Proving Quadrilaterals are Parallelograms
Pacing:This lesson should take one class period
Goal:Students will use triangle congruence postulates and theorems to prove quadrilaterals are parallelograms. This
lesson serves as an application of the concepts learned in the Triangle Congruence lesson.
Be sure to review each proof in the lesson with your class. Have the students perform a Think-Pair-Share by writing
the conditional on the board and having students attempt to prove the statements on their own.
Another Way of Thinking! The proof of, “If a quadrilateral has two pairs of congruent sides, then it is a parallel-
ogram,” can be proven using the SSS Congruence Postulate. Instead of using same side interior angles, use the
Reflexive Property to stateCE=CE.
Rhombuses, Rectangles, and Squares
Pacing:This lesson should take one class period
Goal:This lesson demonstrates another application of triangle congruence. Students are shown important properties
of rhombuses such as bisecting diagonals and opposite angles.
Refer students back to the Venn Diagram or the hierarchy of quadrilaterals. Make sure students understand that
everything that falls within a rhombus possess the same characteristics and properties of a rhombus. Identifying this
key relationship will help students understand this lesson.
Remind students that a diagonal is a segment drawn from one vertex to any non-adjacent vertex.Question to think
about:Will there always be two diagonals for any quadrilateral? What is it is non-convex?
Arts and Crafts Time!Using patty paper and a pencil, have students trace a rectangle. Instruct students to fold the
rectangle so the lower left angle fits on top of the upper right angle, thus forming a diagonal. Open the fold and
repeat the process on the other diagonal. Overlay the patty paper onto a coordinate grid and have students work
through the distance formula to determine the lengths of the diagonals.
Discuss the proof of this as a class, using the patty paper rectangle for further illustration, if necessary.
Be sure students can “take apart” a biconditional into its two separate statements. This may require more practice
on behalf of your students before they can determine if the biconditional is true.
Additional examples:Separate these biconditionals into a conditional and its converse
a. The rain will fall if and only if it is cloudy.If the rain will fall, then it is cloudy. If it is cloudy, then the rain
will fall.
b. An animal is a mammal if and only if it has whiskers.If an animal is a mammal, then it has whiskers. If an
animal has whiskers, then it is a mammal.
c. An object is a circle if an only if it is the set of points equidistant from a single point.If an object is a circle,
then it is the set of points equidistant from a single point. If an object is the set of points equidistant from a
single point, then it is a circle.Trapezoids
Pacing:This lesson should take one class period
Chapter 1. Geometry TE - Teaching Tips