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Common Core For Geometry

Worksheets for Geometry, Module 1, Lesson 28

Student Outcomes

- Students complete proofs that incorporate properties of parallelograms.

**Properties of Parallelograms**

Classwork

**Opening Exercise**

a. If the triangles are congruent, state the congruence. b. Which triangle congruence criterion guarantees part 1? c. ππΊ corresponds with

**Discussion**

How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now prove about the properties of parallelograms?

To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we have assumed other details about parallelograms to be true, too. We assume that:

- Opposite sides are congruent.
- Opposite angles are congruent.
- Diagonals bisect each other.

Let us examine why each of these properties is true.

**Example 1**

If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure. Complete the diagram, and develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why opposite sides and angles of a parallelogram are congruent.

Given: _____

Prove: _____

Construction: Label the quadrilateral π΄π΅πΆπ·, and mark opposite sides as parallel. Draw diagonal π΅π·.

**Example 2**

**If a quadrilateral is a parallelogram, then the diagonals bisect each other.** Complete the diagram, and develop an
appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a
parallelogram bisect each other. Remember, now that we have proved opposite sides and angles of a parallelogram to
be congruent, we are free to use these facts as needed (i.e., π΄π· = πΆπ΅, π΄π΅ = πΆπ·, β π΄ β
β πΆ, β π΅ β
β π·).

Given: _____

Prove: _____

Construction: Label the quadrilateral π΄π΅πΆπ·. Mark opposite sides as parallel. Draw diagonals π΄πΆ and π΅π·.

Now we have established why the properties of parallelograms that we have assumed to be true are in fact true. By extension, these facts hold for any type of parallelogram, including rectangles, squares, and rhombuses. Let us look at one last fact concerning rectangles. We established that the diagonals of general parallelograms bisect each other. Let us now demonstrate that a rectangle has congruent diagonals.

**Example 3**

**If the parallelogram is a rectangle, then the diagonals are equal in length.** Complete the diagram, and develop an
appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a rectangle
are congruent. As in the last proof, remember to use any already proven facts as needed.

Given: _____

Prove: _____

Construction: Label the rectangle πΊπ»πΌπ½. Mark opposite sides as parallel, and add small squares at the vertices to indicate 90Β° angles. Draw diagonals πΊπΌ and π»π½

**Converse Properties:** Now we examine the converse of each of the properties we proved. Begin with the property, and
prove that the quadrilateral is in fact a parallelogram.

**Example 4**

If both pairs of opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. Draw an appropriate diagram, and provide the relevant Given and Prove for this case.

Given: _____

Prove: _____

Construction: Label the quadrilateral π΄π΅πΆπ·. Mark opposite angles as congruent. Draw diagonal π΅π·. Label the measures of β π΄ and β πΆ as π₯Β°. Label the measures of the four angles created by π΅π· as πΒ°, π Β°, π‘Β°, and π’Β°

**Example 5**

If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram. Draw an appropriate diagram, and provide the relevant Given and Prove for this case.

Given: _____

Prove: _____

Construction: Label the quadrilateral π΄π΅πΆπ·, and mark opposite sides as equal. Draw diagonal π΅π·.

**Example 6**

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Draw an appropriate diagram, and provide the relevant Given and Prove for this case. Use triangle congruence criteria to demonstrate why the quadrilateral is a parallelogram.

Given: _____

Prove: _____

Construction: Label the quadrilateral π΄π΅πΆπ·, and mark opposite sides as equal. Draw diagonals π΄πΆ and π΅π·.

**Example 7**

If the diagonals of a parallelogram are equal in length, then the parallelogram is a rectangle. Complete the diagram, and develop an appropriate Given and Prove for this case.

Given: _____

Prove: _____

Construction: Label the quadrilateral πΊπ»πΌπ½. Draw diagonals πΊπΌ and π»π½.

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