Geometry
Unit 7: Reasoning and Parallel Lines
(Do Not Lose This)
Name:________________________________________
Syllabus
Lesson 1: Parallel Lines and Related Angles
M. April 4 – T. April 5
Learning Goals:
 Classify Angle Pairs
 Understand Angle Relationships
 Prove Angle Relationships
Classroom Activities:
 Individual Notes over Chapter 7 Section 1: pg. 363 (Instructions in class)
 Whole Class notes over Corresponding Angles Postulate, Alternate Interior Angles
Theorem, Same-side Interior Angles Theorem.
 Jigsaw Proofs for Alternate Interior Angles Theorem and Same-side Interior Angles
Theorem.
Practice Problems: pg. 366 #1-13, 16, 17a, 21-28
Due: Tuesday, April 5
Lesson 2: Proving Lines Parallel
T. April 5 – Th. April 7
Learning Goals:
 Prove Lines Parallel
 Use Algebra and Parallel Lines to Solve Problems
Classroom Activities:
 Teacher Led Notes over Converse Theorems
 Flow Proof over Theorem 7-3
 Pairs: Flow Proof over Theorem 7-4
 Individually: Translate Paragraph proof of pg. 373 into Two-column Proof
 Teacher Led Example: Using Algebra to Solve Problems
Practice Problems: pg. 374 #5-14, 17, 22-28
Due: Thursday, April 7
Lesson 3: Constructions
Th. April 7
Learning Goals:
 Construct Parallel and Perpendicular Lines with a Compass and Straightedge
Classroom Activities:
 From Chapter 7 Section 3:
◦
Construct Parallel through a Point Not on a Line
◦
Construct Perpendicular through a Point on a Line
◦
Construct Perpendicular through a Point not on a Line
◦
Construct a Trapezoid using two, given bases.
Practice Problems: None
Quiz:
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Lesson 1 through Lesson 3
Friday. April 8
Classify Angle Pairs
Understand Angle Relationships
Prove Angle Relationships
Prove Lines Parallel
Use Algebra and Parallel Lines to Solve Problems
Construct Parallel and Perpendicular Lines with a Compass and Straightedge
Lesson 4: Parallel Lines and Perspective Drawing
M. April 11
Learning Goals:
 Understand the use of Parallel Lines in Art
Classroom Activities:
 Comparing Pre-Renaissance Art to Renaissance Art
 Drawing 3-D Objects using one-point and two-point perpectives
Practice Problems: None
Lesson 5: Exploring Spherical and Hyperbolic Geometries
T. April 12 – W. April 13
Learning Goals:
 Understand the difference between Euclidean Geometry, Spherical Geometry, and Hyperbolic
Geometry
 Apply Knowledge of the three geometries to solve problems and draw conclusions.
Classroom Activities:
 Teacher Led Discussion over assumptions and how different assumptions lead to different
conclusions
 Compare Euclidean to Spherical Geometries: Terms and definitions
 Both Parallel Postulates
 Two-Models of Hyperbolic Geometry and Properties of Hyperbolic Geometry
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Comparison Table of Properties of Euclidean, Spherical, and Hyperbolic Geometries
Practice Problems:
“Exploring Spherical and Hyperbolic Geometries” WS
Reviewing Reasoning and Parallel Lines
Due: Thursday, April 14
Th. April 14
Classroom Activities
 Small group work on End of Chapter Review
 Complete pg. 399 #1-9, 12-18
Test:
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Unit 7 Reasoning and Parallel Lines
Friday, April 15
Classify Angle Pairs
Understand Angle Relationships
Prove Angle Relationships
Prove Lines Parallel
Use Algebra and Parallel Lines to Solve Problems
Understand the use of Parallel Lines in Art
Understand the difference between Euclidean Geometry, Spherical Geometry, and Hyperbolic
Geometry
 Apply Knowledge of the three geometries to solve problems and draw conclusions.
Use this space to write questions or misunderstandings
from the lessons or practice problems
Self Assessment
My Copy
Scale: 1 – 4
Name:___________________________
I will be able to:
Learning Goals
Date(s) of
Introduction
Level of
Knowledge
(Pre)
Level of
Knowledge
After Lesson
and Practice
Level of
Knowledge
After Quiz
Classify Angle Pairs
Understand Angle
Relationships
Prove Angle Relationships
Prove Lines Parallel
Use Algebra and Parallel
Lines to Solve Problems
Construct Parallel and
Perpendicular Lines with a
Compass and Straightedge
Understand the use of
Parallel Lines in Art
Understand the difference
between Euclidean
Geometry, Spherical
Geometry, and Hyperbolic
Geometry
Apply Knowledge of the
three geometries to solve
problems and draw
conclusions.
Quiz 1 Grade ____
Test Grade: ____
Level of
Knowledge
After Test
Self Assessment
Mr. Gaertner's Copy
Scale: 1 – 4
Name:___________________________
Students will be able to will be able to:
Learning Goals
Date(s) of
Introduction
Level of
Knowledge
(Pre)
Level of
Knowledge
After Lesson
and Practice
Level of
Knowledge
After Quiz
Classify Angle Pairs
Understand Angle
Relationships
Prove Angle Relationships
Prove Lines Parallel
Use Algebra and Parallel
Lines to Solve Problems
Construct Parallel and
Perpendicular Lines with a
Compass and Straightedge
Understand the use of
Parallel Lines in Art
Understand the difference
between Euclidean
Geometry, Spherical
Geometry, and Hyperbolic
Geometry
Apply Knowledge of the
three geometries to solve
problems and draw
conclusions.
Quiz 1 Grade ____
Test Grade: ____
Level of
Knowledge
After Test
Ohio's Benchmarks and Standards
A. Formally define geometric figures
1. Formally define and explain key aspects of geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median, altitude, midsegment);
c. points of concurrency related to triangles (centroid, incenter, orthocenter, and
circumcenter);
d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment,
inscribed angle).
E. Draw and construct representations of two- and three-dimensional geometric objects
using a variety of tools, such as straightedge, compass and technology.
H. Establish the validity of conjectures about geometric objects, their properties and
relationships by counter-example, inductive and deductive reasoning, and critiquing
arguments made by others.
03. Make, test and establish the validity of conjectures about geometric properties and
relationships using counterexample, inductive and deductive reasoning, and paragraph or
two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle similarity and congruence;
c. prove theorems involving properties of lines, angles, triangles and quadrilaterals;
d. test a conjecture using basic constructions made with a compass and straightedge or
technology.
B. Apply mathematical knowledge and skills routinely in other content areas and practical
situations.
D. Apply reasoning processes and skills to construct logical verifications or counterexamples to test conjectures and to justify and defend algorithms and solutions.
G. Write clearly and coherently about mathematical thinking and ideas
H. Locate and interpret mathematical information accurately, and communicate ideas,
processes and solutions in a complete and easily understood manner.
Exploring Spherical and Hyperbolic Geometries
Name:_________________________
Draw a sketch to illustrate each property of spherical geometry.
1. There are pairs of points on a sphere
through which more than one line can be
drawn.
2. A triangle can have more than one right
angle.
Draw a Poincare-Disc Model to demonstrate each
3. A triangle.
4. Through line l, there exists two lines n
and m through point P not on line l such
that n and m do not intersect line l.
Draw a counterexample to show that the following properties of Euclidean geometry
are not true in spherical geometry.
5. If a triangle contains a right angle, then
the other two angles in the triangle are
complementary.
6. Through a point not on line l, there
exists one and only one line perpendicular
to l.
7. Complete #14 on pg. 396.