* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Art`s Geometry Notes
Survey
Document related concepts
Multilateration wikipedia , lookup
Noether's theorem wikipedia , lookup
History of geometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Four color theorem wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Line (geometry) wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Euler angles wikipedia , lookup
Integer triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Transcript
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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 1 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 2 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 3 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 4 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 5 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 6 of 34 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 • C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 7 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 8 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 9 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 10 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 11 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 12 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 13 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 14 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 15 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 16 of 34 • • 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) C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 17 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 18 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 19 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 20 of 34 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 21 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 22 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 23 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 24 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 25 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 26 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 27 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 28 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 29 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 30 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 31 of 34 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. C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 32 of 34 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 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 33 of 34 Start 344 C:\TEXT\Math\HighSchool\Geometry\ArtsGeometrySummary.doc Page 34 of 34