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

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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).