Chapter 1 Essentials of Geometry
1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures so you can use geometry terms in the real world. Essential Question: How do you name geometric figures? Common Core: CC.9-12.G.CO.1
Vocabulary: Point: a point has no dimension. It is represented by a dot. A Line: A line has one dimension. Through any two points, there is exactly one line. B l A
B M A C Plane: a plane has two dimensions. It is represented by a shape that looks like piece of paper, but extends without end. Through any 3 points not on the same line, there is exactly one plane. Use three points to name a plane.
EXAMPLE 1 Name points, lines, and planes a. Give two other names for PQ and for plane R. b. Name three points that are collinear. c. Name four points that are coplanar.
Rays and line segments are parts of lines. A has a definite beginning and end. A line segment is part of a line containing two endpoints and all points between them. A line segment is named using its endpoints. The line segment is named segment AB or segment BA A B The symbol for segment AB is AB
Rays and line segments are parts of lines. A has a definite starting point and extends without end in one direction. A B RAY: The starting point of a ray is called the. A ray is named using the endpoint first, then another point on the ray. The ray above is named ray AB. The symbol for ray AB is AB
are two rays that are part of a the same line and have only their endpoints in common. Y X Z XY and XZ are. The figure formed by opposite rays is also referred to as a.
1) Name two segments. Possible Answers: A B C D 2) Name a ray. Possible Answers: U
Intersecting planes practice:
Homework: 1.1 Exercises Concepts: #1-11, 12-38 even, 40 43 Regular: # 1 11, 12-20 even, 21-43 Honors: # 1 44
1.2 Use Segments and Congruence Objective: Use segment postulates to identify congruent segments Essential Question: What are congruent segments? Common Core: CC.9-12.G.CO.1 CC.9-12.G.CO.7
Postulate: A rule that is accepted without proof. aka: axiom
The distance between two points A and B on a number line can be found by using the Ruler Postulate. Ruler Postulate The points of a line can be put into one-toone correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers. B b measure = a b A a
Congruent Statements In geometry, two segments with the same length are called Definition of Congruent Segments Two segments are congruent if and only if they have the same length 6/20/2011 LSowatsky 15
In the figures at the right, AB is congruent to BC, and PQ is A B C congruent to RS. The symbol is used to represent congruence. AB BC, and PQ RS. R
Measuring Segments and Angles Use the number line to determine if the statement is True or False. Explain you reasoning. RS TY R S T Y x -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 11 Because RS = 4 and TY = 5, RS TY So, RS is not congruent to TY, and the statement is false.
Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC 6/20/2011 LSowatsky 18
If AB = 25, find the value of x. Then find AN and NB. Use the Segment Addition Postulate to write an equation. AN + NB = AB Segment Addition Postulate
Homework: 1.2 Exercises Concepts: 1 30, 32 34 Regular: #1 30, 32-35 Honors: #1 36 *** skip #3-#5 (measuring)
1.3 Use Midpoint and Distance Formulas Objective: Find the lengths of segments in the coordinate plane Essential Question: How do you find the midpoint and distance of two points on the coordinate plane? Common Core: CC.9-12.G.GPE.7
You will learn to find the coordinates of the midpoint of a segment. A C B AB The midpoint of a line segment,, is the point C that bisects the segment. A C B -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 C = [3 + (-5)] 2 = (-2) 2 = -1
