Geometry Chapter 1 Note-Taking Guide Name ________________________

Foundations of Geometry Per ____ Date _________________

1.1 The most basic figures in geometry are undefined terms. Undefined terms cannot be defined by using other figures and are the building blocks of geometry. The undefined terms are point, line, and plane.

1.1 A point (pt or pts) names a location and has NO size. It is represented by a dot and named by a CAPITOL letter.

1.1 A line is a straight path that has NO thickness and extends FOREVER (the extending forever is represented by arrows on each end of the line). It is named in two different ways:

1. Any two points on the line with a line symbol above it. (If a line has more than two points on it, pick just two to name the line)

2. A lower case cursive letter near the line with the word line in front of it.

1.1 Collinear points are points that all lie on the SAME LINE. (Co means same, linear means line) Non-collinear points are points that do NOT lie on the same line. Three points that are non-collinear will form a triangle. If you are trying to determine if more than 3 points are non-collinear, determine if any three of the them form a triangle, if so then they are non-collinear.

1.1 A plane is a flat surface with only two dimensions, length and width. It has NO thickness and extends forever. It is usually drawn as a shaded parallelogram with an upper case cursive letter in the corner. A plane can be named in two ways:

1. The word plane (or pl. for short) with the upper case cursive letter that is in the corner. 2. The word plane (or pl. for short) with three non-collinear points.

1.1 Coplanar points are points that lie on the same plane. (Co means same, planar means plane) Non-Coplanar points are points that do NOT lie on the same plane.

Name three coplanar points: Name four coplanar points: Name four non-coplanar points: How many planes are in the diagram?

1.1 A line segment (or segment or seg) is the part of the line consisting of two points (called endpoints) and all the points between them. A line segment is named by the endpoints with the segment symbol above it.

1.1 A ray is a part of a line that starts at an endpoint and extends forever in one direction. A ray is always named endpoint FIRST and then any other point on the ray with a ray symbol above it. The ray symbol used when naming the ray will ALWAYS point to the right even if the ray itself points to the left.

1.1 Opposite rays are two rays that have the same endpoint and form a line.

1.1 Postulate (also known as an axiom) is a statement that is accepted as true without proof. Postulates about points, lines, and planes help describe geometric properties.

1.1 Line Uniqueness Postulate: Through any two points there is exactly one line

1.1 Plane Uniqueness Postulate: Through any three non-collinear points there is exactly one plane containing them

1.1 Continuity Postulate: If two points lie in a plane, then the line containing those points lies in the plane.

1.1 Line Intersection postulate: If two unique lines intersect, then they intersect at exactly ONE point.

1.1 Plane Intersection Postulate: If two unique planes intersect, then they intersect to form exactly ONE line.

1.1 Name 2 planes in the diagram: Where do the two planes intersect? Name a line in plane T: Name a line in plane N:

Name the intersection of those two lines: Name three collinear points: Name three non-collinear points: Name four coplanar points: Name four non-coplanar points: Name a pair of opposite rays: Name one ray in two different ways:

1.2 A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on the ruler. This number is called a coordinate. Ruler postulate: points on a line can be put into a one-to-one correspondence with the real numbers (which are called coordinates).

1.2 The distance between two points is the absolute value of the difference of the coordinates (.

AB = |𝑎 − 𝑏| = |𝑏 − 𝑎| If the two points are connected by a line segment, then the length (measure) of the line segment is the same as the distance between the two points.

𝑚𝐴𝐵̅̅ ̅̅ = 𝐴𝐵 = |𝑎 − 𝑏| = |𝑏 − 𝑎| The measure (m) of a segment is equal to the distance between its two endpoints.

𝑚𝐴𝐶̅̅ ̅̅ = AB =

1.2 Congruent segments (≅ 𝑠𝑒𝑔𝑠) are two segments that are equal in measure (same length). Segments that are congruent are marked with tick marks. Segments that have the same markings are congruent (equal in measure). “measures are equal, segments are congruent”

1.2 Constructions are a way of creating a figure that is more precise and easier than using measuring tools. Constructions using a compass and straight edge have been used for hundreds of years. Recently, constructions using patty (tracing) paper have become popular also.

1.2 Constructing congruent segments: 1. Start by drawing a line to create the congruent segment on. 2. Create your starting point on the line.

3. Use your compass to measure the segment you want to “copy” by putting the sharp end on

one endpoint and the pencil on the other endpoint.

