Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 1
Revised Fair 2014-2015
Lesson 1 - Basic Terms of Geometry (p.10)
Basic Terms of Geometry v Point – ____________________________________________________________________
How do you name a point? ____________________________________________________ v Space - ____________________________________________________________________ v Line - _____________________________________________________________________
How do you name a line? ______________________________________________________ v Collinear Points - ___________________________________________________________
– Identifying Collinear Points a. Are points E, F, and C collinear? b. Are points F, P, and C collinear?
c. Name line m in three other ways d. Why do you think arrowheads are used when drawing a line or naming a line such as EF ?
– Naming a Plane a. Name the plane represented by the front of the ice cube? b. Name the plane represented by the top of the ice cube?
Basic Postulates of Geometry v Postulate 1-1 – Through any two points there is exactly one line.
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 2
Revised Fair 2014-2015
v Postulate 1-2 – If two lines intersect, then they intersect in exactly one point.
What are the three ways to solve a System of Equations?
a.________________________ b.________________________ c.________________________ How many solutions are there in the
System of Equations shown in the graph? v Postulate 1-3 – If two planes intersect, then they intersect in exactly one line.
- Finding the Intersection of Two Planes a. What is the intersection of plane HGFE and plane BCGF? b. What two planes intersect in BF ? c. What is the intersection of plane ADCB and DH ? v Postulate 1-4 – Through any three non-collinear points there is exactly one plane.
- Using Postulate 1-4 a. Name another point that is in the same plane as points A, B, and C; then shade the plane.
b. Name another point that is in the same plane as points E, H, and C; then shade the plane.
Algebra I Review
A B C D
E F
G H
A B C D
E F
G H
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 3
Revised Fair 2014-2015
Lesson 2 - Segments, Rays, Parallel Lines and Planes (p. 17)
Identifying Segments and Rays v Segment – ________________________________________________________________
What is the symbol for a segment? ______________________________________________ How do you name a segment? __________________________________________________
v Ray - ___________________________________________________________________
What is the symbol for a ray? _________________________________________________ How do you name a ray? _____________________________________________________
v Opposite Rays - ___________________________________________________________
What is the symbol for a line? _________________________________________________ How do you name a line? _____________________________________________________
– Naming Segments and Rays Name the segments and rays in the figure at the right. a. Name three segments.
b. Name four rays. c. and form a line. Are they opposite rays? Explain.
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 4
Revised Fair 2014-2015
Recognizing Parallel Figures v Parallel Lines – _____________________________________________________________
What is the symbol for parallel? ________________________________________________ How do you name parallel lines? ________________________________________________
v Skew Lines - _______________________________________________________________
What is the symbol for skew? ________________________________________________ How do you name skew lines? ________________________________________________
– Identifying Parallel and Skew Segments a. Name all labeled segments that are parallel to . b. Name all labeled segments that are skew to .
v Parallel Planes – ____________________________________________________________
How do you name parallel planes?_______________________________________________
- Identifying Parallel Planes a. Name the pairs of parallel planes in the figure below.
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 5
Revised Fair 2014-2015
Lesson 3 - Measuring Segments and Angles (p. 25)
Finding Segment Lengths v Postulate 1-5 (Ruler Postulate) – The points of a line can be put into one-to-one
correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.
v Congruent Segments - _______________________________________________________
What is the symbol for congruence?______________________________________________ How do you write that two segments are congruent? ________________________________
– Comparing Segment Lengths a. Find AB and BC . v Postulate 1-6 (Segment Addition Postulate) – If three points A, B, and C are collinear and
B is between A and C, then ACBCAB =+ .
– Using the Segment Addition Postulate a. If DT =60, find the value of x. Then find DS and ST . b. If EG=100, find the value of x. Then find EF and FG .
E F G
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 6
Revised Fair 2014-2015
v Midpoint – ________________________________________________________________ What does a midpoint do to a segment?______________________________________
- Finding Lengths C is the midpoint of . Find AC , CB , and AB .
Finding Angle Measures v Angle - _________________________________________________________
What is the symbol for an angle? ________________________________________________
How do you name an angle? _____________________________ How are angles measured?_______________ How do you represent this?_______
Naming Angles a. Name 1Ð in two other ways. b. Would it be correct to name any of the angles EÐ ?
Explain. v Postulate 1-7 (Protractor Postulate) – Let and be opposite rays in a plane. , ,
and all the rays with endpoint O that can be drawn on one side of AB can be paired with the real numbers from °0 to °180 so that…
a. is paired with °0 and is paired with °180 . b. If is paired with x and is paired with y, then m yxCOD -=Ð .
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 7
Revised Fair 2014-2015
v You can classify angles according to their measures.