M is the midpoint of RT. Find RM, MT, and RT. Use the definition of midpoint to write an equation. RM = MT Definition of midpoint
Find the midpoint, C(x, y), of a segment on the coordinate plane. Consider the x-coordinate: 10 9 8 7 y x = 1 x = 9 A y = 7 6 5 4 y C(x, y) Consider the y-coordinate: 3 2 1 0 x -1 x y = 3 B -2-2 -1 1 2 3 4 5 6 7 8 9 10 0
The Midpoint Formula The coordinates of the midpoint M of AB with endpoints A(x 1, y 1 ) and B(x 2, y 2 ) are M x 1 x 2 1 2, y y 2 2 y ( x1, y1) x x, y y 2 2 1 2 1 2 ( x, y ) 2 2 O x
Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B. x-coordinate of B y-coordinate of B 10 y 9 B(-1, 8) 8 7 6 5 4 3 2 B(x, y) is somewhere over there. midpoint C(3, 5) A(7, 2) 1 0 x -1-2 -2-1 1 2 3 4 5 6 7 8 9 10 0
The Distance Formula The distance d between two points A (x 1, y 1 ) and B (x 2, y 2 ) is d 2 2 x x y y 2 1 2 1 6/20/2011 LSowatsky 27
Example: Find the distance between R (-2, 6) and S (6, -2) to the nearest tenth. Note: be careful using your calculator use at the very end. 6/20/2011 LSowatsky 28
Homework: 1.3 Exercises Concepts: #1, 2 24 even, 25 39 odd, 48, 49 Regular: #1, 2 24 even, 25 39 odd, 48-52 Honors: #1, 2 22 even, 23, 24 42 even, 48-52
1.4 Measure and Classify Angles Objective: Name, measure, and classify angles to identify congruent angles. Essential Question: How do you identify if an angle is acute, obtuse, right, or straight? Common Core: CC.9-12.G.CO.1 CC.9-12.G.CO.7
There is another case where two rays can have a common endpoint. This figure is called an. S Some parts of angles have special names. The common endpoint is called the, and the two rays that make up the sides of the angle are called the sides of the angle. R vertex side T
There are several ways to name this angle. 1) Use the vertex and a point from each side. or The vertex letter is always in the middle. S 2) Use the vertex only. If there is only one angle at a vertex, R then the angle can be named with that vertex. vertex 1 side T 3) Use a number.
Definition of Angle An angle is a figure formed by two noncollinear rays that have a common endpoint. E 2 D Symbols: DEF FED E 2 F Naming Angles (Video)
1) Name the angle in four ways. A C 1 2) Identify the vertex and sides of this angle. vertex: sides: B
Protractor Postulate For every angle, there is a unique positive number between 0 and 180 called the degree measure of the angle. A m ABC = n and 0 < n < 180 B n C
You can use a protractor to measure angles and sketch angles of given measure. 1) Place the center point of the protractor on vertex R. Align the straightedge with side RS. 2) Use the scale that begins with 0 at RS. Read where the other side of the angle, RQ, crosses this scale. Q R S
Find the measurement of: m SRQ = m SRJ = m SRH m QRG = m SRG = m GRJ = H G J Q R S
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 obtuse angle 90 < m A < 180 A right angle m A = 90 A acute angle 0 < m A < 90
Classify each angle as acute, obtuse, or right. 110 90 40 Obtuse Right Acute 50 130 75 Acute Obtuse Acute
The measure of B is 138. Solve for x. 5x - 7 B The measure of H is 67. Solve for y. H 9y + 4 Given: (What do you know?) B = 5x 7 and B = 138 Given: (What do you know?) H = 9y + 4 and H = 67
For any angle PQR, if A is in the interior of PQR, then m PQA + m AQR = m PQR. Angle Addition Postulate Q 1 2 P A m 1 + m 2 = m PQR. R There are two equations that can be derived using this postulate. m 1 = m PQR m 2 m 2 = m PQR m 1 These equations are true no matter where A is located in the interior of PQR.
