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Unit 1, Lesson 13

Proofs about Parallelograms

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Integrated Math 2

13.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

To Gather

Internet-enabled device

Activity Narrative

The purpose of this Warm-up is to elicit conjectures about rectangles and parallelograms, which will be useful when students write proofs of these conjectures in a subsequent activity. While students may notice and wonder many things about these images, the relationships between the diagonals of a parallelogram and the diagonals of a rectangle is the important discussion point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, then restate their observation with more precise language in order to communicate more clearly.

This activity is designed to be done digitally because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting and interacting with the applet of the parallelogram and rectangle would be helpful during the Launch as students notice and wonder.

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If conjectures about diagonals do not come up during the conversation, ask students to discuss this idea.

13.2

Activity

Instructional Routines

Draw It MLR1: Stronger and Clearer Each Time

Materials

To Copy (from Blackline Masters)

Brace Yourself! Long Strips

Brace Yourself! Short Strips

To Gather

1-inch strips cut from card stock with evenly spaced holes Geometry toolkits (HS)Metal paper fasteners Rulers

13.3

Activity

Instructional Routines

MLR2: Collect and Display Take Turns

Materials

None

Activity Narrative

Working backward from what we are trying to show in order to see how that relates to the given information is a useful skill in geometry. This activity makes that strategy explicit by having partners take turns saying, “I would know that were true if . . . .” This is one of the first cases where students might find and use overlapping triangles in a proof. Students can use triangles and , or they can study the angles in triangles and and prove that angles and must be complementary (where is the intersection of and ). This method will require some algebraic manipulation of expressions for the angles.

Monitor for students who are focused on the four non-overlapping triangles formed by the diagonals, and for students who are focused on the two larger overlapping triangles formed by the diagonals.

<p>Two rectangles, each labeled A B C D. Left rectangle has 4 colors, right rectangle has different line shading.<br>
</p>

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing that students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—including, especially, everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

This activity uses the Collect and Display math language routine to advance students in developing their mathematical language.

Lesson Synthesis

Remind students of the term “converse.” Ask students what the converse of “If a quadrilateral is a parallelogram, then its diagonals bisect each other” would be. (If a quadrilateral's diagonals bisect each other, then it is a parallelogram.) We have proven both the statement and its converse. Introduce the phrase “if and only if,” and explain that we use it when the statement and its converse are true. Display these statements for all to see:

A quadrilateral is a parallelogram if its diagonals bisect each other.
A quadrilateral is a parallelogram only if its diagonals bisect each other.

Ask multiple students to put the second statement in their own words. (There can’t be any parallelograms with diagonals that don’t bisect each other; all quadrilaterals have diagonals that bisect each other; if it’s a parallelogram, then the diagonals bisect each other).

Use some more everyday examples, and ask students which are examples of if-and-only-if statements (the statement and its converse are true) and which are not. For example:

A person is a parent if and only if they are a mother. (This is not an if-and-only-if statement. A father can be a parent too. It’s true that if they’re a mother, then they’re a parent, but it’s not true that if they’re a parent, then they must be a mother.)
A person is vegan if and only if they don’t eat animal products. (This is as close to true as things get in the real world, since it’s the definition of being vegan.)

Invite students to come up with their own examples and nonexamples. Ask students why it is useful to know a quadrilateral is a parallelogram if and only if its diagonals bisect each other. (Now we can use either part as the given information, so it's like adding two theorems to the reference chart.)

to access Cool-down.

Student Lesson Summary

A quadrilateral is a parallelogram if and only if its diagonals bisect each other. The “if and only if” language means that both the statement and its converse are true. So we need to prove:

If a quadrilateral has diagonals that bisect each other, then it is a parallelogram.
If a quadrilateral is a parallelogram, then its diagonals bisect each other.

To prove part 1, make the statement specific: If quadrilateral has diagonals and that intersect at such that is congruent to and is congruent to , then side is parallel to side , and side is parallel to side .

<p>Quadrilateral EFGH. Diagonals EG and HF intersect at point Y.</p>

We could prove triangles and are congruent by the Side-Angle-Side Triangle Congruence Theorem. That means that corresponding angles in the triangles are congruent, so angle is congruent to . This means that alternate interior angles formed by lines and are congruent, so lines and are parallel. We could also make an argument that shows triangles and are congruent. Then, angles and are congruent, which means that lines and must be parallel.

To prove part 2, make the statement specific: If parallelogram has side parallel to side and side parallel to side , and diagonals and that intersect at , then we are trying to prove that is the midpoint of and of .

<p>Quadrilateral ABCD. Diagonals AC and DB intersect at point X.</p>

We could use a transformation proof. Rotate parallelogram by using the midpoint of diagonal as the center of the rotation. Then show that the midpoint of diagonal is also the midpoint of diagonal . That point must be since it is the only point on both line and line . So must be the midpoints of both diagonals, meaning the diagonals bisect each other.

We have proved that any quadrilateral with diagonals that bisect each other is a parallelogram, and that any parallelogram has diagonals that bisect each other. Therefore, a quadrilateral is a parallelogram if and only if its diagonals bisect each other.

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Lesson 13

Proofs about Parallelograms

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSG-CO.C.11

Instructional Routines

Materials

To Gather

Activity Narrative

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSG-CO.B.8

HSG-CO.C.9

Addressing

HSG-CO.C.11

Building Toward

Standards Alignment

Building On

HSG-CO.B.8

HSG-CO.C.9

Addressing

HSG-CO.C.11

Building Toward