Overview
A required course, which follows Algebra I, Geometry, is an introduction to geometric concepts and reasoning through formal proof and informal investigation and discovery. Students learn to visualize and analyze geometric relationships in two and three dimensions and to develop inductive and deductive reasoning skills.
Objectives
The student will:
A. Use
trigonometric functions to solve triangle problems
B. Define
circles, spheres, arcs, tangents, radii, chords of a circle, and central angles
and apply theorems and problems related to them
C. Solve
problems and prove statements involving inscribed angles and angles formed by
chords, secants, and tangents
D. Know and use
the formulas for the areas of a rectangle, parallelogram, triangle, rhombus,
and regular polygon
E. Find the
ratio of the areas of two triangles
F. Apply the
relationships of scale factors, perimeters and areas of similar figures
G. Find the
lateral areas, total areas, and volumes of right prisms, regular pyramids,
right cylinders, and right cones
H. Find the
area and the volume of a sphere
I. Apply the
properties of similar solids
Content
A. Points, Lines, Planes, and Angles
1. Use the terms equidistant, point, line, plane, collinear,
coplanar, and intersection.
2. Draw representations of points, lines, and planes.
3. Use symbols for lines, segments, rays, and distances.
4. Find distances.
5. State and use the Segment Addition Postulate.
6. Name angles and find their measures.
7. State and use the Angle Addition Postulate.
8. Use postulates and theorems relating points, line, and planes.
B. Deductive Reasoning
1. Recognize the hypothesis and the conclusion of an if-then
statement.
2. State the converse of an if-then statement.
3. Use a counterexample to disprove an if-then statement.
4. Understand the meaning of if and only if.
5. Use properties from algebra and properties of congruence in
proofs.
6. Use the Midpoint Theorem and the Angle Bisector Theorem.
7. Apply the definitions of complementary and supplementary angles.
8. State and use the theorem about vertical angles.
9. Apply the definitions and theorems about perpendicular lines.
10. State and apply the theorems about angles supplementary to, or
complementary to, congruent angles.
11. Plan proofs and then writes them in two-column
form.
C. Parallel Lines and Planes
1. Distinguish between intersecting lines, parallel lines, and skew
lines.
2. State and apply theorems about the intersection of two parallel
planes by a third plane.
3. Identify the angles formed when two lines are cut by a
transversal.
4. State and apply postulates and theorems about parallel lines.
5. State and apply theorems about a parallel and a perpendicular to a
given line through a point outside a line.
6. Classify triangles according to sides and to angles.
7. State and apply theorems and corollaries about the sum of the
measures of the angles of a triangle.
8. State and apply theorems about the measure of an exterior angle of
a triangle.
9. Recognize and name convex polygons and regular polygons.
10. Find the measures of interior angles and exterior angles of convex
polygons.
11. Use inductive reasoning.
D. Congruent Triangles
1. Identify the corresponding parts of congruent figures.
2. Prove two triangles congruent by using the SSS Postulate, the SAS
Postulate, and the ASA Postulate.
3. Deduce information about segments and angles after proving that
two triangles are congruent.
4. Apply theorems and corollaries about isosceles triangles.
5. Use the AAS Theorem to prove two triangles congruent.
6. Use the HL Theorem to prove two right triangles congruent.
7. Prove that two overlapping triangles are congruent.
8. Prove two triangles congruent by first proving two other triangles
congruent.
9. Apply the definitions of the median and the altitude of a triangle
and the perpendicular bisector of a segment.
10. State and apply theorems about a point on the perpendicular bisector
of a segment, and the converse.
11. State and apply theorems about a point on the
bisector of an angle, and the converse.
E. Quadrilaterals
1. Apply the definition of a parallelogram and the theorems about
properties of a parallelogram.
2. Prove that certain quadrilaterals are parallelograms.
3. Apply theorems about parallel lines.
4. Apply the midpoint theorems for triangles.
5. Apply the definitions and identify the special properties of a
rectangle, a rhombus, and a square.
6. Determine when a parallelogram is a rectangle, rhombus, or square.
7. Apply the definitions and identify the properties of a trapezoid
and an isosceles trapezoid.
F. Inequalities in Geometry
1. Apply properties of inequality to positive numbers, lengths of
segments, and measures of angles.
2. State and use the Exterior Angle Inequality Theorem.
3. State the contra positive and inverse of an if-then statement.
4. Understand the relationship between logically equivalent
statements.
5. Write indirect proofs in paragraph form.
6. State and apply the inequality theorems and corollaries for one
triangle.
7. State and apply the inequality theorems for two triangles.
G. Similar Polygons
1. Express a ratio in simplest form.
2. Solve for an unknown term in a given proportion.
3. Express a given proportion in an equivalent form.
4. State and apply the properties of similar polygons.
5. Use the AA Similarity Postulate to prove triangles similar.
6. Use similar triangles to deduce information about segments or
angles.
