Chapter 8

1. Chapter 8 Introductory Geometry Section 8.3 Triangles, Quadrilateral and Circles

2. 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

3. 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

4. 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 non-Euclidean geometry to compare results.

5. 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 rhombus square trapezoid rectangle The blue arrows mark pairs of parallel sides. The green angles mark right angles. The orange segments mark sides that are congruent.

6. 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. 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. cord diameter radius radius center center diameter of a circle circle center Central angle of a circle