4. Place the sharp point of the compass on your starting point, then lightly draw a curve on the

line to create the endpoint for the congruent segment.

1.2 Betweeness: in order for you to say that a point B is between two points A and C, all three points

must lie on the same line and 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶. Segment Addition Postulate: If B is between points A and C, then 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶

1.2 SAMPLE: Given that M is between N and O. If NM = 17, MO = 3x – 5, and NO = 5x + 2, then find x and the measure of each segment.

1.2 SAMPLE: Construct a segment congruent to 𝑀𝑁̅̅ ̅̅ ̅. Construct a segment that measures MN + PQ

1.2 A midpoint is a point that bisects, or divides a segment into two congruent segments. If M is the

midpoint of 𝐴𝐵̅̅ ̅̅ , then 𝐴𝑀̅̅̅̅̅ ≅ 𝑀𝐵̅̅ ̅̅̅. The shorthand way for writing midpoint is _____. The shorthand way for writing segment bisector is _________________.

1.2 Given: A is the midpoint of 𝐶𝑇̅̅̅̅ , 𝑚𝐶𝐴̅̅ ̅̅ = 5𝑥, 𝑚𝐴𝑇̅̅ ̅̅ = 3𝑥 + 4. Find the measure of each segment.

1.2 Constructing a midpoint or segment bisector: 1. Place the sharp point of the compass on one endpoint. 2. Make sure the pencil is farther than half but less the whole length of the segment. 3. Draw a semi-circle that starts above the endpoint, crosses the segment and ends below the

endpoint. 4. DO NOT CHANGE THE COMPASS SETTING! 5. Place the sharp point of the compass on the other endpoint. 6. Draw a semi-circle that starts above the endpoint, crosses the segment and ends below the

endpoint. 7. The semi-circles will intersect at two places. Use a ruler to connect those two points and draw

a line. 8. The line will cross the original segment at the midpoint.

1.2 Construct the midpoint of 𝑃𝑄̅̅ ̅̅

1.3 An angle is a figure formed by two rays (sides) with a common endpoint (vertex). An angle can be

named in three ways, all of them require the word angle or the angle symbol or then

1. Three letters: a point on one side, the vertex, a point on the other side. **preferred 2. One letter: the vertex (this can only be used if only ONE angle exists with that vertex) 3. One number: some complex diagrams will place a number (1-9 usually) near the vertex in the

interior region of the angle. It is important to note that this number is NOT a measure just a naming tool!

1.3 The measure of an angle measures the rotation of one side to the other around the vertex. This rotation is measured in degrees. We usually use a protractor to measure an angle.

1.3 Protractor Postulate: Given 𝐴𝐵 ⃡ and a point O on 𝐴𝐵 ⃡ , all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180.

1.3 Measuring an angle using a protractor:

1.3 Types of angles based on measure: 1. Acute Angle: measures more than 0 degrees and less than 90 degrees 2. Right Angle: measures 90 degrees 3. Obtuse Angle: measures more than 90 degrees and less than 180 degrees 4. Straight Angle: measures 180 degrees and looks like a straight line

1.3 Congruent angles s are two angles that are equal in measure. So ______________________

can be rewritten as ________________________. Like congruent segments, angles have special markings to show which angles are congruent. These marks are called arc marks.

1.3 Constructing congruent angles: 1. Draw a starting line. Create a point on the line that will be the vertex of your new angle.

2. Go to the original angle you are trying to “Copy”. Draw an arc so it crosses both sides of the

angle. DO NOT MOVE THE SETTING ON THE COMPASS!

2. 3. 4. 3. Go to the starting line and vertex point, use the setting to draw an arc. 4. Go back to the original angle. Remember where the arc crossed the two sides? Put the sharp

point on the crossing on one side and place the pencil on the crossing for the other side (I draw a little arc). DO NOT MOVE THE SETTING ON THE COMPASS!

5. Go to the starting line, put the sharp point where the arc crosses the starting line. Draw an arc that crosses the first arc drawn.

5. 6. 6. Connect the vertex point to the point where the two arcs cross. This arc should be a “copy” or

congruent to the original angle.

