Wayne County Public Schools
Foundations of Geometry
WCPS Mathematics Standard Course of Study
Foundations of Geometry maintains the study of algebraic concepts and develops the study of geometric concepts. It is
designed for students who need additional preparation before they take Geometry. Beginning with the basic elements of
Geometry, the course allows time for students to develop reasoning skills for using definitions, properties, postulates, and
theorems to solve problems. Students will move from an inductive approach to deductive methods of reasoning in their
study of triangles. Appropriate technology, from manipulatives to graphing calculators, should be used regularly for
instruction and assessment. Foundations of Geometry will prepare the student for Geometry.
This course does not count as a math credit for graduation, but it will count as an elective credit for graduation.
Major Concepts/Skills
 Inductive Reasoning
 Basic Elements of Geometry
 Logic and Deductive Reasoning
 Properties and Relationships of Angles
 Properties and Relationships of Lines
 Parallel and Perpendicular Lines
 Congruent Triangles
 Properties of and Relationships within Triangles
 Similar Triangles
 Right Triangles
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
Concepts/Skills to Maintain
▪ Ratios and proportions
▪ Real Numbers
▪ Evaluating Algebraic Expressions
▪ Solving Equations
▪ Graphing Linear Functions
▪ Slope
▪ Pythagorean Theorem
▪ Perimeter and Area of
a Rectangle, a Triangle, or a Square
▪ Circumference and Area of a Circle
1
Wayne County Public Schools
Foundations of Geometry
WCPS Mathematics Standard Course of Study
Number &
Operations
The learner will perform
operations with real numbers &
expressions to solve problems.
1.01 Develop number
sense for real
numbers.
a. Simplify
operations with
real numbers.
b. Simplify radical
expressions.
1.02 Find and simplify
the ratio of two
numbers.
1.03 Use proportions to
solve problems.
1.04 Use proportions to
determine the
geometric mean of
two positive
numbers.
Measurement & Geometry
Algebra
The learner will use properties & relationships in geometry and measurement concepts to
solve problems. The learner will use inductive and deductive reasoning to solve problems.
While other polygons will be studied, a major emphasis will be placed on Triangles.
The learner will use algebra to recognize
and describe patterns, to locate and
describe geometric figures in the
coordinate plane, and to graph lines.
2.01 Use inductive reasoning to make conjectures. Inductive reasoning is a
process that includes looking for patterns and making conjectures.
a. Look for a pattern.
b. Make a conjecture.
c. Verify the conjecture.
d. Find a counterexample
3.01 Review/maintain algebraic
skills:
a. Evaluate algebraic
expressions.
b. Solve algebraic equations.
c. Plot points on a coordinate
plane.
d. Use Algebraic Properties of
Equality.
2.02 Understand and use the basic elements of geometry:
a. Points, lines, and planes.
1. Collinear points
2. Coplanar points
3. Line segments
4. Rays
b. Measure line segments.
1. Use the Ruler Postulate to find the distance between 2 points.
2. Use the Segment Addition Postulate to find distances of segments
that lie on the same line.
3. Use the Distance Formula to find the distance between 2 points in a
coordinate plane.
c. Measure angles.
1. Use the Protractor Postulate to measure angles.
2. Use the Angle Addition Postulate to measure angles.
d. Divide a segment or an angle into two equal parts.
1. Use the Midpoint Formula to find the coordinates of the midpoint
of 2 points on a line.
2. Analyze the angle bisector of angle to determine angle measures.
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
2
3.02 Recognize and describe
patterns:
a. Number patterns;
b. Patterns in the coordinates of
points on the coordinate
plane;
c. Visual patterns.
3.03 Identify parallel lines in the
coordinate plane.
a. Find slopes of parallel lines.
b. Use slopes of lines to
identify parallel lines in a
coordinate plane.
e. Use relationships among special pairs of angles to find angle measures.
