Essentials of Geometry Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

Basic Definitions ERHS Math Geometry Mr. Chin-Sung Lin

Definition ERHS Math Geometry Mr. Chin-Sung Lin A definition is a statement of the precise meaning of a term A good definition must be expressed in words that have already been defined or in words that have been accepted as undefined

Postulate ERHS Math Geometry Mr. Chin-Sung Lin A postulate is an accepted statement of fact

Undefined Terms: Set, Point, Line & Plane ERHS Math Geometry Mr. Chin-Sung Lin

Undefined Terms Set ERHS Math Geometry Mr. Chin-Sung Lin A collection of objects such that it is possible to determine whether a given object belongs to the collection or not

Undefined Terms Point ERHS Math Geometry Mr. Chin-Sung Lin A point indicates place or location and has no size or dimensions A point is represented by a dot and named by a capital letter A C D E B

Line ERHS Math Geometry Mr. Chin-Sung Lin A line is a set of continuous points that form a straight path that extends without ending in two opposite directions A line has no width AB Undefined Terms

Line ERHS Math Geometry Mr. Chin-Sung Lin A line is identified by naming two points on the line. The notation AB is read as “line AB” Points that lie on the same line are collinear AB Undefined Terms

Plane ERHS Math Geometry Mr. Chin-Sung Lin A plane is a set of points that form a flat surface that has no thickness and extends without ending in all directions A plane is represented by a “window pane” R Undefined Terms

Plane ERHS Math Geometry Mr. Chin-Sung Lin A plane is named by writing a capital letter in one of its corners or by naming at least three non-colinear points in the plane Points and lines in the same plane are coplanar A B C Undefined Terms

Postulate ERHS Math Geometry Mr. Chin-Sung Lin Through any two points there is exactly one line AB Undefined Terms

Postulate ERHS Math Geometry Mr. Chin-Sung Lin If two lines intersect, then they intersect in exactly one point P Undefined Terms

Postulate ERHS Math Geometry Mr. Chin-Sung Lin If two planes intersect, then they intersect in exactly a line Undefined Terms

Properties of Real Numbers ERHS Math Geometry Mr. Chin-Sung Lin

Addition & Multiplication Operation Properties ERHS Math Geometry Mr. Chin-Sung Lin  Closure  Commutative Property  Associative Property  Identity Property  Inverse Property  Distributive Property  Multiplication Property of Zero

Closure ERHS Math Geometry Mr. Chin-Sung Lin  Closure property of addition The sum of two real numbers is a real number a + b is a real number  Closure property of multiplication The product of two real numbers is a real number a  b is a real number

Commutative Property ERHS Math Geometry Mr. Chin-Sung Lin  Commutative property of addition Change the order of addition without changing the sum a + b = b + a  Commutative property of multiplication Change the order of multiplication without changing the product a  b= b  a

Associative Property ERHS Math Geometry Mr. Chin-Sung Lin  Associative property of addition When three numbers are added, the sum does not depend on which two numbers are added first (a + b) + c = a + (b + c)  Associative property of multiplication When three numbers are multiplied, the product does not depend on which two numbers are multiplied first (a  b)  c = a  (b  c)

Identity Property ERHS Math Geometry Mr. Chin-Sung Lin  Additive identity When 0 is added to any real number a, the sum is a a + 0 = a and 0 + a = a  Multiplicative identity When 1 is multiplied to any real number a, the product is a a  1 = a and 1  a = a

Inverse Property ERHS Math Geometry Mr. Chin-Sung Lin  Additive inverses Two real numbers are additive inverses, if their sum is 0 a + (-a) = 0  Multiplicative inverses Two real numbers are multiplicative inverses, if their product is 1 a  (1/a) = 1 (for all a ≠ 0)

Distributive Property ERHS Math Geometry Mr. Chin-Sung Lin  Multiplication distributes over addition a  (b + c) = a  b + a  c (a + b)  c = a  c + b  c

Multiplication Property of Zero ERHS Math Geometry Mr. Chin-Sung Lin  Zero has no multiplicative inverse  Zero product property a  b = 0 if and only if a = 0 or b = 0

Exercise ERHS Math Geometry Mr. Chin-Sung Lin Identify the additive and multiplicative inverses of the following nonzero real numbers:  9  -6  d  -b  (3 – b)

Exercise ERHS Math Geometry Mr. Chin-Sung Lin Identify the additive and multiplicative inverses of the following nonzero real numbers:  9-9  -66  d-d  -bb  (3 – b)(b – 3)

Exercise ERHS Math Geometry Mr. Chin-Sung Lin Identify the additive and multiplicative inverses of the following nonzero real numbers:  9-91/9  -66-1/6  d-d1/d  -bb-1/b  (3 – b)(b – 3)1/(3-b)

