Chapter 8
Introductory Geometry
Section 8.3
Triangles, Quadrilateral and Circles
Triangles
Adjectives that describe triangles can either refer to the number of sides that are
congruent or the size of the largest interior angle of the triangle.
The adjectives that refer to the number of congruent sides are scalene, isosceles
and equilateral.
Scalene Triangles are triangles that have none of the sides congruent to each
other.
Isosceles Triangles have at least two of the sides congruent to each other.
Equilateral Triangles are triangles that have all three sides congruent.
Below are some examples of the triangles I mentioned here.
Scalene Triangles
Isosceles Triangles
Equilateral Triangles
The adjectives that refer to the measures of the interior angles of the triangle are
acute, right and obtuse.
Acute Triangles are triangles whose interior angles all measure less than 90.
Right Triangles are triangles in which one of the interior angles measures
exactly 90 (i.e. a right angle). The remaining two angles must be
complementary angles.
Obtuse Triangles are triangles that have one of its interior angles measure
greater than 90.
Below are some examples of the triangles I have mentioned.
Acute Triangles
Right Triangles
Obtuse Triangles
van Hiele Levels
In Holland two high-school mathematics teachers Dina and Pierre van Hiele
developed a theory about learning geometry. They said that there are 5 stages (or
levels) to learning geometry with each stage increasing the amount of complexity
and detail that are in geometric figures.
1. Recognition – Students learn to recognize shapes by their appearance.
They do not analyze parts or components of the figures.
2. Description – Students are able to describe the parts (such as sides and
angles) that make up a shape and tell if they are congruent or not. Students
use the parts to name or describe the figure such as right triangle or rhombus.
3. Relationships – Students begin to use deduce relationships between
shapes to categorize them. They learn that some figures share certain
properties such as all squares are rhombuses.
4. Formal Deduction – This is usually what is part of the high school
geometry curriculum. Students use logical reasoning, axioms, definitions to
write proofs and establish geometric results.
5. Rigor – Students work in different axiomatic systems such as finite or nonEuclidean geometry to compare results.
Quadrilaterals
Quadrilaterals are 4 sided polygons. The descriptions (i.e. definitions) of some of
the most common quadrilateral illustrate both the second and third van Hiele levels.
Trapezoids are quadrilaterals that have exactly one pair of parallel sides.
Parallelograms are quadrilaterals in which each pair of opposite sides are
parallel.
Rectangles are parallelograms that have four right angles.
Rhombuses are parallelograms with four congruent sides.
Squares are rectangles that have four congruent sides.
Below are examples of each of the following. Which is which?
parallelogram
rectangle
square
rhombus
trapezoid
The blue arrows mark pairs of parallel sides. The green angles mark right angles.
The orange segments mark sides that are congruent.
The Venn Diagram to the right
shows how the common
quadrilaterals, trapezoid,
parallelogram, rectangle, rhombus
and square are categorized. This
reasoning represents the third van
Hiele level
parallelograms
rectangles
rhombuses
trapezoids
squares
Circles
A circle is the set of all points in a plane that are the
same distance (called the radius) from a given point
(called the center).
A cord of a circle is a line segment that goes from
one point on the circle to another not through the
center.
radius
radius
center
center
diameter
of a circle
circle
A diameter is a line segment that goes from one
point on the circle to another through the center.
A central angle is an angle whose vertex is at the
center of the circle.
center
Central angle of a circle