line Def: Transversal line
• A line that intersects two or more coplanar lines at different points. Line l in the
diagram is a transversal
•
•
Interior - The area between lines g and h is called the interior.
Exterior - The area not between lines g and h is called the exterior
Line Postulate: Minimum number of points
A line contains at least two points.
Line Postulate: Two point Postulate:
There is exactly one line through any two (distinct) points
Line Postulate: Line Postulate:
There is exactly one line through any two (distinct) points
Line Postulate: Line/Plane Postulate:
If two points line in a plane, the line joining them lies in the plane, too.
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Line Postulate: Line Intersection Postulate:
The intersection of any two distinct lines will be a single point.
Line Postulate: Parallel Line Postulate
If two parallel lines are cut by a transversal, then corresponding angles are
congruent.
Line Postulate: Converse Parallel Line Postulate
If two lines are cut by a transversal and corresponding angles are congruent, then
the lines are parallel.
Line Postulate: Parallel Line Postulate (Alternate)
Given a line and a point not on the line, there is exactly one line parallel to the
given line that goes through that point.
Line Postulate: Perpendicular Line Postulate
Given a line and a point not on the line, there is exactly one line perpendicular to
the given line that goes through that point.
Line Theorem: Parallel Lines Property
If line l is parallel to line m, and line m is parallel to line n, then lines l and n are
also parallel.
Line Theorem: Converse with Perpendicular Transversals
If a transversal forms right angles on two different coplanar lines, those two lines
are parallel.
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Line Theorem: Parallels are equidistant
If two lines are parallel, then all points on one line are equidistant from the other
line.
Line Theorem: Parallels and congruent transversal segments
If three parallel lines cut off congruent segments on one transversal, they cut off
congruent segments on every transversal.
Line Theorem: Parallel bisecting one side bisects the other
A line containing the midpoint of a triangle’s side and parallel to another side
passes through the midpoint of the third side.
Line Theorem: Segment joining midpoints is parallel to third side and half as long
A segment joining the midpoints of two sides of a triangle is parallel to the third
side and half as long as the third side.
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Angles
Angle Def: the angle that intercepts, the segment that subtends
Segment DE subtends angle DCE. Angle DCE intercepts segment DE.
Segment AB subtends angle ACB. Angle ACB intercepts segment AB.
Angle Def: Acute angle
An angle whose measure is greater than 0 but less than 90 degrees. Also, see
obtuse angle.
Angle Def: Obtuse angle
An angle that measures greater than 90◦ and less than 180◦
Angle Def: Right angle
A right angle measures exactly 90◦.
Angls Def: Congruent angles
Two angles are congruent if they have the same measure (ex., degrees, radians).
Angle Def: Straight angle
An angle whose measure is 180 degrees, forming a line with its sides
Angls Def: Complementary angles
mnemonic: Complementary and Corner
A pair of angles whose sum of measures is 90D
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Angle Def: Supplementary angles
mnemonic: Supplementary and Straight both begin with S
Two angles whose measures, when added together, equal 180 degrees
Angle Def: Vertical angles
Two angles that share a common vertex and whose sides form 2 lines
Angle Def: Corresponding, Adjacent, Vertical, Alternate, Consecutive, Same-side
The transversal line l intersects two other lines, g and h
•
•
•
•
•
Interior - The area between lines g and h is called the interior.
Exterior - The area not between lines g and h is called the exterior
Corresponding angles – In the same position relative to both lines that the
transversal crosses. Examples: ∠ 1 and ∠ 7; ∠ 2 and ∠ 4; ∠ 8 and ∠ 6; and ∠ 3
and ∠ 5.
Adjacent angles – Share a side and do not overlap. Examples: ∠ 1 and ∠ 2; ∠ 2
and ∠ 3; ∠ 4 and ∠ 7; and ∠ 8 and ∠ 1.
Vertical angles – Nonadjacent angles made by intersection of two lines.