Measuring and Classifying Angles Find the measure of each angle. Classify each angle as acute, right, obtuse, or straight.
v Postulate 1-8 (Angle Addition Postulate) -
Using the Angle Addition Postulate a. What is m TSWÐ if m RSTÐ =50 and m RSWÐ =125. b. What is m DEGÐ =145, find m GEFÐ . c. v Congruent Angles - _________________________________________________
How do you write that two angles are congruent
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 8
Revised Fair 2014-2015
Lesson 4 - The Coordinate Plane (p. 43)
Finding Distance on the Coordinate Plane
v The Distance Formula – The distance d between two points ( )11,yxA and ( )22,yxB is
– Finding Distance a. Find the distance between T (5, 2) and R (-4, -1) to the nearest tenth.
– Real-World Connection a. Each morning Juanita takes the “Blue Line” subway from Oak Station
to Jackson Station. As the map shows, Oak Stations is 1 mile west and 2 miles south of City Plaza. Jackson Station is 2 miles east and 4 miles north of City Plaza. Find the distance Juanita travels between Oak Station and Jackson Station.
Quadrant ( , )
Quadrant ( , )
Quadrant ( , )
Quadrant ( , )
-axis
-axis
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 9
Revised Fair 2014-2015
Finding the Midpoint of a Segment
v To find the midpoint of a segment, you simply average or find the mean of the coordinates of the endpoints.
v The Midpoint Formula – The coordinates of the midpoint M of with endpoints A (x1,
y1) and B (x2, y2) are the following:
- Finding the Midpoint a. QS has endpoints Q (3, 5) and S (7, -9). Find the coordinates of its midpoint M.
Finding an Endpoint a. The midpoint of is M (3, 4). One endpoint is A (-3, -2). Find the coordinates of the other endpoint B.
b. The midpoint of XY has coordinates (4, -6). X has coordinates (2, -3). Find the coordinates of Y.
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 10
Revised Fair 2014-2015
Lesson 5 - Lines in the Coordinate Plane (p. 152)
Graphing Lines v Slope-Intercept Form - ________________________________________________
- Graphing Lines in Slope-Intercept Form
a. Graph the line 243
+= xy
v Standard Form of a Linear Equation - ___________________________________
– Graphing Lines Using Intercepts a. Graph 6x + 3y = 12
b. Graph -2x + 4y = -8
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 11
Revised Fair 2014-2015
- Transforming to Slope-Intercept Form a. Rewrite the equation in slope-intercept form 4x – 2y = 9.
b. Rewrite the equation in slope-intercept form -5x + y = -3 v Point-Slope Form - ____________________________________________________
Writing Equations of Lines
- Using Point-Slope Form a. Write the equation of a line through the point P(-1, 4) with m=3.
- Writing an Equation of a Line Give Two Points a. Write an equation of a line through A(-2, 3) and B(1, -1).
b. Write an equation of a line through P(5, 0) and Q(7, -3).
- Equations of Horizontal and Vertical Lines a. Write the equations for the horizontal line and the vertical line that contain P(3, 2)
Find slope 1st!
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 12
Revised Fair 2014-2015
Lesson 6 - Slopes of Parallel and Perpendicular Lines (p. 158)
Slope and Parallel Lines v Slopes of Parallel Lines • If two non-vertical lines are parallel, their slopes are equal. • If the slopes of two distinct non-vertical lines are equal, the lines are parallel. • Any two vertical lines are parallel.
- Checking for Parallel Lines a. Are lines l1 and l2 parallel?
v Slope-intercept form allows you to compare slopes easily in order to determine whether
slopes are parallel.
– Determining Whether Lines are Parallel a. Are the lines 4y – 12 x = 20 and y = 3x -1 parallel? Explain.
- Writing Equations of Parallel Lines a. Write an equation for the line parallel to y = -4x + 3 that contains (1, -2).
b. Write an equation for the line parallel to y = -x + 4 that contains (-2, 5).
Unit 1 – Tools of Geometry
Lesson 1 – Points, Lines, & Planes Page 13
Revised Fair 2014-2015
Slope and Perpendicular Lines v Slopes of Perpendicular Lines • If two non-vertical lines are perpendicular, the product of their slopes is -1. • If the slopes of two lines have a product of -1, the lines are perpendicular. • Any horizontal line and any vertical lines are perpendicular.
- Checking for Perpendicular Lines a. Are lines l1 and l2 perpendicular? Explain.
- Writing an Equations of Perpendicular Lines c. Write an equation of a line perpendicular to y = -3x – 5 that contains (-3, 7).
d. Write an equation of a line perpendicular to 5y – x = 10 that contains (15, -4).