7. Use the SAS Similarity Theorem and the SSS Similarity Theorem to
prove triangles similar.
8. Apply the Triangle Proportionality Theorem and its corollary.
9. State and apply the Triangle Angle-Bisector Theorem.
H. Right Triangles
1. Determine the geometric mean between two numbers.
2. State and apply the Pythagorean Theorem.
3. State and apply the converse of the Pythagorean Theorem and
related theorems about obtuse and acute triangles.
4. Determine the lengths of two sides of a 45o-45o-90o
or a 30o-60o-90o triangles when the length of
the third side is known.
5. Define the tangent, sine, and cosine ratios for an acute angle.
6. Solve right triangle problems by using the tangent, sine, and
cosine ratios.
I. Circles
1. Define a circle, a sphere, and terms related to them.
2. Recognize inscribed polygons and circles and circumscribed
polygons and circles.
3. Apply theorems that relate tangents and radii.
4. Define and apply properties of arcs and central angles.
5. Apply theorems about the chords of a circle.
6. Solve problems and prove statements involving inscribed angles and
angles formed by chords, secants, and tangents.
7. Solve problems involving lengths of chords, secant segments, and
tangent segments.
J. Areas of Plane Figures
1. Understand the area postulates and what is meant by the area of a
polygon.
2. Know and use the formula for the area of rectangles,
parallelograms, triangles, rhombuses, and trapezoids.
3. Know and use the formula for the areas of regular polygons.
4. Know and use the formulas for the circumferences and areas of
circles.
5. Know and use the formulas for arc lengths and the areas of sectors
of a circle.
6. Find the ratio of the areas of two triangles.
7. Understand and apply the relationship between scale factors,
perimeters, and areas of similar figures.
8. Use areas to solve problems involving geometric probability.
K. Areas and Volumes of Solids
1. Identify the parts of prisms.
2. Find the lateral areas, total areas, and volumes of right prisms.
3. Identify the parts of pyramids.
4. Find lateral areas, total areas, and volumes of regular pyramids.
5. Find the lateral areas, total areas, and volumes of right
cylinders and right cones.
6. Find the area and the volume of a sphere.
7. State and apply the properties of similar solids.
L. Coordinate Geometry (as time allows)
1. State and apply the distance formula.
2. State and apply the general equation of a circle.
3. State and apply the slope formula.
4. Use slope to determine whether two lines are parallel,
perpendicular, or neither.
5. Understand the basic properties of vectors.
6. State and apply the midpoint formula.
7. Identify the slope and y-intercept of the line specified by a
given equation.
8. Draw the graph of the line specified by a given equation.
9. Determine the intersection of two lines.
10. Write the equation of a line when given either one point and the
slope of the line, or two points on the line.
11. Given a polygon, choose a convenient placement of coordinate axes
and assign appropriate coordinates.
12. Prove statements by using coordinate geometry methods.
Methodology
The students check their homework answers and ask questions. Most explanations of new material begin with lecture as the students copy the information from the board or receive note handouts. Then the teacher and students engage in class discussion about the topic and may participate in a “hands-on” activity for further understanding. Students then are asked to work several practice problems, many times in small groups. Students are assigned homework each night.
Evaluation
Major tests are given at the end of each chapter. Announced
quizzes, pop quizzes, notebook checks, and special project assignments are
given periodically within each chapter. Daily
homework assignments are checked for completeness every day.
Resources
Textbook Resources:
Jurgensen, Ray C., Richard G.
Brown, and John W. Jurgensen. Geometry: Teacher’s Edition.
Jurgensen, Ray C., Richard G.
Brown, and John W. Jurgensen. Geometry: Study Guide for Reteaching
and Practice.
Jurgensen, Ray C., Richard G.
Brown, and John W. Jurgensen. Geometry: Resource Book.
Jurgensen, Ray C., Richard G.
Brown, and John W. Jurgensen. Geometry: Tests.
McDougal Littell, 1997.
Jurgensen, Ray C., Richard G.
Brown, and John W. Jurgensen. Geometry: Practice Masters.
Supplemental Resources:
Abbott, Edwin A.
Flatland: A Romance of Many Dimensions.
Bass, Laurie E., et al.
Geometry: Tools for a Changing
World.
Bennett, Dan. Exploring
Geometry With The Geometer’s Sketchpad.
Serra, Michael. Discovering Geometry: An Inductive Approach.
Serra, Michael. Patty Paper Geometry.
Technological Resources:
Geometer’s Sketchpad.
TI-83 Graphing Calculator.
TI-83 Graphing Calculator View Screen Panel.
Student Resources:
Jurgensen, Ray C., Richard G.
Brown, and John W. Jurgensen. Geometry.
TI-83+ Graphing Calculator.