Exercise ERHS Math Geometry Mr. Chin-Sung Lin Identify the properties in the following operations:  6  (1/6) = 1  7 + (4 + a) = (7 + 4) + a  3  4 = 4  3  7  (x + 2) = 7  x + 7  2  = 12

Exercise ERHS Math Geometry Mr. Chin-Sung Lin Identify the properties in the following operations:  6  (1/6) = 1(multiplicative inverses)  7 + (4 + a) = (7 + 4) + a (associative)  3  4 = 4  3(commutative)  7  (x + 2) = 7  x + 7  2(distributive)  = 12(additive identity)

Lines & Line Segments ERHS Math Geometry Mr. Chin-Sung Lin

Distance between Tow Points Distance ERHS Math Geometry Mr. Chin-Sung Lin The distance between two points on the real number line is the absolute value of the difference of the coordinates of the two points AB =| a – b | = | b – a | AB ab

Order of Points Betweenness ERHS Math Geometry Mr. Chin-Sung Lin B is between A and C if and only if A, B and C are distinct collinear points (on ABC) and AB + BC = AC AB = | b – a | = b – a BC = | c – b | = c – b AB + BC = (b – a) + (c – b) = c – a = AC AB ab C c

Line Segment Segment ERHS Math Geometry Mr. Chin-Sung Lin A segment is a subset, or a part of a line consisting of two endpoints and all points on the line between them Symbol: AB AB

Line Segment Length or Measure of a Line Segment ERHS Math Geometry Mr. Chin-Sung Lin The length or measure of a line segment is the distance between its endpoints, i.e., the absolute value of the difference of the coordinates of the two points AB = |a - b| = |b - a| Symbol: AB AB

ERHS Math Geometry Mr. Chin-Sung Lin AB represents segment AB AB represents the measure of AB Line Segment Length or Measure of a Line Segment AB

Line Segment Congruent Line Segments ERHS Math Geometry Mr. Chin-Sung Lin Congruent segments are segments that have the same measure AB CD

Line Segment Congruent Line Segments ERHS Math Geometry Mr. Chin-Sung Lin AB CD AB CD, the segments are congruent AB = CD, the measures/distances are the same ≅

Midpoints & Bisectors ERHS Math Geometry Mr. Chin-Sung Lin

Line Segment Midpoint of a Line Segment ERHS Math Geometry Mr. Chin-Sung Lin The midpoint of a line segment is a point of that line segment that divides the segment into two congruent segments AB M

Line Segment Midpoint of a Line Segment ERHS Math Geometry Mr. Chin-Sung Lin AM MBor AM = MB AM = ( 1 / 2 ) AB orMB = ( 1 / 2 ) AB AB = 2AMor AB = 2MB AB M ≅

Line Segment Midpoint of a Line Segment ERHS Math Geometry Mr. Chin-Sung Lin Coordinate of the midpoint of AB is (a + b)/2 Midpoint is the average point AB M ab

Line Segment Bisector of a Line Segment ERHS Math Geometry Mr. Chin-Sung Lin The bisector of a line segment is any line or subset of a line that intersects the segment at its midpoint A B M C D E F

Line Segment Adding/Subtracting Line Segments ERHS Math Geometry Mr. Chin-Sung Lin A line segment, AB is the sum of two line segments, AP and PB, if P is between A and B AB = AP + PBAP = AB – PBPB = AB - AP AB P ab

Rays & Angles ERHS Math Geometry Mr. Chin-Sung Lin

Half-Lines and Rays On one side of a point ERHS Math Geometry Mr. Chin-Sung Lin Two points, A and B, are on one side of a point P if A, B, and P are collinear and P is not between A and B PBA

Half-Lines and Rays Half-Line ERHS Math Geometry Mr. Chin-Sung Lin A half-line consists of the set of all points on one side of a point of division, not including that point (endpoint) PBA Half-line

Half-Lines and Rays Ray ERHS Math Geometry Mr. Chin-Sung Lin A ray is the part of a line consisting of a point on a line and all the points on one side of the point (endpoint) A ray consists of an endpoint and a half-line AB

Half-Lines and Rays Ray ERHS Math Geometry Mr. Chin-Sung Lin A ray AB is written as AB, where A needs to be the endpoint AB

Half-Lines and Rays Opposite Rays ERHS Math Geometry Mr. Chin-Sung Lin The opposite rays are two collinear rays with a common endpoint, and no other point in common Opposite rays always form a line A

Lines Parallel Lines ERHS Math Geometry Mr. Chin-Sung Lin Lines that do not intersect may or may not be coplanar Parallel lines are coplanar lines that do not intersect Segments and rays are parallel if they lie in parallel lines AB CD