Examples: ∠ 1 and ∠ 3; ∠ 2 and ∠ 8; ∠ 4 and ∠ 6, and ∠ 5 and ∠ 7.
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•
•
Alternate interior angles - In the interior on opposite sides of the transversal.
Examples: ∠ 3 and ∠ 7; ∠ 8 and ∠ 4.
Alternate exterior angles - In the exterior on opposite sides of the transversal.
Examples: ∠ 1 and ∠ 5; ∠ 2 and ∠ 6.
Consecutive interior angles – In the interior and next to each other. Examples:
∠ 8 and ∠ 7; ∠ 3 and ∠ 4. Also called Same-side interior angles.
Angle Def: Linear Pair
Two supplementary adjacent angles whose non-common sides form a line
Angle Def: Adjacent angles
Two nonstraight and nonzero angles that have a common side in the interior of the
angle formed by the noncommon sides
Angle Def: Alternate exterior angles
Two exterior angles on alternate sides of the transversal (not on the same parallel
line)
Angle Def: Alternate interior angles
Two interior angles on alternate sides of the transversal (not on the same parallel
line)
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Angle Theorem: Vertical Angles Theorem
If two angles are vertical angles then they are congruent.
Angle Postulate: Linear Pair Postulate:
If two angles are a linear pair, then they are supplementary.
Angle Postulate: Protractor Postulate
For every angle there is a number between 0 and 180 that is the measure of the
angle in degrees.
Angle Postulate: Protractor Postulate:
For every angle there is a number between 0 and 180 that is the measure of the
angle in degrees.
Angle Postulate: Angle Addition Postulate
The measure of any angle can be found by adding the measures of the smaller
angles that comprise it.
Angle Postulate: Angle Bisector Postulate
Every angle has exactly one bisector.
Angle Postulate: Linear Pair Postulate:
If two angles are a linear pair, then they are supplementary.
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Angle Postulate: Corresponding Angles Postulate
If the lines crossed by a transversal are parallel, then corresponding angles will be
congruent.
Angle Postulate: Converse of Corresponding Angles Postulate
If corresponding angles are congruent when two lines are crossed by a transversal,
then the two lines crossed by the transversal are parallel.
Angle Postulate: Converse of Interior Angles Postulate
If two consecutive interior angles made by two lines and a transversal add up to
180◦; the two lines that form the consecutive angles are parallel.
Angle Postulate: Angle Addition Postulate
The measure of any angle can be found by adding the measures of the smaller
angles that comprise it.
Angle Postulate: Angle Bisector Postulate
Every angle has exactly one bisector.
Angle Postulate: Bisector Postulate
See Angle Bisector Postulate.
Angle Theorem: Right Angle
If two angles are right angles, then the angles are congruent.
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Angle Theorem: Supplements of the Same Angle
If two angles are both supplementary to the same angle (or congruent angles) then
the angles are congruent.
Angle Theorem: Complements of the Same Angle
If two angles are both complementary to the same angle (or congruent angles)
then the angles are congruent.
Angle Theorem: Alternate Interior Angles Theorem
Alternate interior angles formed by two parallel lines and a transversal will
always be congruent.
Angle Theorem: Alternate Exterior Angles Theorem
If two parallel lines are crossed by a transversal, then alternate exterior angles are
congruent.
Angle Theorem: Consecutive Interior Angles Theorem
If two parallel lines are crossed by a transversal, then consecutive interior angles
are supplementary
Angle Theorem: Converse of Alternate Interior Angles Theorem
If two lines are crossed by a transversal and alternate interior angles are
congruent, then the lines are parallel.
Angle Theorem: Converse of the Alternate Exterior Angles Theorem
If two lines are crossed by a transversal and the alternate exterior angles are
congruent, then the lines crossed by the transversal are parallel.
Angle Theorem: Converse of Alternate Interior Angles
If two lines are crossed by a transversal and alternate interior angles are
congruent, then the lines are parallel.
