7.2: Properties of Trapezoids and Isosceles Trapezoids
This activity is intended to supplement Geometry, Chapter 6, Lesson 5.
In this activity, you will explore:
- the properties of trapezoids, including isosceles trapezoids
To begin, open up a blank Cabri Jr. document.
Let’s start with a line segment,
Create a line through
Hide the parallel line and complete the trapezoid. Measure the interior angles. Are the base angles at
To verify that the lines are parallel, construct lines through
Construct the points of intersection of the new lines with
Label the new points on
Try to drag point
In order to construct an isosceles trapezoid, start with a line segment,
Press TRACE/F4 and select the Reflection option. Click on the perpendicular line through
A new point will appear on the line segment on the right side. Label this point
Construct perpendicular lines through
Construct a perpendicular to
Hide the parallel and perpendicular lines and points
Measure the base angles—the interior angles at \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, \begin{align*}C\end{align*} and \begin{align*}D\end{align*}. Which angles are congruent? Did you expect those angles to be congruent? Can you prove that they are?
Construct and measure the lengths of the two diagonals \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{BC}\end{align*}. Should they be congruent? Can you prove that they are?
Drag point \begin{align*}C\end{align*} and watch the angles and sides. Are all of the properties established above preserved?
However, one property that is common to all trapezoids is that the line segment connecting the non-parallel sides is also parallel to these sides and its length is half the sum of the parallel sides. Construct a trapezoid \begin{align*}ABCD\end{align*}. Construct the midpoints at \begin{align*}E\end{align*} and \begin{align*}F\end{align*} and construct \begin{align*}\overline{EF}\end{align*}.
Measure the lengths of \begin{align*}\overline{CD}\end{align*}, \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{AB}\end{align*}. How could you prove that \begin{align*}\overline{EF}\end{align*} is parallel to \begin{align*}\overline{CD}\end{align*} and \begin{align*}\overline{AB}\end{align*}?
Use the Calculate tool to find the sum of the lengths of \begin{align*}\overline{CD}\end{align*} and \begin{align*}\overline{AB}\end{align*}.
Place the number “2” on the screen and divide the sum by 2 using the calculate tool. Will this result always equal the length of \begin{align*}\overline{EF}\end{align*}? Can you prove this result?
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