Lines Skew Lines ERHS Math Geometry Mr. Chin-Sung Lin Skew lines do not lie in the same plane They are neither parallel nor intersecting A B CD

Basic Definition of Angles ERHS Math Geometry Mr. Chin-Sung Lin

Basic Definition ERHS Math Geometry Mr. Chin-Sung Lin  Definition of Angles  Naming Angles  Degree Measure of Angles

Definition of Angles ERHS Math Geometry Mr. Chin-Sung Lin An angle is the union of two rays having the same endpoints The endpoint is called the vertex of an angle; the rays are called the sides of the angle Vertex: A Sides: AB and AC B A C 1

Naming Angles ERHS Math Geometry Mr. Chin-Sung Lin  Three letter: CAB or BAC  A number (or lowercase letter) in the interior of angle: 1  A single capital letter (its vertex): A A B C interior of angle exterior of angle 1

Naming Angle ERHS Math Geometry Mr. Chin-Sung Lin O Y X

Naming Angle -  XOY or  YOX ERHS Math Geometry Mr. Chin-Sung Lin O Y X

Degree Measure of Angles ERHS Math Geometry Mr. Chin-Sung Lin Let OA and OB be opposite rays in a plane. OA, OB and all the rays with endpoints O that can be drawn on one side of AB can be paired with the real numbers from 0 to 180 in such a way that: 1.OA is paired with 0 and OB is paired with 180 ABO

Degree Measure of Angles ERHS Math Geometry Mr. Chin-Sung Lin 2.If OC is paired with x and OD is paired with y, then, the degree measure of the angle: m COD = | x – y | ABO D C

Degree Measure of Angles ERHS Math Geometry Mr. Chin-Sung Lin If OC is paired with 60 and OD is paired with 150, then, the degree measure of the angle: m COD = ? ABO D C

Degree Measure of Angles ERHS Math Geometry Mr. Chin-Sung Lin If OC is paired with 60 and OD is paired with 150, then, the degree measure of the angle: m COD = | 60 – 150 | = | -90 | = 90. ABO D C

Type of Angles by Measures ERHS Math Geometry Mr. Chin-Sung Lin  Straight Angle  Obtuse Angle  Right Angle  Acute Angle

Straight Angle ERHS Math Geometry Mr. Chin-Sung Lin A straight angle is an angle that is the union of opposite rays m AOB = 180 ABO

A Degree ERHS Math Geometry Mr. Chin-Sung Lin A degree is the measure of an angle that is 1/180 of a straight angle ABO

Obtuse Angle ERHS Math Geometry Mr. Chin-Sung Lin An obtuse angle is an angle whose degree measure is greater than 90 and less than < m DOE < 180 EO D

Right Angle ERHS Math Geometry Mr. Chin-Sung Lin A right angle is an angle whose degree measure is 90 m GHI = 90 IH G

Acute Angle ERHS Math Geometry Mr. Chin-Sung Lin An acute angle is an angle whose degree measure is greater than 0 and less than 90 0 < m DOE < 90 EO D

Congruent Angles ERHS Math Geometry Mr. Chin-Sung Lin Congruent angles are angles that have the same measure DOE = ABC m DOE = m ABC ~ CB A EO D

Bisector of an Angle ERHS Math Geometry Mr. Chin-Sung Lin A bisector of an angle is a ray whose endpoint is the vertex of the angle, and that divides the angle into two congruent angles If OC is the bisector of AOD m AOC = m COD DO A C

Calculate Angle ERHS Math Geometry Mr. Chin-Sung Lin If mAOB = 120, OC is an angle bisector, then mAOC = ? B A O C

Calculate Angle ERHS Math Geometry Mr. Chin-Sung Lin If mAOB = 120, OC is an angle bisector, then mAOC = 60 B A O C

Calculate Angle ERHS Math Geometry Mr. Chin-Sung Lin If mCOB = 30, OC is an angle bisector, then mAOB = ? B A O C

Calculate Angle ERHS Math Geometry Mr. Chin-Sung Lin If mCOB = 30, OC is an angle bisector, then mAOB = 60 B A O C

Adding Angles ERHS Math Geometry Mr. Chin-Sung Lin A non-straight angle AOC is the sum of two angles AOP and POC if point P is in the interior of angle AOC AOC = AOP + POC Note that AOC may be a straight angle with P any point not on AOC CO A P

Calculate Angle ERHS Math Geometry Mr. Chin-Sung Lin If mAOC = 50, mBOC = 40, then mAOB = ? B A O C

Calculate Angle ERHS Math Geometry Mr. Chin-Sung Lin If mAOC = 50, mBOC = 40, then mAOB = 90 B A O C