Angle Theorem: Converse of the Alternate Exterior Angles
If two lines are crossed by a transversal and the alternate exterior angles are
congruent, then the lines crossed by the transversal are parallel.
Angle Theorem: Converse of Consecutive Interior Angles
If two consecutive interior angles add up to 180◦; the two lines that form the
consecutive angles are parallel.
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Polygons
Polygon Def:
A union of 3 or more segments where each segment intersects 2 other segments,
one at each endpoint; "many sided"
Polygon Def: Convex polygon
A polygon where no line containing a side also contains an interior point
Polygon Def: Types, number of sides
3 sides, triangle; 4, quadrilateral; 5, pentagon; 6, hexagon
8, octagon, 10, decagon, n, n-gon
Polygon Def: Diagonal
A diagonal is a segment connecting two nonconsecutive vertices.
Polygon Def: Regular
A regular polygon is both equiangular and equilateral.
Polygon Def: Polygonal region
The union of a polygon and its interior.
Polygon Theorem: Sum of Interior Angles formula
N = number of sides. Sum of Interior Angles is ( N − 2) *180
Note: N-2 = number of triangles the polygon splits into
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Polygon Theorem: Sum of Exterior Angles
The sum of exterior angles is 360D
Polygon Def: Similar polygons
Two polygons are similar if corresponding angles are congruent and
corresponding sides are in proportion.
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Triangles
Triangle Def:
A polygon with three sides. Varieties include equilateral, isoceles, right , scalene,
obtuse.
Triangle Def: Vertex angle
The angle that is opposite the base of the triangle.
Triangle Def: Altitude
A line drawn from an angle perpendicular to the opposite side.
Triangle Def: Median
The median is a segment from a vertex to the midpoint of the opposite side.
Triangle Def: Scalene triangle
A triangle with no congruent sides
Triangle Def: Isosceles triangle
A triangle with at least two congruent sides
Triangle Def: Equilateral triangle
A triangle where all sides are congruent
Triangle Def: Acute triangle
A triangle where all angles are acute
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Triangle Def: Obtuse triangle
A triangle that contains an obtuse angle
Triangle Def: Right triangle
A triangle that contains a right angle
Triangle Def: Equiangular triangle
A triangle where all angles are congruent
Triangle Def: Interior and exterior angles
Note: exterior angles shown below are “clockwise” exterior angles. If triangle
sides were extended in other direction we’d get “counterclockwise” exterior
angles.
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Triangle Postulate: Side-Side-Side (SSS)
If three sides in one triangle are congruent to the three corresponding sides in
another triangle, then the triangles are congruent to each other.
Triangle Postulate: Angle-Side-Angle (ASA)
If two angles and the included side in one triangle are congruent to two angles and
the included side in another triangle, then the two triangles are congruent.
Triangle Postulate: Side-Angle-Side (SAS)
If two sides and the included angle in one triangle are congruent to two sides and
the included angle in another triangle, then the two triangles are congruent.
Triangle Postulate: Note: SSA is NOT a postulate. (May fail)
But HL congruence is a theorem. So SSA works if the angle is a right angle.
Triangle Theorem: Triangle Sum
The sum of the measures of the interior angles in a triangle is 180º
Triangle Theorems: Exterior Angles in a Triangle
In a triangle, the measure of an exterior angle is equal to the sum of the remote
interior angles.
Note: the theorem implies that the measure of an exterior angle is greater than the
measure of either remote interior angle.
Triangle Theorem: Third Angle
If two angles in one triangle are congruent to two angles in another triangle, then
the third pair of angles are also congruent.
Triangle Theorem: Angle-Angle-Side (AAS) Congruence
If two angles and a non-included side in one triangle are congruent to two
corresponding angles and a non-included side in another triangle, then the
triangles are congruent.
Triangle Theorem: HL Congruence
If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and
leg in another right triangle, then the two triangles are congruent.
