Geometry Unit 2

Points Lines Planes

There are three undefined terms in geometry: Point Line Plane *They are undefined because they have to be explained using examples and descriptions.

Point Simply a location Drawn as a dot and named with a capital letter A point has neither shape nor size Say "point A" A

Line is made up of points and has neither thickness or width Drawn with arrows on both ends Labeled with capital letters representing points or a lowercase script letter say "line AB" or "line l " write AB or l There is exactly one line through any two points l A B

Plane a flat surface made up of points Has no edges or sides, but is drawn as a 4-sided figure There is exactly one plane through any 3 noncollinear points. Name a plane with 3 noncollinear letters or one capital script letter. Plane XYZ, Plane ZXY, Plane T Say "Plane XYZ" or "Plane T " X Y Z T

Other important terms: Collinear points: points all in the same line Which points are collinear? Non-collinear points: Points not all in the same line. B A C

Other important terms: Coplanar Points: Points all in the same plane Non-Coplanar Points: points not all in the same plane C B A H E G F D

Space: a bound-less, 3-dimensional set of all points. Can contain lines and planes.

Line Segment  Line Segment: The part of a line that connects two points.  It has definite end points.  Adding the word "segment" is important, because a line normally extends in both directions without end.  Write AB B A

Ray  Ray: A line with a start point but no end point (it goes to infinity)  Write: AB A B

Angle Pair Relationships

Angle Pair Relationship Essential Questions How are special angle pairs identified?

___________ are two rays that are part of a the same line and have only their endpoints in common. Opposite rays X Y Z XY and XZ are ____________. opposite rays The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees. straight angle

There is another case where two rays can have a common endpoint. R S T This figure is called an _____. angle Some parts of angles have special names. The common endpoint is called the ______, vertex and the two rays that make up the sides of the angle are called the sides of the angle. side

R S T vertex side There are several ways to name this angle. 1) Use the vertex and a point from each side. SRT or TRS The vertex letter is always in the middle. 2) Use the vertex only. R If there is only one angle at a vertex, then the angle can be named with that vertex. 3) Use a number. 1 1

Definition of Angle An angle is a figure formed by two noncollinear rays that have a common endpoint. E D F 2 Symbols: DEF 2 E FED

B A 1 C 1) Name the angle in four ways. ABC 1 B CBA 2) Identify the vertex and sides of this angle. Point B BA andBC vertex: sides:

W Y X 1) Name all angles having W as their vertex. 1 2 Z 1 2 2) What are other names for ? 1 XWY or YWX 3) Is there an angle that can be named ? W No! XWZ

Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle. Types of Angles A right angle m A = 90 acute angle 0 < m A < 90 A obtuse angle 90 < m A < 180 A

Classify each angle as acute, obtuse, or right. 110° 90° 40° 50° 130° 75° Obtuse Obtuse Acute Acute Acute Right

When you “split” an angle, you create two angles. D A C B 1 2 The two angles are called _____________ adjacent angles  1 and  2 are examples of adjacent angles. They share a common ray. Name the ray that  1 and  2 have in common. ____ adjacent = next to, joining.

Definition of Adjacent Angles Adjacent angles are angles that: M J N R 1 2  1 and  2 are adjacent with the same vertex R and common side A) share a common side B) have the same vertex, and C) have no interior points in common

Determine whether  1 and  2 are adjacent angles. No. They have a common vertex B, but _____________ no common side 1 2 B 1 2 G Yes. They have the same vertex G and a common side with no interior points in common. N 1 2 J L No. They do not have a common vertex or ____________ a common side The side of  1 is ____ The side of  2 is ____

Determine whether  1 and  2 are adjacent angles. No. 2 1 Yes. 1 2 X D Z In this example, the noncommon sides of the adjacent angles form a ___________. straight line These angles are called a _________ linear pair

Definition of Linear Pairs Two angles form a linear pair if and only if (iff):  1 and  2 are a linear pair. A) they are adjacent and B) their noncommon sides are opposite rays C A D B 1 2

In the figure, and are opposite rays. 1 2 M 4 3 E H T A C 1) Name the angle that forms a linear pair with  1.  ACE  ACE and  1 have a common side the same vertex C, and opposite rays and 2) Do  3 and  TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays.

Definition of Complementary Angles 30° A B C 60° D E F Two angles are complementary if and only if (iff) The sum of their degree measure is 90. m  ABC + m  DEF = = 90

30° A B C 60° D E F If two angles are complementary, each angle is a complement of the other.  ABC is the complement of  DEF and  DEF is the complement of  ABC. Complementary angles DO NOT need to have a common side or even the same vertex.

