Basics of Geometry

Basics of Geometry

Points, Lines and Planes

PLANES, LINES AND POINTS A A PointA B A Line Segment AB, AB or BA A B C A B C Ray AB, AB , AC is the initial point Line AB, AB or BA or AC, CA or linem m

C A D B Collinear Points: points that line on the same line Points A, B and D are collinear Points A, B and C are not collinear Any two points are collinear. Two points define a line. Opposite rays: Collinear rays with the same initial point, extending in opposite directions . and are opposite rays BA BD Between two points: In order to say that a point is between two other points all three points must be collinear. B is between A and D C is not between A and D

M D C PlaneM or plane ABC or plane BCD A B E Coplanar points: points that line on the same plane. A, B, C, D are all coplanar. A, B, C, E are not coplanar Any three points are coplanar. Any three points define a plane. Plane ABE exists although it is not shown.

Two or more figures intersect if they have one or more points in common. Two lines intersect as a point Two planes intersect at a line

Draw and label one diagram that includes all of the following: a. Plane R, points X, Y and Z coplanar, but not collinear. b. XY and ZX c. AB non-coplanar to plane R, but intersecting plane R at point X A R Z X Y B

1.3 Segments and Their Measures

Postulates: rules that are accepted Without proof Theorems: rules that are proven

Ruler Postulate The points on a line can be matched one to one with real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. Find AB: AB= 3-9 or 9-3 AB=6 B A 9 3

Segment Addition Postulate: If B is Between A and C, then AB + BC = AC If AB + BC = AC then B is between A and C 12 7 5 C C B A A B is between A and C B B is not between A and C

Ex: if DE=2, EF=5, and DE=FG, find FG, DF, DG, & EG. D E F G FG=2 DF=7 DG=9 EG=7

Ex. Y is between X and Z. Find XY, YZ and XZ if: XY= 3x + 4 YZ= 2x + 5 XZ= 9x - 3 (3x + 4) + (2x +5) = 9x - 3 5x + 9 = 9x - 3 12 = 4x x = 3 YZ = 2(3) + 5 YZ = 11 XZ = 9(3) -3 XZ = 24 XY = 3(3) + 4 XY = 13

Congruent Segments: If two line segments have the same lengths they are said to be congruent. The symbol for congruence is ≅ If AB = 5 and CD = 5 then, AB = CD, distances are equal AB ≅ CD, segments are congruent

The Distance Formula If A ( x₁, y₁) and B ( x₂, y₂) are points in a coordinate plane then the distance between A and B is: B AB = √ ( x₂ - x₁ )² + ( y₂ - y₁ )² A

Using the Distance Formula GH = √ 8² + 4² GH = √ (11 -3)² + (6 - 2)² (11, 6) H GH = √ 64 + 16 GH = √ 80 GH = 4√ 5 (3, 2) G

The Distance Formula comes from the Pythagorean Theorema² + b² = c² where a and b are the legs of a right triangle and c is the hypotenuse. AB is the hypotenuse of a right triangle (x₂ - x₁)² = a, the length of the horizontal leg (y₂ - y₁)² = b, the length of the vertical leg B (x₂ , y₂) c b A a (x₁ , y₁)

Using the Pythagorean Theorem c b a (11,2) a² + b² = c² (11, 6) H √a² + b² = c √∣ 11-3 ∣² + ∣ 6-2 ∣² = c √8² + 4² = c √64 + 16 = c (3, 2) G √80 = c 4√5 = c

J (2,5), K (7,11) Find JK a is the distance between 2 and 7, a=5 b is the distance between 5 and 11, b=6 JK = √5² + 6² RS = √4² + 10² RS = √116 RS = 2√29 JK = √61 R (5,12), S (9, 2) Find RS a is the distance between 5 and 9, a=4 b in the distance between 12 and 2, b=10

Angles and Their Measures

An angle consists of two different rays that have the same initial point. The rays are the sides of the angle. The initial point is the vertex of the angle. C side vertex A side B

Naming Angles Angles are named using three points , the vertex and a point on each ray which makes up the angle. The vertex is always the second point listed. If an angle stands alone it can be named by its vertex only. J H ∠JHI or ∠IHJ or ∠H I

Naming Angles J ∠JHI or ∠IHJ H ∠JHK or ∠KHJ I ∠IHK or ∠KHI K You should not name any of these angles as H because all three angles have H as their vertex. The name H would not distinguish one angle from the others.

Measuring Angles The measure of ∠ A is denoted by m∠ A. The measure of an angle can be approximated using a protractor, using units called degrees(°). B For instance, ∠ BAC has a measure of 50°, which can be written as m∠ BAC = 50°. C A

Congruent Angles Angles that have the same measure are called congruent angles If ∠BAC and ∠DEC both measure 75˚ then m ∠BAC = m∠DEC, measures are equal ∠BAC ≅ ∠DEC , angles are congruent

ProtractorPostulate Consider a point B on one side of AC. The rays of the form AB can be matched one to one with the real numbers from 1-180. The measure of BAC is equal to the absolute value of the difference between the real numbers for AC and AB. B C A

Applying the Protractor Postulate C D B A m∠ CAB = 60˚ , m∠ BAD = 130˚ m∠ CAD = ∣ 130˚ - 60˚ ∣ or ∣ 130˚ - 60˚ ∣ = 70˚

B C A A point is interior of an angle if it is between the sides of the angle Point C is interior of angle A A point is exterior of an angle if it is not on the angle or in its interior Point B is exterior of angle A