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Triangle Theorem: Base Angles
If two sides of a triangle are congruent, then their opposite angles are also
congruent. In other words, the base angles of an isosceles triangle are congruent.
Triangle Theorem: Converse of Base Angles
If two angles in a triangle are congruent, then the sides opposite them will also be
congruent.
Triangle Theorem: Midsegment Theorem
The segment that joins the midpoints of a pair of sides of a triangle is:
1. parallel to the third side.
2. half as long as the third side.
Triangle Theorem: Perpendicular Bisector Theorem:
If a point is on the perpendicular bisector of a segment, then it is equidistant from
the endpoints of the segment.
Triangle Theorem: Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then the point is on the
perpendicular bisector of the segment.
Triangle Theorem: Concurrency of Perpendicular Bisectors
The perpendicular bisectors of the sides of a triangle intersect in a point that is
equidistant from the vertices.
Triangle Theorem: Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides
of the angle.
Triangle Theorem: Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and equidistant from the sides, then it lies
on the bisector of the angle.
Triangle Theorem: Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle intersect in a point that is equidistant from the
three sides of the triangle.
Triangle Theorem: Angle opposite to longer side is larger
In ΔRST , if RT > RS then m∠RST > m∠RTS
Triangle Theorem: Side opposite to larger angle is longer
In ΔRST , if m∠RST > m∠RTS then RT > RS
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Triangle Theorem: Triangle inequality
Sum of lengths of any two sides is greater than length of remaining side.
Triangle Theorem: SAS inequality
Given ΔABC and ΔDEF , AB ≅ DE and BC ≅ EF . If m∠ABC > m∠DEF then
AC > DF
Triangle Theorem: SAS inequality
Given ΔABC and ΔDEF , AB ≅ DE and BC ≅ EF . If AC > DF then
m∠ABC > m∠DEF
Triangle postulate: AA similarity
If two angles of ΔABC are congruent to two angles of ΔDEF , then the triangles
are similar.
Triangle Theorem: SAS similarity
Given ΔABC and ΔDEF , if m∠ABC = m∠DEF and AB ≅ DE and
BC ≅ EF , then the triangles are similar.
Triangle Theorem: SSS similarity
If the sides of two triangles are in proportion then the triangles are similar.
Triangle Theorem: Parallel proportionality
If a line parallel to one side of a triangle intersects the other two sides, then it
divides the sides proportionally.
Triangle Theorem: Angle bisector
If a ray bisects an angle, it divides the opposite side into segments proportional to
the triangle’s other two sides.
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Quadrilaterals
Quadrilateral Def:
A four-sided polygon; see rhombus, parallelogram, square, rectangle, trapezoid,
isoscoles trapezoid, kite
Quadrangle Def:
A four-sided polygon; see quadrilateral
Quadrilateral Def: Parallelogram
A quadrilateral with both pairs of opposite sides parallel
Parallelogram Theorem: Opposite sides are congruent
The opposite sides of a parallelogram are congruent.
Parallelogram Theorem: Opposite angles are congruent
The opposite angles of a parallelogram are congruent.
Parallelogram Theorem: Diagonals
The diagonals of a parallelogram bisect each other.
Quadrilateral Theorem: Congruent pairs of opposite sides
If both pairs of opposite sides of a quadrilateral are congruent, it is a
parallelogram.
Quadrilateral Theorem: Congruent and parallel opposite sides
If a pair of opposite sides of a quadrilateral are congruent and parallel, it is a
parallelogram.
Quadrilateral Theorem: Congruent pairs of opposite angles
If both pairs of opposite angles of a quadrilateral are congruent, it is a
parallelogram.
Quadrilateral Theorem: Bisecting diagonals
If the diagonals of a quadrilateral bisect each other, it is a parallelogram.
Quadrilateral Def: Median of a Parallelogram
The median is a segment that joins the midpoints of the legs.