1. Vertical angles.
2. Linear pairs.
3. Complementary angles.
4. Supplementary angles.
2.03 Find the perimeter and the area of a rectangle, a triangle, or a square;
find the circumference and the area of a circle.
2.04 Use logic and deductive reasoning to draw conclusions and solve
problems. Deductive reasoning is a process that uses facts, definitions,
and accepted properties in a logical order to write a logical argument.
a. Recognize conditional statements.
b. Identify Point, Line, and Plane Postulates.
c. Rewrite a postulate or a statement in if-then form.
d. Write the inverse, converse, and contrapositive of a given postulate
or a given statement.
e. Recognize and use definitions.
f. Recognize and use biconditional statements.
g. Use symbolic notation to represent logical statements.
h. Form conclusions by applying the laws of logic to true statements.
There are two laws of deductive reasoning:
1. Use the Law of Detachment.
2. Use the Law of Syllogism.
i. Reason with Properties from Algebra.
j. Use the algebraic properties of equality in geometry.
2.05 Apply properties, definitions, postulates, and theorems of angles and
lines to solve problems.
a. Justify statements about congruent segments.
1. Reflexive Property of Segment Congruence.
2. Symmetric Property of Segment Congruence.
3. Transitive Property of Segment Congruence.
4. Segment Relationships: Midpoint; Segment Addition Postulate
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
3
3.04 Identify perpendicular lines in
the coordinate plane.
a. Find slopes of perpendicular
lines.
b. Use slopes of lines to
identify perpendicular lines
in a coordinate plane.
3.05 Apply algebraic concepts to
confirm properties of geometric
figures in the coordinate plane.
b. Use angle congruence properties.
1. Reflexive Property of Angle Congruence.
2. Symmetric Property of Angle Congruence.
3. Transitive Property of Angle Congruence.
4. Right Angle Congruence Theorem.
c. Use properties about special pairs of angles to solve problems.
1. Congruent Supplements Theorem.
2. Congruent Complements Theorem.
3. Linear Pair Postulate.
4. Vertical Angles Theorem.
d. Identify relationships between lines: parallel (Parallel Postulate),
perpendicular (Perpendicular Postulate), skew.
e. Identify angles formed by transverals: corresponding angles;
alternate interior angles; alternate exterior angles; consecutive
interior angles (same side interior angles).
f. Use results about perpendicular lines to solve problems.
g. Use results about parallel lines and transversals to solve problems.
1. Postulates involving parallel lines:
a. Corresponding Angles Postulate
b. Linear Pair Postulate (revisited)
2. Theorems involving parallel lines:
a. Alternate Interior Angles Theorem
b. Consecutive Interior Angles Theorem.
c. Alternate Exterior Angles Theorem.
d. Perpendicular Transversal Theorem
e. Vertical Angles Theorem (revisited)
3. Postulate involving transversals: Corresponding Angles Converse
4. Theorems involving transversals:
a. Alternate Interior Angles Converse
b. Consecutive Interior Angles Converse
c. Alternate Exterior Angles Converse
2.06 Apply properties, definitions, and theorems of congruent triangles to
solve problems .
a. Classify triangles by their sides and angles.
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
4
b. Find angle measures in triangles.
1. Triangle Sum Theorem
2. Exterior Angle Theorem
3. Corollary to the Triangle Sum Theorem
c. Identify congruent figures & corresponding parts.
1. Third Angles Theorem.
d. Explain why two triangles are congruent: corresponding angles and
corresponding sides
e. Use the Theorem: Properties of Congruent Triangles to solve problems.
1. Reflexive Property of Congruent Triangles.
2. Symmetric Property of Congruent Triangles.
3. Transitive Property of Congruent Triangles.
f. Explain why triangles are congruent using the Side-Side-Side (SSS)
Congruence Postulate & the Side-Angle-Side (SAS) Congruence Postulate.
g. Explain why triangles are congruent using the Angle-Side-Angle
(ASA) Congruence Postulate and the Angle-Angle-Side (AAS)
Congruence Theorem.
h. Use properties of isosceles and equilateral triangles.
