GKT101: General Knowledge for Teachers – Math

Example 1

All squares are rectangles, but not all rectangles are squares. How is this possible?

Rectangles are defined as quadrilaterals with four right angles. Squares are defined as quadrilaterals with four right angles and four congruent sides. Because all squares have four right angles and satisfy the definition for rectangles, they can all also be called rectangles. On the other hand, not all rectangles have four congruent sides, so not all rectangles can also be called squares.

Example 2

Draw a square. Draw in the diagonals of the square. Make at least one conjecture about the diagonals of the square.

To make a conjecture means to make an educated guess. There are a few conjectures you might make about the diagonals of a square. In other lessons, you will learn how these conjectures may be proven true.

Here are some possible conjectures:

1. diagonals of a square are congruent

2. diagonals of a square are perpendicular

3. diagonals of a square bisect each other (cut each other in half)

4. diagonals of a square bisect the angles (cut the $\begin{align*}90^\circ\end{align*}$angles in half)

 

Example 3

A quadrilateral has four congruent sides. What type of quadrilateral must it be? What type of quadrilateral could it be?

It must be a rhombus and therefore also a parallelogram. It could be a square.

Example 4

Solve for $\begin{align*}x\end{align*}$ (picture not drawn to scale).

 

This is a parallelogram so opposite sides are congruent.

$\begin{align*}3x+1 &=5x-12 \\ 2x &=13 \\ x &=6.5\end{align*}$