15° H 75° I Some examples of complementary angles are shown below. m  H + m  I = 90 m  PHQ + m  QHS = 90 50° H 40° Q P S 30° 60° T U V W Z m  TZU + m  VZW = 90

Definition of Supplementary Angles If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Two angles are supplementary if and only if (iff) the sum of their degree measure is ° A B C 130° D E F m  ABC + m  DEF = = 180

105° H 75° I Some examples of supplementary angles are shown below. m  H + m  I = 180 m  PHQ + m  QHS = ° H 130° Q P S m  TZU + m  UZV = ° 120° T U V W Z 60° and m  TZU + m  VZW = 180

Recall that congruent segments have the same ________. measure _______________ also have the same measure. Congruent angles

Definition of Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure 50° B V  B   V iff m  B = m  V

1 2 To show that  1 is congruent to  2, we use ____. arcs Z X To show that there is a second set of congruent angles,  X and  Z, we use double arcs. X  ZX  Z m  X = m  Z This “arc” notation states that:

When two lines intersect, ____ angles are formed. four There are two pair of nonadjacent angles. These pairs are called _____________. vertical angles

Definition of Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines Vertical angles:  1 and  3  2 and  4

Theorem 3-1 Vertical Angle Theorem Vertical angles are congruent m n  1   3  2   4

Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°. 130° x°

Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° 125° x – 10 = 125. x = 135.

Suppose  A   B and m  A = 52. Find the measure of an angle that is supplementary to  B. A 52° B 1  B +  1 = 180  1 = 180 –  B  1 = 180 – 52  1 = 128°

1) If m  1 = 2x + 3 and the m  3 = 3x + 2, then find the m  3 2) If m  ABD = 4x + 5 and the m  DBC = 2x + 1, then find the m  EBC 3) If m  1 = 4x - 13 and the m  3 = 2x + 19, then find the m  4 4) If m  EBG = 7x + 11 and the m  EBH = 2x + 7, then find the m  1 x = 17;  3 = 37° x = 29;  EBC = 121° x = 16;  4 = 39° x = 18;  1 = 43° A B C D E G H

Distance & Midpoint Formulas

Applying Distance and Midpoint Formulas The distance d between any two points (x1, y1) and (x2, y2) in a coordinate plane is:, (x2 y2)

Ex. 1 Find the distance between (-1, 3) and (5, 2). Let (x1, y1) = (-1, 3) and (x2, y2) = (5, 2) This means that the distance between the points is It doesn't matter which ordered pair isand (x1, y1) (x2, y2) units

Ex. 1b You try: Find the distance between (-3, 1) and (2, 3). Let (x1, y1) = (-3, 1) and (x2, y2) = (2, 3) units

Sometimes you will be asked to find a missing coordinate using the distance formula. Ex. 2 The distance between (3, -5) and (7, b) is 5 units. Find the value of b. Plug in values and simplify Don't forget how to square a binomial! Combine like terms square both sides to get rid of radical set both sides = 0 by subtracting 25 Factor the trinomial Apply zero product property and solve b = -8 or b = -2

Ex. 2b The distance between (4, a) and (1, 6) is 5 units. Find the value of a.

The midpoint of a line segment is the point on the segment that is equidistant (same distance) from the endpoints. Use the midpoint formula to find the midpoint of a line segment: Notice there is a comma in the formula indicating that midpoint is an ordered pair.

Ex. 3 What is the midpoint of the line legment with endpoints (-1, -2) and (3, -4)? Fill in values and simplify The midpoint is the ordered pair (1, -3)

Ex. 3b What is the midpoint of the line legment with endpoints (-3, -1) and (7, -5)?

Practice 1. Find the distance between the two points a. (2, -2), (6, 1) b. (-6, 7), (2, 9) 2. The distance d between 2 points is given. Find the value of b. a. (b, -6), (-5, 2); d = 10 b. (13, -3), (b, 2); d = Find the midpoint of the line segment with the given endpoints. a. (6, -3), (4, -7) b. (-50, -75), (8, 9)

INSERT NOTES ON Parallel Lines and Transversals

TRANSFORMATIO NS

 Def: moving figures on a coordinate plane  Pre-image: the image you start with EX: ∆ABC  Image: resulting figure after transformation EX: ∆A’B’C’  Algebraic Notation: (x, y) ⟶(x-1, y+5)

Isometry  A congruent transformation  Three of the four transformations we discuss will be isometries

TRANSLATIONS  Def: a transformation that moves all points of a figure the same direction, the same distance.  (Can be described in words or algebraic notation)

EXAMPLE  A triangle is located at A(0,0) B(-3,4) C (5,1). Show the resulting endpoints (vertices) for a transformation 5 units right and 2 units down.  Algebraic notation: (x+5, y-2)  A’=(0 + 5,0 – 2) = (5, -2)  B’= (-3 + 5, 4 – 2) = (2, 2)  C’ = (5 + 5, 1 – 2) = (10, -1)

CLASSWORK!  Odd problems on Classwork WS.