1.3 Angle addition postulate: If a point is in the interior of an angle then ray that connects the vertex to the interior point will create two small angles whose measures will add to be the measure of the original angle.

m SQPm RQ m R PS Q

115 , 48 , find .m DEG m DEF m FEG

1.3 A ray that divides an angle into two congruent angles is called an angle bisector.

Given: bisects KM JKL

1.3 Given: bisects KM JKL

4 6m JKM x

7 12m LKM x

Find m JKL

1.3 Constructing an angle bisector:

1. Draw an arc of any size so it crosses both sides of the angle. 2. From one of those intersections draw another arc inside of the angle. KEEP THE COMPASS

SETTING! 3. From the other intersection draw the arc inside the angle so it crosses the last arc. 4. Connect the intersection of those arcs to the vertex of the angle.

1.3 Copy the angle to the left

Bisect the angle to the right

1.4 Adjacent angles are two angles in the same plane with a common vertex and a common side but do

not overlap interior regions. (adj. s )

1.4 Linear Pair of angles is a pair of adjacent angles whose non-common sides are opposite rays.

1.4 Complementary angles are two angles whose measures add to 90 degrees (angles do not have to be adjacent to be complementary)

Comp s

1.4 Supplementary Angles are two angles whose measures add to 180 degrees (angles do not have to be adjacent to be supplementary, but a linear pair is automatically supplementary since they form a straight angle is 180 degrees).

Supp s

1.4 Vertical Angles are formed by two lines intersecting, the two angles that are not adjacent are called vertical angles.

1.4 Name the relationship between the angles:

1. 5 and 7

2. 5 and 8

3. 5 and 6

4. 7 and 6

1.4 An angle is 10° more than 3 times the measure of its complement. Find the measure of the complement.

An angle is 3 degrees less than twice the measure of its complement. Find the measure of its complement.

1.6 The coordinate plane is a plane that is divided into four regions by a horizontal axis (x-axis) and a vertical axis (y-axis). The location, or coordinate, of a point is given by an ordered pair (x,y).

1.6 The midpoint of any segment (or two points) in the coordinate plane can be calculated by the

midpoint formula: 1 2 1 2,2 2

x x y ymp

1.6 Find the midpoint between (-2, -1) and (4, 2) The midpoint of a segment is (-1, 1) and one endpoint is (-6, -1). Find the coordinate of the other endpoint. The midpoint of a segment is (4, -3) and one endpoint is (2, 2). Find the coordinate of the other endpoint.

1.6 In a coordinate plane, the distance (or length of a segment) d between two points (x1, y1) and (x2, y2)

can be calculated by distance formula: 2 2

2 1 2 1d x x y y

1.6 Find mME if M = (3, 2) and E = (-3, -1).

1.6 A right triangle and its parts: A right triangle is a triangle with one right angle The hypotenuse is the side across from the right angle The legs are the remaining sides, the sides that form the right angle.

1.6 The Pythagorean theorem is the relationship between the sides of a right triangle. The sum of the legs squared is equal to the hypotenuse squared. The Pythagorean theorem is also how the distance formula is derived.

2 2 2

2 2

c a b

c a b

1.6 Find AB and CD. Determine if AB CD .

Find FE and GH. Determine if FE GH .

1.7 A transformation is a change in the position, size, or shape of a figure. The original figure is called the preimage. The changed figure (resulting figure) is called the image. A transformation maps the preimage to the image. Arrow (→) notation is used to describe a transformation, and primes (‘) are used to label the image.

1.7 A Translation (slide) is a transformation in which all the points of a figure move the same distance in the same direction. Arrow notation for coordinates will be indicated by adding or subtracting values to the original coordinates. (x, y) → (x + a, y + b)

' ' 'ABC A B C

1.7 A Reflection (flip) is a transformation across a line called the line of reflection. Each point and its image are the same distance from the line of reflection. The most common lines of reflection are the x-axis and the y-axis. Arrow notation for coordinates will be indicated by changing the sign of one of the coordinates. (x, y) → (±x, ±y)

' ' 'ABC A B C

1.7 A Rotation (turn) is a transformation about a point P called the center of rotation. The center of rotation does not move or turn but everything else will rotate around that point. Each point and its image are the same distance from P. The most common rotations are 90 degrees clockwise and 90 degrees counterclockwise. Arrow notation for coordinates will show the x and y values interchanging and changing sign.

EFGH →E’F’G’H’

1.7 Determine each transformation. Use arrow notation to describe the transformation.

1.7 Find the coordinates for image of JKLM after the translation (x, y) → (x - 2, y + 4). Draw the image.

1.7