1. Base Angles Theorem
2. Converse of the Base Angles Theorem
i. Use properties of right triangles.
1. Hypotenuse-Leg (HL) Congruence Theorem.
j. Place geometric figures in a coordinate plane.
1. Use Distance Formula to measure distances or to locate points.
2. Use Midpoint Formula to measure distances or to locate points.
2.07 Apply properties, definitions, and theorems of triangles to solve
problems.
a. Use properties of perpendicular bisectors.
1. Perpendicular Bisector Theorem
2. Converse of the Perpendicular Bisector Theorem
b. Use properties of angle bisectors to identify equal distances.
1. Angle Bisector Theorem
2. Converse of the Angle Bisector Theorem
c. Use properties of medians of a triangle; the point of concurrency of
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
5
the 3 medians is called the centroid of the triangle..
1. Theorem: Concurrency of Medians of a Triangle
d. Use properties of altitudes of a triangle; the 3 lines containing the
altitudes are concurrent and intersect at a point called the orthocenter
of the triangle.
1. Theorem: Concurrency of Altitudes of a Triangle.
e. Identify the midsegments of a triangle.
1. Midpoint Formula.
2. Distance Formula.
3. Theorem: Midsegment Theorem.
f. Use the properties of midsegments of a triangle.
g. Use triangle measurements to decide which side is longest or which
angle is largest.
1. Theorem: If one side of a triangle is longer than another side,
then the angle opposite the longer side is larger than the angle
opposite the shorter side.
2. Theorem: If one angle of a triangle is larger than another angle,
then the side opposite the larger angle is longer than the side
opposite the smaller angle.
3. Theorem: Exterior Angle Inequality
h. Use the Triangle Inequality.
1. Theorem: Triangle Inequality
i. Use the Hinge Theorem and its converse to compare side lengths and
angle measures.
2.08 Identify similar triangles and use similar triangles to solve real-life
problems.
a. Postulate: Angle-Angle (AA) Similarity Postulate
2.09 Use similarity theorems to explain why two triangles are similar.
a. Theorem: Side-Side-Side (SSS) Similarity Theorem
b. Theorem: Side-Angle-Side (SAS) Similarity Theorem
2.10 Use proportionality theorems about similar triangles to calculate segment
lengths.
a. Theorem: Triangle Proportionality Theorem
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
6
b. Theorem: Converse of the Triangle Proportionality Theorem
c. Theorem: If 3 parallel lines intersect 2 transversals, then they divide
the transversals proportionally.
d. Theorem: If a ray bisects an angle of a triangle, then it divides the
opposite side into segments whose lengths are proportional to the
lengths of the other 2 sides.
2.11 Solve problems involving similar right triangles.
a. Proportions in right triangles
b. Theorem: If the altitude is drawn to the hypotenuse of a right
triangle, then the two triangles formed are similar to the original
triangle and to each other.
2.12 Use a geometric mean to solve problems.
Geometric Mean Theorem: In a right triangle, the altitude from the
right angle to the hypotenuse divides the hypotenuse into two
segments. The length of the altitude is the geometric mean of the
lengths of the two segments.
2.13 Use the Pythagorean Theorem to solve problems:
a. Theorem: In a right triangle, the square of the length of the
hypotenuse is equal to the sum of the squares of the lengths of the
legs.
b. Find side lengths of a right triangle.
c. Determine whether the side lengths of a triangle form a Pythagorean
triple.
2.14 Use the Converse of the Pythagorean Theorem to solve problems:
If the square of the length of the longest side of a triangle is equal
to the sum of the squares of the lengths of the other two sides,
then the triangle is a right triangle.
Text: McDougal Littell Geometry © 2004
Final Revision November 2009
7