Reflections  Def: A transformation where a figure is flipped over a point, line, or plane. The size of the object does not change. Typically it is reflected over: x-axis, y-axis, y=x, or y=-x.

REFLECTIONS  What do you notice?  Every point is the same distance from the central line !... and...  The reflection has the same size as the original image

How Do I Do It Myself?  Just approach it step-by-step. For each corner of the shape:  1. Measure from the point to the mirror line (must hit the mirror line at a right angle)right angle

How Do I Do It Myself?  2. Measure the same distance again on the other side and place a dot.  3. Then connect the new dots up!

EXAMPLE  ∆ABC A(0,4) B(4, 0) and C (-1, -1) is reflected over the x- axis.  (Change the sign of your y values)

EXAMPLE  ∆DEF over the y-axis: D(2, 1), E (8, 1) and F (6, 4).  (Change the sign of your x-coordinate)

Example  Reflect A(3, 0) B(6, 0) C (4, 2) over y=x.  (Change x and y coordinates)

example  Reflect ∆ABC over y=5. A (0, 0) B(3, 1) C (0, 3)

CLASSWORK!  Even problems on Classwork WS.

Homework!  WS #1-14 ALL

TRANSFORMATIO NS

Rotations  Def: moving a figure a certain degree about a fixed point called the center of rotation. Typically we use 90°, 180 °, 270 °, 360 °. If it is not stated whether your rotation is clockwise or counterclockwise, then assume it is counterclockwise.  ns-in-math.php ns-in-math.php

EXAMPLE  Where the image of ∆ABC after a rotation of 90° CC?  (Same as 270 ° Clockwise)  Algebraic Representation: (-y, x)

EXAMPLE  Rotate ∆DEF 270 ° CC about the origin. D (3, -1), E (6, 5) and F (-8, 10)  (Same as 90 ° Clockwise)

Other rotations  180 ° algebraic rule: (-x, -y)  360 ° algebraic rule: (x, y)

CLASSWORK!  Problems 1-7 on Classwork WS.

Homework!  WS #1-7 all

TRANSFORMATIO NS

Dilations  Def: a transformation where a figure is reduced or enlarged by a scale factor “r”  Algebraic notation: (rx, ry)

EXAMPLE  The ∆ABC is located A (-4, 5) B (0, -2) C (6, 8). It undergoes a dilation where r = 3. Where is the image located?  Algebraic rule: (3x, 3y)  A’ (3 · -4, 3 · 5) = (-12, 15)  B’ (3 · 0, 3 · -2) = (0, -6)  C’ (3 · 6, 3 · 8) = (18, 24)

EXAMPLE  The pre-image of ∆ABC is located at (6, 6) (0, 1) and (-6, 2). Dilate the figure with r = ½. Where is the resulting image located?  Algebraic Rule: (½x, ½y)  A’ = (½ · 6, ½ · 6) = (3, 3)  B’ = (½ · 0, ½ · 1) = (0, ½)  C’ = (½ · -6, ½ · 2) = (-3, 1)

EXAMPLE  The image of ABC after a dilation of ¼ is represented by the A’(1, -2), B’(½, 4), and C’(-8, 24). Where is the pre-image located?

CLASSWORK!  Alice in Wonderland WS.

HomeWORK! Dilations WS.

TRANSFORMATIO NS

Composition of motion  Composition of transformations – Two or more sequential transformations of a figure.  The 1 st maps onto the 2 nd and the 2 nd maps onto the 3 rd.  EX. A A’ A” etc.

HOW TO DO IT?  a) Find the vertices of figure ABC under the first transformation and sketch the figure A’B’C’.  b) Perform the next transformation on A’B’C’ and label the result A”B”C”.

EXAMPLE

Example

Graph the image of GHI after the following transformations: Translate (x, y) (x -3, y + 6) Rotate 90 o counterclockwise.

EXAMPLE

HomeWORK!