Portland Community College | Portland, Oregon

Course Number:
MTH 75
Course Title:
Introduction to Formal Geometry
Credit Hours:
Lecture Hours:
Lecture/Lab Hours:
Lab Hours:
Special Fee:

Course Description

Topics include: inductive and deductive reasoning, geometric constructions, line and angle properties, triangle properties, polygon properties, circles, transformations, area, volume, Pythagorean Theorem, similarity, and geometric proofs. Results communicated in oral and written form. Prerequisites: MTH 60. Audit available.

Intended Outcomes for the course

  • To recognize and apply geometric properties to solve a variety of application problems.

  • To discover geometric properties.

  • To develop an understanding of proof.

Outcome Assessment Strategies

Assessment shall include:

  1.  At least two in-class examinations.

  2. At least one writing assignment and

  3. At least two of the following additional measures:  

  1. Take-home examinations.

  2. Graded homework.

  3. Quizzes.

  4. Group projects.

  5. In-class activities.

  6. Attendance.

  7. Portfolios.

  8. Individual projects.

  9. Individual student conference.

Course Content (Themes, Concepts, Issues and Skills)


  1.  Language of Geometry

  2. Lines, Angles, & their properties

  3. Properties of Polygons

  4. Properties of Circles

  5. Area

  6. Volume

  7. Similarity & Congruence

  8. Geometric Reasoning

  9. Trigonometry


1.0              LINE & ANGLE PROPERTIES 

1.1          Define a postulate and show how we can use it to describe a point and a line. Explain what a definition is and use it to define a line segment, a ray, and an angle.

1.1.1        Find the midpoint of a line.

1.1.2        Find the slope of a line

1.1.3        Define complementary, supplementary, vertical, adjacent angles

1.2          Describe angles based on their measure

1.3          Recognize and apply the properties of parallel lines

1.3.1        Define parallel lines, perpendicular lines, transversal

1.3.2        Apply the special angle properties of parallel lines cut by transversals


2.1      Triangles €“ Define a triangle and look at all the properties associated with triangles.

2.1.1        Classifying Triangles by Angles and Sides €“ Scalene, isosceles, equilateral, acute, right, obtuse, equiangular.

2.1.2        Properties of Triangles- Triangle Sum Theorem

2.1.3        Triangle Congruence Theorems €“ Define what congruent polygons are and use it to determine if triangles are congruent by using the SAS, ASA, congruence postulates, and the AAS, SSS congruence theorems.

2.1.4        Right Triangles- Use the Pythagorean Theorem to solve problems.  

2.2      Quadrilaterals - Define a quadrilateral and look at all the properties associated with quadrilaterals. Explain the difference and similarities between parallelograms, rectangles, rhombuses, squares, kites, and trapezoids.  

2.2.1        Parallelograms €“ Define a parallelogram. Understand the theorems that determine if a quadrilateral is a parallelogram. Perform test to show that a quadrilateral is a parallelogram. Understand and show that diagonals of a parallelogram bisect each other, opposite sides of a parallelograms are equal, the opposite angles of a parallelogram are equal and its consecutive angles are supplementary.

2.2.2        Rectangles €“ Define a rectangle. Show that diagonals of a rectangle are equal. Show that consecutive sides are perpendicular.

2.2.3        Rhombuses- Define a rhombus. -Show that diagonals of rhombus are perpendicular. - Show all rhombuses are parallelograms.

2.2.4        Squares €“ Define a square. Determine that a square is a rhombus, a rectangle and a parallelogram.

2.2.5        Kites €“ Define a kite. €“ Determine that two sets of consecutive sides are congruent.

2.2.6        Trapezoids- Define a trapezoid. €“ Determine that the base angles of an isosceles trapezoid are equal €“ Determine that the diagonals of an isosceles trapezoid are equal.

3.0              PROPERTIES OF CIRCLES

3.1        Circle Terminology €“ Define a circle and radius.

3.2        Discover Properties           

3.2.1        Lines and Segments -Define a chord, and the diameter of a circle. €“Discover properties of chords.  Define what a tangent line is with respect to a circle.

3.2.2        Arcs and Angles  Arc Measure €“ define what a semicircle is, what minor and major arcs are.  Define central angles, inscribed angles, and intercepted arcs. Look at theorems related to circles.

3.2.3        €“ Define congruent circles, and concentric circles.

3.3        Measurement

3.3.1        Define the circumference of a circle and use the equation C = 2 r to find the circumference.

3.3.2        Explain the definition of pi ( Circumference/Diameter = )

3.3.3        Define arc length.

4.      AREA & VOLUME €“

For each of the following geometric shapes find the area, surface area and volume as applicable.  

4.1        Area of Parallelograms, Rectangles, and Rhombi

4.2        Area of Triangles, Trapezoids, and Kites

4.3        Area of Circles

4.4        Surface Area of Prisms, Pyramids, & Cylinders

4.5        Volume of Prisms and Cylinders

4.6        Volume of Pyramids and Cones

4.7        Surface Area and Volume of a Sphere


5.1        Define ratio and proportion.  

5.2        Similar Triangles

5.2.1        Perform indirect measurement with similar triangles.

5.2.2        Understand and investigate theorems giving criteria for two triangles being similar.

5.3        Trigonometry

5.3.1        Use the ratios of a right triangle to find the missing angle or side.

5.3.2        Perform indirect measurement with similar right triangles.  


6.1      Deductive Reasoning

6.1.1        Investigate forms of valid reasoning.

6.1.2        Investigate symbols associated with logic.

6.1.3        Investigate different approaches to mathematical proofs.

6.1.4        Define what a conjecture, theorem, postulate, and a corollary are.

6.2      Proofs

6.2.1        Formal Proof Prove two triangles congruent. Prove one theorem about circles Prove one theorem about quadrilaterals.  

7.0      TECHNOLOGY €“ Geometer€™s Sketchpad  

7.1      Using the tools

7.1.1        Selection Tool (select, translate, rotate)

7.1.2        Point Tool

7.1.3        Straight Edge Tool (line, ray, segment)

7.1.4        Compass Tool

7.1.5        Labeler (labels, text)

7.2      Using the menus

7.2.1        Measurement

7.2.2        Constructions & Construction Help

7.2.3        Transformations