Lesson 7 Quadrilaterals: Beyond Definition Practice Understanding

Jump Start

Give the name of a geometric figure that has the following characteristics:

1.

A type of quadrilateral with no lines of symmetry.

2.

A type of quadrilateral whose diagonals are its only lines of symmetry.

3.

A type of quadrilateral where the lines through the midpoints of opposite sides are lines of symmetry.

Learning Focus

Relate attributes of special quadrilaterals to symmetry.

What else might be true about parallelograms, rectangles, squares, or rhombuses other than the characteristics given about them in their definitions?

How might I be convinced that certain characteristics must occur in every member of a special class of quadrilaterals?

Open Up the Math: Launch, Explore, Discuss

1.

We have found that many different quadrilaterals possess lines of symmetry and/or rotational symmetry. In the following chart, write the names of the quadrilaterals that are being described in terms of their symmetries.

Based on the symmetries we have observed in various types of quadrilaterals, we can make claims about other features and properties that the quadrilaterals may possess.

a graphic describing the different terms of symmetries fro different quadrilaterals ▪ 180° rotation▪ 180° rotation▪ 2 lines of symmetry(diagonals)▪ 180° rotation▪ 2 lines of symmetry(through midpoint of sides)▪ 90° and 180° rotation▪ 4 lines of symmetry (diagonals & through midpoint of sides)

2.

A rectangle is a quadrilateral that contains four right angles.

Based on what you know about transformations, what else can we say about rectangles besides the defining property that “all four angles are right angles?” Make a list of additional properties of rectangles that seem to be true (we call such statements conjectures) based on the transformation(s) of the rectangle onto itself. You will want to consider properties of the sides, the angles, and the diagonals. Then justify why the properties would be true for this specific example of a rectangle using transformational symmetry.

A rectangle

3.

A parallelogram is a quadrilateral in which opposite sides are parallel.

Based on what you know about transformations, what else can we say about parallelograms besides the defining property that “opposite sides of a parallelogram are parallel?” Make a list of additional properties of parallelograms that seem to be true based on the transformation(s) of the parallelogram onto itself. You will want to consider properties of the sides, angles, and diagonals. Then justify why the properties would be true for this specific example of a parallelogram using transformational symmetry.

A parallelogram

Pause and Reflect

4.

A rhombus is a quadrilateral in which all four sides are congruent.

Based on what you know about transformations, what else can we say about a rhombus besides the defining property that “all sides are congruent?” Make a list of additional properties of rhombuses that seem to be true based on the transformation(s) of the rhombus onto itself. You will want to consider properties of the sides, angles, and diagonals. Then justify why the properties would be true for this specific example of a rhombus using transformational symmetry.

A rhombus

5.

A square is both a rectangle and a rhombus.

Based on what you know about transformations, what can we say about a square? Make a list of properties of squares that seem to be true based on the transformation(s) of the squares onto itself. You will want to consider properties of the sides, angles, and diagonals. Then justify why the properties would be true for this specific example of a square using transformational symmetry.

A square

Ready for More?

An isosceles trapezoid is a quadrilateral with one pair of parallel sides and the non-parallel sides are congruent, as shown in figure . Can you find a way to demonstrate that the following statements are true?

  • The diagonals do not bisect each other.

  • The diagonals are congruent.

  • and are supplementary, that is, .

A coordinate plane with x- and y- axis of 1-unit increments with a isosceles with diagonals and vertices at A(-5,0), B(-3,4), C(3,4), D(5,0) x–5–5–5555y–5–5–5555000

Takeaways

Our Conjectures about Properties of Quadrilaterals (based on experimentation and reasoning with rigid transformations)

In the following chart, write the names of the quadrilaterals that are being described in terms of their features and properties, and then record any additional features or properties of that type of quadrilateral you may have observed. Be prepared to share reasons for your observations.

a graphic describing the different features and properties of different quadrilaterals ▪ 2 pairs of parallel sides▪ 2 pairs of parallel sides▪ all four sides are congruent▪ 2 pairs of parallel sides▪ all four angles are congruent▪ 2 pairs of parallel sides▪ all four sides are congruent▪ all four angles are congruent

What do you notice about the relationships between quadrilaterals based on their characteristics and the structure of the above chart?

How are the charts at the beginning and end of this task related? What do they suggest?

Vocabulary

Lesson Summary

In this lesson, we used rigid transformations to examine properties of the sides, angles, and diagonals in parallelograms, rectangles, rhombuses, and squares. We learned that some quadrilaterals can be classified in terms of the properties they share with other quadrilaterals, such as congruent opposite sides or angles.

Retrieval

1.

These two figures are congruent. Find the measure of the missing sides and angles.

A quadrilateral with angle B labeled 83 degrees, angle C labeled 119, angle A labeled 87 and angle D. Line segment BC labeled 6 cm. A quadrilateral with angle B', angle C', angle A', and angle D' labeled 71 degrees. Line segment A'B' 6 cm, line segment A'D' 8 cm, and line segment C'D' 5.4 cm.

2.

These two figures are similar. Find the measure of the missing sides and angles.

A triangle with angle A labeled 60 degrees, angle B labeled 40 degrees and angle C. Line segment AB labeled 7 cm and line segment AC labeled 5 cm. A triangle with angle A', B', and angle C' labeled 80 degrees. Line segment B'C' 12.8 cm and line segment A'B' 14 cm