Angle Addition Postulate J H M K If M is in the interior of ∠ JHK, then m∠ JHM + m∠ MHK = m∠ JHK

Draw a sketch using the following information D is interior of ∠ABC C is interior of ∠DBE m ∠ABC = 75˚ m ∠DBC = 45˚ m ∠ABE = 100˚ Find the m ∠ABD , m ∠CBE and m ∠DBE Ans→

D is interior of ∠ABC C is interior of ∠DBE m ∠ABC = 75˚ m ∠DBC = 45˚ m ∠ABE = 100˚ D 45˚ 75˚ 100˚ A Find the m ∠ABD , m ∠CBE and m ∠DBE E m∠ABC - m∠DBC = m∠ABD m∠ABE - m∠ABC = m∠CBE C 75˚ - 45˚ = 30˚ 100˚ - 75˚ = 25˚ B m∠ABD = 30˚ m∠CBE = 25˚ m∠DBC + m∠CBE = m∠DBE 45˚ + 25˚ = 70˚ m∠DBE = 70˚

Classifying Angles Angles are classified according to their measures. Angles have measures greater than 0° and less than or equal to 180°

Adjacent Angles Two angles are adjacent if they share a common vertex and side, but no common interior points. J H M K ∠JHM and ∠MHK are adjacent angles ∠JHM and ∠JHK are not adjacent angles

Segment and Angle Bisectors

Bisecting a Segment Bisect: to divide into two equal parts. The midpoint of a segment is the point that bisects the segment, dividing the segment into two congruent segments. B A C If C is the midpoint of AB, then AC ≅ CB A B Congruent segments are indicated using marks through the segments. A segment bisector is a segment, ray, line or plane that intersects a segment at its midpoint. Y C XY is a bisector of AB X

Midpoint Formula To find the coordinates of the midpoint of a segment you find the mean of the x coordinates and the y coordinates of the endpoints. B A The midpoint of AB = The midpoint of AB = B (10, 6) A ( 3,2 )

Finding the coordinates of an endpoint The midpoint of GH is M ( 7, 5 ). One endpoint is H ( 15, -1 ). Find the coordinates of point G. G ( x₂ , y₂ ) M ( 7 , 5 ) H ( 15 , -1 ) The x coordinate of G The y coordinate of G G is at ( -1, 11 )

Practice Problems JK has endpoints J ( -1, 7 ) and K ( 3, -3 ) find the coordinates of the midpoint ( 1, 2 ) NP has midpoint M ( -8, -2 ) and endpoint N ( -5, 9 ). Find the coordinates of P P ( -11, -13 )

Angle Bisector An angle bisector is a ray that divides an angle into two congruent adjacent angles. D A B C If BD is a bisector of ∠ ABC, then ∠ ABD ≅ ∠ DBC Congruent angles are indicated by separate arcs on each angle

Practice Problems KL is a bisector of ∠ JKM. Find the two angle measures not given in the diagram J J L L 37⁰ 85⁰ K K M M m ∠ JKL = m ∠ LKM = 37⁰ m ∠ JKL = 85/2 = 42½⁰ m ∠ JKM = 2 x 37 = 74⁰ m ∠ LKM = 85/2 = 42½⁰

Practice Problem KL is a bisector of ∠ JKM. Find the value of x J 6x – 11 = 10x – 51 -11 = 4x -51 40 = 4x x = 10 L ( 6x – 11 )⁰ K M ( 10x – 51 )⁰ Check 6 (10) – 11 = 10 (10) -51 60 - 11 = 100 – 51 49 = 49 END

Angle Pair Relationships

Ruler Postulate The points on a line can be matched one to one with real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. Find AB: AB= 3-9 or 9-3 AB=6 B A 9 3

Vertical Angles: angles whose sides form opposite rays 1 2 4 3 ∠ 1 and ∠ 3 are vertical angles ∠ 2 and ∠ 4 are vertical angles

Linear Pair: two adjacent angles whose noncommon sides are opposite rays 1 2 4 3 ∠ 2 and ∠ 3 are a linear pair ∠1 and ∠ 2 are a linear pair ∠ 4 and ∠ 1 are a linear pair ∠ 3 and ∠ 4 are a linear pair

110˚ 1 2 70˚ 4 70˚ 3 110˚ If m ∠ 1 = 110° , what is the m ∠ 2 and m ∠ 3 and m ∠ 4? What conclusions can you draw about vertical angles? What conclusions can you draw about linear pairs?

Vertical angles are congruent The sum of the measures of the angles that form a linear pair is 180°

Practice Problem ( 4x + 15 )° ( 5x + 30 )° ( 3y + 15 )° ( 3y - 15 )° Find the value of x and y ( 4x + 15 ) + ( 5x + 30 ) = 180 ( 3y + 15 ) + ( 3y - 15 ) = 180 9x + 45 = 180 6y = 180 9x = 135 Y = 60 X = 15

Complementary angles: two angles whose sum is 90°. Each of the angles is called the complement of the other. Complementary angles can be adjacent or nonadjacent 67° 23° 67° 23° Adjacent complementary angles Nonadjacent complementary angles

Supplementary angles: two angles whose sum is 180°. Each of the angles is called the supplement of the other. Supplementary angles can be adjacent or nonadjacent 50° 130° 50° 130° Adjacent supplementary angles Nonadjacent supplementary angles

Basics of Geometry