Quadrilateral Def: Rectangle
A parallelogram with four right angles.
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Rectangle Theorem: Diagonals are congruent
The diagonals of a rectangle are congruent.
Quadrilateral Def: Rhombus
A parallelogram with four equilateral sides (like a tilted square).
Rhombus Theorem: Diagonals are perpendicular
The diagonals of a rhombus are perpendicular.
Rhombus Theorem: Diagonals bisect angles
The diagonals of a rhombus bisect the angles.
Quadrilateral Def: Trapezoid
A quadrilateral that has at least one pair of parallel sides.
Trapezoid Theorem: Congruent base angles
The base angles of a trapezoid are congruent.
Trapezoid Def: Median
The median of a trapezoid is the segment that joins the midpoints of the legs.
Trapezoid Theorem: Median is parallel and has average length
The median of a trapezoid is parallel to the bases and has length equal to the
average of the bases.
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Circles
Circle Def: Radius, diameter, circumference
A circle is all the points a fixed distance (the radius) from a single point.
Circle Def: Congruent circles
Two circles with the same radius are congruent.
Circle Def: Chord, secant, tangent, point of tangency
A chord is a line segment starting at one point and
ending at another point on the circle.
A chord that goes through the center of the circle is
called the diameter of the circle. The diameter is
twice as long as the radius.
A secant is a line that cuts through the circle and
continues infinitely in both directions.
A tangent line is defined as a line that touches the
circle at exactly one point. This point is called the point of tangency.
Circle Def: Sector
Part of a circle containing its center and an arc
Circle Def: Central angle
An angle whose vertex is at center and whose sides are radii.
Circle Def: Intercepts and subtends
A central angle intercepts the arc. The intercepted arc subtends its central angle.
Circle Postulate: Arc addition
The measure of two adjacent arcs equals the measure of the individual arcs.
Circle Def: Minor arc
An arc whose endpoints form an angle less than 180 degrees with the center of the
; see major arc. The word “arc” by default usually means minor
circle; written
arc.
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Circle Def: Major arc
An arc whose endpoints form an angle over 180 degrees with the center of the
circle; written
- the extra letter is used to distinguish it from a minor arc.
Circle, Def: Measure of an arc
The measure of an arc is the measure of its central angle.
Circle Theorem: Tangent to a circle
At the point of tangency, the tangent line is at right angles to
the radius.
Circle Theorem: Tangent to a circle converse
If a line is perpendicular to the radius of a circle at its outer
endpoint, then the line is tangent to the circle.
Circle Theorem: Tangent Segments from a Point
If two segments from the same exterior point are tangent
the circle, then they are congruent.
to
Circle Def: circumscribed and inscribed polygon
Circle Theorem: Three points determine a circle
Through any three noncolinear points there is a unique circle.
Circle Def: Inscribed angle
An angle whose vertex in on the circle and whose sides contain
chords of the circle.
Circle Theorem: Inscribed angle, intercepted arc
The measure of an inscribed angle equals ½ the measure of the intercepted arc.
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Solids
Solid Def: Parallelpiped
A prism whose opposite faces are all parallelograms and congruent (in pairs)
Solid Def: Opposite faces
Faces that lie in parallel planes
Solid Def: Polyhedron
A three-dimensional surface which is the union of polygonal regions and has no
holes
Solid Def: Right cylinder
A cylinder whose direction of sliding is perpendicular to the plane of the base
Solid Def: Pyramid
The surface of a conic solid whose base is a polygon; see regular pyramid
Solid Def: Right cone
A cone whose axis is perpendicular to the plane containing its base
Solid Def: Right prism
A prism whose direction of sliding is perpendicular to the plane of the base
Solid Def: Slant height
The length of a lateral edge of a conic solid
Solid Def: Small circle
The circle formed by the intersection of a sphere and a plane that doesn't contain
the center
Solid Def: S.A.
Surface area
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