• Lesson
9-12
2

Instead of considering the diagonals within a quadrilateral, this lesson provides a unique opportunity: students start with the diagonals and deduce the type of quadrilateral that surrounds them. Using an applet, students explore certain characteristics of diagonals and the quadrilaterals that are associated with them.

Group students into pairs. Give each pair markers and something to write on (e.g. whiteboard) that they can later display to the whole class.

Ask students to draw two perpendicular line segments, connect the endpoints, and describe the quadrilateral that is formed.

[Some pairs will likely assume the diagonals bisect each other and will sketch squares, while other pairs will not assume the diagonals bisect each other and will sketch non-square kites.]

Group students' quadrilaterals according to their type. This is a great opportunity to prompt students to think critically about the relationships between the diagonals and their quadrilaterals. The goal is not to press for right answers, but to encourage students to think about the relationships they will see when manipulating the applet in the next activity. Prompt students:

• Describe the properties for this diagonal.
• Do the diagonals bisect each other?
• Are there other possible ways the diagonals could intersect?

Explain to students that they will be using a program that shows how diagonals are related to their quadrilaterals. Launch the Diagonals to Quadrilaterals I tool, either as a whole-class activity or as pairs of students working on a computer.

Ask students to try a quadrilateral. Have them click on "Show Perpendicular Diagonals" and practice dragging the points on the diagonals to change the shape of the quadrilaterals. Ask students what types of quadrilaterals are possible, and have them list the possibilities on their whiteboards. Students may see that the initial shape is a general quadrilateral; others may drag the points to create a kite, a rhombus, or even what appears to be a square. Remind students that their assumptions about what looks like a kite, a rhombus, or a square, may not necessarily be correct. We cannot verify what we see without measures or constructions. Do not spend a lot of time examining all quadrilateral shapes on the students’ list; these possibilities come up again as students work through the applets.

Next, distribute the Diagonals and Quadrilaterals activity sheet. Note that the class just answered the question for perpendicular diagonals; have students record the answer on the table. Students should note that for perpendicular diagonals, the quadrilateral can be a general shape or, in specific cases, it can be a kite, rhombus, or square.

Prompt students to explore the effects of adding conditions to the diagonals. Click on an appropriate button in the applet to examine other relationships between perpendicular diagonals. For the last three shapes, students will need to use a second tool, Diagonals to Quadrilaterals II. For each quadrilateral, students will describe the type (or types) of quadrilateral and explain their reasoning.

Students return to a whole-group setting. As pairs give their answers, other students are responsible for questioning the pair for their reasoning as well as for clarity. Probe students for responses about their conclusions. For instance, you might ask:

• How do you know the quadrilateral is a rhombus?

[Students may respond that the diagonals appear to bisect each other so we can get congruent triangles like GKM and HKM using SAS.]

As students describe their findings, record their results on the Diagonals and Quadrilaterals overhead. This lists categories of quadrilaterals (some will overlap; for example, rhombi will fall in both the "general" and the "parallelograms" categories). Mark each cell with an "A" (all quadrilaterals in this category can be created given the conditions on the diagonals) or "S" (some quadrilaterals can be created).

 For sample results and conclusions for the record sheet, see the Diagonals and Quadrilaterals Key.

Assessments

1. The first objective, describe the relationships among the diagonals and types of figures, may be assessed in a journal entry. A possible prompt is:
The conditions we place on the diagonals of a quadrilateral tell us the type(s) of quadrilateral we have. Describe the types of conditions we might put on diagonals of a quadrilateral. Explain how these conditions lead us to particular types of quadrilaterals.

2. An alternative way to assess this situation may be a matching or short-answer task. For example:
The diagonals of quadrilateral MATH are perpendicular and they bisect each other. What type of quadrilateral is MATH? Answer in words and draw one or more sketches to illustrate your answer.
By having students combine words and sketches, we have a better idea of whether vocabulary is getting in the way of their expression of valid ideas about relationships among diagonals and their quadrilaterals.
3. To assess the second objective, students need a new set of geometric objects and relationships from which to deduce types. In pairs, have students complete Triangle Time Activity Sheet. Students will examine the relationship between the altitude of a triangle and the side it intersects. They use this to figure out what type of triangle it is.

Extensions

1. Quadrilaterals inscribed in a circle are called cyclic quadrilaterals. Many general quadrilaterals can be inscribed in a circle. Which, if any, of the quadrilaterals we found in this lesson (kite, parallelogram, rhombus, square, rectangle) are cyclic? Justify your choices.
[Some kites, all rhombi, squares, and rectangles are cyclic. Since the quadrilaterals have four sides, a pair of adjacent sides will subdivide the circle into two arcs. Since the squares and rectangles have right angles at each vertex, opposite angles will be inscribed in a semicircle. The two semicircles for each square or rectangle will constitute a circle. Since the opposite angles of non-square rhombi are not supplementary, then the non-square rhombus is not cyclic. The same reasoning is true for the parallelogram. Kites are cyclic only if one pair of opposite angles is supplementary. Since adjacent sides of a kite must be congruent, opposite angles created by a pair of non-congruent sides will be congruent and therefore will be supplementary when each of these opposite angles is a right angle.]

2. Have students work in pairs on the Triangle Time activity sheet. As a whole group, students create a chart similar to the one created for the quadrilaterals. In summary, discuss the following: "When we looked at the quadrilaterals, we saw that adding conditions to the diagonals led to a more specific type of quadrilateral. Do you think that adding conditions to the altitude and side of a triangle intersecting the altitude will lead to a more specific type of triangle? Explain."
[Explanations should include that bisecting leads to isosceles triangles, perpendicular adds nothing, congruent eliminates some of the usual named types but does narrow the possibilities in other ways (e.g., triangles whose areas are one-half of the square of the length of the side).]

Questions for Students

1. Why does it make sense that knowing the diagonals of a quadrilateral are perpendicular is not sufficient to show that the quadrilateral is a rhombus?
[The diagonals can be perpendicular without bisecting each other; thus, the quadrilateral may not be a rhombus.]
2. Explain using diagonals why a square is both a rhombus and a rectangle.
[A rhombus must have diagonals that are both perpendicular and bisecting each other. A rectangle must have diagonals that are congruent and bisect each other. Since a square is both a rhombus and a rectangle, its diagonals are congruent perpendicular bisectors.]
3. Explain using diagonals why a square is always a rhombus but a rhombus is not always a square.
[A square has diagonals that are congruent perpendicular bisectors. A rhombus has diagonals that are perpendicular bisectors. Thus, the diagonals of a square fulfill the requirements for the diagonals of a rhombus: perpendicular bisectors. However, the diagonals of a rhombus need not be congruent. So, the diagonals of a rhombus do not fulfill one of the requirements for the diagonals of a square: congruent.]

Teacher Reflection

• How does giving conditions about the diagonals and asking students to draw conclusions about the quadrilaterals, or giving students conditions on an altitude and the side it intersects and asking students to conclude things about types of triangles, parallel what students must do in the proof exercises often asked in high school geometry?
• How did students express their thinking when using words? Using the applet? Using sketches? Using hand motions?

### Learning Objectives

Students will:
• Identify the types of quadrilateral possible, based on information about the diagonals.
• Deduce characteristics of a polygon based on relationships among components of the polygon.