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1.6: Quadrilaterals

Created by: CK-12

Interior Angles

Pacing: This lesson should take one class period

Goal: Students will use the Triangle Sum Theorem to derive the Polygonal Sum Theorem by dividing a convex polygon into triangles.

Inquiry Based Learning! Analyzing a pattern is another method to looking at the polygonal sum theorem. Begin by setting up the below chart. Ask students to fill in the second column, asking if the number of side lengths can form a polygon. Ask students to then complete the obvious interior angle sums such as triangle and quadrilateral. Encourage students to see a pattern and create its function. Students should see that the “starting” sum is 180 and each subsequent polygonal sum is 180\;\mathrm{degrees} greater.

Vocabulary! Reiterate to students that a diagonal is drawn from any vertex to a non-adjacent vertex.

Extension! Does this theorem work for non-convex polygons? Pose this question to students as you draw several non-convex polygons on the board. Since the polygon is non-convex, the “indented” angle will always be obtuse, showing this theorem will only work for convex polygons.

Additional Example: The sum of the interior angles of an n-gon is 3,960\;\mathrm{degrees.} What is n? n = 24

Exterior Angles

Pacing: This lesson should take one class period

Goal: This lesson introduces students to exterior angles of polygons. The Linear Pair Theorem is used to determine the measures of exterior angles.

Stress to students that there are two possibilities for exterior angles and it will be extremely important to label the angle correctly.

Additional Examples: Using what you have learned thus far, determine the measure of \angle FGH. Students will use the Polygonal Sum Theorem to determine the sum of the interior angles in the heptagon is 900^\circ. Dividing by 7, each interior angle has a measure of 128.57^\circ. \angle FGH forms a linear pair with \angle AGF and are supplementary. Therefore, the measure of angle FGH = 51.43\;\mathrm{degrees}.

Classifying Quadrilaterals

Pacing: This lesson should take one to two class periods

Goal: Students are introduced to the most common quadrilaterals and relationships they share. A Venn diagram is provided as a visual to allow students to visualize how a quadrilateral such as a square fits with a rectangle, parallelogram, and trapezoid.

Have students take the Venn diagram and transfer it to a hierarchy, showing the most general quadrilateral to the most specific quadrilateral.

Discussion! Begin a discussion with students regarding trapezoids and parallelograms. Some textbooks describe a trapezoid as, “a quadrilateral with at least one pair of parallel sides.” Discuss this possible definition with your students. How would the Venn diagram change if this definition were accepted as true? Should it be accepted as true? Why is the definition of a trapezoid provided in this text stating “exactly one pair of parallel sides?”

Additional Examples: Ask students to answer the following questions, either on a personal whiteboard, journal entry, or in a Think-Pair-Share group.

  1. Always, sometimes, never. All rhombi are squares. Sometimes. A square is a special type of rhombus.
  2. Always, sometimes, never. All rhombi are parallelograms. Always.
  3. Always, sometimes, never. Parallelograms are trapezoids. This answer depends upon the discussion of your class.

Pythagorean’s Theorem AGAIN! Reiterate to your students the connection between Pythagorean’s Theorem and the distance formula. This is especially helpful when determining the lengths of segments of a quadrilateral to determine its appropriate classification.

Trapezoids and Parallel Lines! Encourage your students to make the connection between consecutive interior angles of parallel lines and a trapezoid. A trapezoid is really a pair of parallel lines cut by two transversals. Therefore, consecutive interior angles are supplementary (according to Euclid’s’ 5^{\mathrm{th}} Postulate).

Using Parallelograms

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to familiarize students with properties special to parallelograms. These properties are useful when proving a figure is a parallelogram. These properties also hold for rectangles, rhombi, and squares – those quadrilaterals that are “included” within parallelograms in the Venn diagram.

In-class Activity! Instead of using string, your students can also use raw spaghetti noodles. Be sure the segments are equal in length; otherwise, the models may not illustrate a parallelogram appropriately.

Making Connections! Make as many connections as possible. This will help your students see how geometrical concepts fit together. For example, students have learned parallel lines are equidistant from each other. Connect this to a parallelogram

Flash Fast Game! Have your students create flashcards with quadrilateral names on one side and important information or properties on the reverse. Have various types of quadrilaterals, both abstract and real world, ready to show students. Once students believe they have classified the quadrilateral, they are to hold up the appropriate name. You can keep score or use this as a summative assessment.

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 parallelogram,” can be proven using the SSS Congruence Postulate. Instead of using same side interior angles, use the Reflexive Property to state CE = 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

  1. 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.
  2. 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.
  3. 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

Goal: This lesson introduces students to the special properties of trapezoids, especially those of the isosceles trapezoid.

Real Life Application! Show a photograph of the John Hancock Center, located in Chicago, Illinois. This structure was the first skyscraper to be built using exterior tube technology, instead of using internal beams as support. Each face of the John Hancock Center is comprised of six isosceles trapezoids.

Try This! Have each student draw three different isosceles trapezoids. Exchange papers with one student. Ask students to check one trapezoid to be sure it is isosceles by measuring non-base sides for congruence and equivalent diagonals. Have correcting students “sign off” on their opinions. Switch papers a second a third time, repeating the process.

Kites

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to present special properties of kites.

Visualization! Have students draw two isosceles triangles sharing the same base. By erasing the shared base, they have just drawn a kite! The shared base represents one diagonal of the kite.

Vocabulary! Using the phrase “ends of the kite” can be misleading for students. The ends do not always mean the endpoints of the longer diagonal. Be sure students can identify the ends of non-convex kites, and non-traditional kites (as shown below).

Beat the clock! Have students draw and cut out two copies of a scalene triangle. In one minute, have students form as many polygons as possible, drawing a sketch of each they form. You can also use tangrams for this activity.

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CK.MAT.ENG.TE.1.Geometry.1.6

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