Solve for x ERHS Math Geometry Mr. Chin-Sung Lin OC is an angle bisector. If mAOB = 60, mCOB = 2x, then x = ? B A O C

Solve for x ERHS Math Geometry Mr. Chin-Sung Lin OC is an angle bisector. If mAOB = 60, mCOB = 2x, then x = 15 B A O C

Perpendicular Lines ERHS Math Geometry Mr. Chin-Sung Lin Perpendicular lines are two lines that intersect to form right angles C O A

Distance from a Point to a Line ERHS Math Geometry Mr. Chin-Sung Lin Distance from a point to a line is the length of the perpendicular from the point to the line C O A

Triangles ERHS Math Geometry Mr. Chin-Sung Lin

Polygons ERHS Math Geometry Mr. Chin-Sung Lin A polygon is a closed figure in a plane that is the union of line segments such that the segments intersect only at their endpoints and no segments sharing a common endpoint are collinear

Triangles ERHS Math Geometry Mr. Chin-Sung Lin A triangle is a polygon that has exactly three sides ∆ ABC Vertex: A, B, C Angle:A, B, C Side:AB, BC, CA Length of side: AB = c, BC = a, AC = b a A CB b c

Type of Triangles by Sides ERHS Math Geometry Mr. Chin-Sung Lin  Scalene Triangles  Isosceles Triangles  Equilateral Triangles

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin A scalene triangle is a triangle that has no congruent sides A C B

Isosceles Triangle ERHS Math Geometry Mr. Chin-Sung Lin A isosceles triangle is a triangle that has two congruent sides AC B

Equilateral Triangle ERHS Math Geometry Mr. Chin-Sung Lin A equilateral triangle is a triangle that has three congruent sides AC B

Parts of an Isosceles Triangle ERHS Math Geometry Mr. Chin-Sung Lin Leg: the two congruent sides Base: the third non-congruent side Vertex Angle: the angle formed by the two congruent side Base Angle: the angles whose vertices are the endpoints of the base AC B Base Leg Base Angle Vertex Angle

Type of Triangles by Angles ERHS Math Geometry Mr. Chin-Sung Lin  Acute Triangle  Right Triangle  Obtuse Triangle  Equiangular Triangle

Acute Triangle ERHS Math Geometry Mr. Chin-Sung Lin An acute triangle is a triangle that has three acute angles AC B

Right Triangle ERHS Math Geometry Mr. Chin-Sung Lin An right triangle is a triangle that has a right angle AC B

Obtuse Triangle ERHS Math Geometry Mr. Chin-Sung Lin An obtuse triangle is a triangle that has an obtuse angle A C B

Equiangular Triangle ERHS Math Geometry Mr. Chin-Sung Lin An equiangular triangle is a triangle that has three congruent angles AC B

Parts of a Right Triangle ERHS Math Geometry Mr. Chin-Sung Lin Leg: the two sides that form the right angle Hypotenuse: the third side opposite the right angle Leg Right Angle AC B Hypotenuse

Included Sides ERHS Math Geometry Mr. Chin-Sung Lin If a line segment is the side of a triangle, the endpoints of that segment is the vertics of two angles, then the segment is included between those two angles AB is included between A and B BC is included between B and C CA is included between C and A AC B

Included Angles ERHS Math Geometry Mr. Chin-Sung Lin Two sides of a triangle are subsets of the rays of an angle, and the angle is included between those sides A is included between AB and AC B is included between AB and BC C is included between BC and AC AC B

Opposite Sides / Angles ERHS Math Geometry Mr. Chin-Sung Lin For each side of a triangle, there is one vertex of the triangle that is not the endpoint of that side A is opposite to BC and BC is opposite to A B is opposite to CA and CA is opposite to B C is opposite to AB and AB is opposite to C AC B

Using Diagrams in Geometry ERHS Math Geometry Mr. Chin-Sung Lin We may assume:  A line segment is part of a line  An intersect point is a point on both lines  Points on a segment are between endpoints  Points on a line are collinear  A ray in the interior of an angle with its endpoint at the vertex of the angle separate the angle into two adjacent angles

Using Diagrams in Geometry ERHS Math Geometry Mr. Chin-Sung Lin We may NOT assume:  One segment is longer, shorter or equal to another one  A point is a midpoint of a segment  One angle is greater, smaller or equal to another one  Lines are perpendicular or angles are right angles  A triangle is isosceles or equilateral  A quadrilateral is a parallelogram, rectangle, square, rhombus, or trapezoid

Q & A ERHS Math Geometry Mr. Chin-Sung Lin

The End ERHS Math Geometry Mr. Chin-Sung Lin