Find m 2 if m XYZ = 86 and m 1 = 22. X W 1 2 Y Z
Find m ABC and m CBD if m ABD = 120. m ABC + m CBD = m ABD C D 2x (5x 6) A B
Congruent Angles Angles with the same measure are If m 1 m 2, then 1 2 Angle Bisector: a ray that divides an angle into two congruent angles 6/20/2011 LSowatsky 44
Homework: 1.4 Exercises Concepts: #1, 2, 3 39odd, 51 Regular: #1, 2 48 even, 51, 52 Honors: #1, 2 50 even, 51, 52
1.5 Describe Angle Pair Relationships Objective: Use special angle relationships to find angle measures Essential Question: How do you identify complementary and supplementary angles? Common Core: CC.9-12.G.CO.1 CC.9-12.G.CO.9
Definition of Congruent Angles Two angles are congruent iff, they have the same. 50 B V 50 B V iff m B = m V 7/7/2011 LSowatsky 47
To show that 1 is congruent to 2, we use. 1 2 To show that there is a second set of congruent angles, X and Z, we use double arcs. X This arc notation states that: Z X Z m X = m Z 7/7/2011 LSowatsky 48
Two angles are complementary if and only if (iff) the sum of their degree measure is 90. Definition of Complementary Angles B A 30 C D 60 E F m ABC + m DEF = 30 + 60 = 90 7/7/2011 LSowatsky 49
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 180. Definition of Supplementary C D Angles A 50 B E 130 F m ABC + m DEF = 50 + 130 = 180 7/7/2011 LSowatsky 50
Definition of Adjacent Angles R Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common 2 1 J 1 and 2 are adjacent with the same vertex R and common side RM N 7/7/2011 LSowatsky 51
When two lines intersect, angles are formed. There are two pair of nonadjacent angles. These pairs are called. 4 1 3 2 7/7/2011 LSowatsky 52
Definition of Vertical Angles Two angles are vertical iff there are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: 4 1 3 2 1 and 3 2 and 4 7/7/2011 LSowatsky 53
Vertical angles are congruent. Vertical Angle Theorem m 1 2 4 3 n 1 3 2 4 7/7/2011 LSowatsky 54
Definition of Linear Pairs Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays A D B 1 2 1 and 2 are a linear pair. BA and BD form AD 1 2 180 7/7/2011 LSowatsky 55
1) If m 1 = 4x + 3 and the m 3 = 2x + 11, then find the m 3 1 4 3 2 2) If m 2 = x + 9 and the m 3 = 2x + 3, then find the m 4 3) If m 2 = 6x - 1 and the m 4 = 4x + 17, then find the m 3 4) If m 1 = 9x - 7 and the m 3 = 6x + 23, then find the m 4 7/7/2011 56
Find the value of x in the figure: The angles are vertical angles. 130 x So, the value of x is 130. 7/7/2011 LSowatsky 57
Find the value of x in the figure: The angles are vertical angles. (x 10) 125 (x 10) = 125. x 10 = 125. x = 135. 7/7/2011 LSowatsky 58
Suppose two angles are congruent. What do you think is true about their complements? 1 2 1 + x = 90 2 + y = 90 x is the complement of 1 x = 90-1 x = 90-1 x = y y is the complement of 2 y = 90-2 Because 1 2, a substitution is made. y = 90-1 x y If two angles are congruent, their complements are congruent. 7/7/2011 LSowatsky 59
Homework: 1.5 Exercises Concepts: #1, 2 52 even Regular: #1, 2 52 even Honors: #1, 2 52 even
1.6 Classify Polygons Objective: Classify polygons Essential Question: How do you classify polygons? Common Core: CC.9-12.G.GMD.4 CC.9-12.G.MG.1
A polygon is a in a plane formed by segments, called sides. A polygon is named by the number of its or. A triangle is a polygon with three sides. The prefix means three.
Prefixes are also used to name other polygons. Prefix tri- quadri- penta- hexahepta- octa- nona- deca- Number of Sides 3 4 5 6 7 8 9 10 Name of Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon
A vertex is the point of intersection of two sides. P Q Consecutive vertices are the two endpoints of any side. A segment whose endpoints are nonconsecutive vertices is a diagonal. U T S R Sides that share a vertex are called consecutive sides.
An equilateral polygon has all congruent. An equiangular polygon has all congruent. A regular polygon is both and. equilateral but not equiangular equiangular but not equilateral regular, both equilateral and equiangular Investigation: As the number of sides of a series of regular polygons increases, what do you notice about the shape of the polygons?
A polygon can also be classified as convex or concave. If all of the diagonals lie in the interior of the figure, then the polygon is. If any part of a diagonal lies outside of the figure, then the polygon is.
Example: Classify the polygon shown at the right by the number of sides. Explain how you know that the sides of the polygon are congruent and that the angles of the polygon are congruent.
Example: A table is shaped like a regular hexagon. The expressions shown represent side lengths of the hexagonal table. Find the length of a side. First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent.
Homework: 1.6 Exercises Concepts: #1 30, 32 36 Regular: #1 30, 32-36 Honors: # 1 30, 32 36, 39, 40
Chapter 1 Test