Lesson 28: Properties of Parallelograms

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 28 M1 GEOMETRY

Student Outcomes . Students complete proofs that incorporate properties of parallelograms.

Lesson Notes Throughout this module, we have seen the theme of building new facts with the use of established ones. We see this again in Lesson 28, where triangle congruence criteria are used to demonstrate why certain properties of parallelograms hold true. We begin establishing new facts using only the definition of a parallelogram and the properties we have assumed when proving statements. Students combine the basic definition of a parallelogram with triangle congruence criteria to yield properties taken for granted in earlier grades, such as opposite sides of a parallelogram are parallel.

Classwork Opening Exercise (5 minutes)

Opening Exercise

a. If the triangles are congruent, state the congruence.

△ 푨푮푻 ≅ △ 푴풀푱

b. Which triangle congruence criterion guarantees part 1?

AAS

c. 푻푮̅̅̅̅ corresponds with

푱풀̅̅̅

Discussion/Examples 1–7 (35 minutes)

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.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 28 M1 GEOMETRY

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: Parallelogram 푨푩푪푫 (푨푩̅̅̅̅ ∥ 푪푫̅̅̅̅, 푨푫̅̅̅̅ ∥ 푪푩̅̅̅̅)

Prove: 푨푫 = 푪푩, 푨푩 = 푪푫, 풎∠푨 = 풎∠푪, 풎∠푩 = 풎∠푫

Construction: Label the quadrilateral 푨푩푪푫, and mark opposite sides as A B parallel. Draw diagonal 푩푫̅̅̅̅̅.

PROOF:

푨푩푪푫 Parallelogram Given D C 풎∠푨푩푫 = 풎∠푪푫푩 If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

푩푫 = 푫푩 Reflexive property

풎∠푪푩푫 = 풎∠푨푫푩 If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

△ 푨푩푫 ≅△ 푪푫푩 ASA

푨푫 = 푪푩, 푨푩 = 푪푫 Corresponding sides of congruent triangles are equal in length.

풎∠푨 = 풎∠푪 Corresponding angles of congruent triangles are equal in measure.

풎∠푨푩푫 + 풎∠푪푩푫 = 풎∠푨푩푪,

풎∠푪푫푩 + 풎∠푨푫푩 = 풎∠푨푫푪 Angle addition postulate

풎∠푨푩푫 + 풎∠푪푩푫 = 풎∠푪푫푩 + 풎∠푨푫푩 Addition property of equality

풎∠푩 = 풎∠푫 Substitution property of equality

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., 푨푫 = 푪푩, 푨푩 = 푪푫, ∠푨 ≅ ∠푪, ∠푩 ≅ ∠푫).

Given: Parallelogram 푨푩푪푫

Prove: Diagonals bisect each other, 푨푬 = 푪푬, 푫푬 = 푩푬

Construction: Label the quadrilateral 푨푩푪푫. Mark opposite sides as parallel. Draw diagonals 푨푪̅̅̅̅ and 푩푫̅̅̅̅̅.

PROOF:

Parallelogram 푨푩푪푫 Given

풎∠푩푨푪 = 풎∠푫푪푨 If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

풎∠푨푬푩 = 풎∠푪푬푫 Vertical angles are equal in measure.

푨푩 = 푪푫 Opposite sides of a parallelogram are equal in length.

△ 푨푬푩 ≅ △ 푪푬푫 AAS

푨푬 = 푪푬, 푫푬 = 푩푬 Corresponding sides of congruent triangles are equal in length.

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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.

Students may need a reminder that a rectangle is a parallelogram with four right angles.

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: Rectangle 푮푯푰푱

Prove: Diagonals are equal in length, 푮푰 = 푯푱

Construction: Label the rectangle 푮푯푰푱. Mark opposite sides as parallel, and add small squares at the vertices to indicate ퟗퟎ° angles. Draw diagonals 푮푰̅̅̅̅ and 푯푱̅̅̅̅.

PROOF:

Rectangle 푮푯푰푱 Given

푮푱 = 푰푯 Opposite sides of a parallelogram are equal in length.

푮푯 = 푯푮 Reflexive property

∠푱푮푯, ∠푰푯푮 are right angles. Definition of a rectangle

△ 푮푯푱 ≅ △ 푯푮푰 SAS

푮푰 = 푯푱 Corresponding sides of congruent triangles are equal in length.

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: Quadrilateral 푨푩푪푫 with 풎∠푨 = 풎∠푪, 풎∠푩 = 풎∠푫

Prove: Quadrilateral 푨푩푪푫 is a parallelogram.

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Construction: Label the quadrilateral 푨푩푪푫. Mark opposite angles as congruent.

̅̅̅̅̅ A o B Draw diagonal 푩푫. Label the measures of ∠푨 and ∠푪 as 풙°. Label xo t the measures of the four angles created by 푩푫̅̅̅̅̅ as 풓°, 풔°, 풕°, and 풖°. u o

PROOF: o r o s o x Quadrilateral 푨푩푪푫 with 풎∠푨 = 풎∠푪, 풎∠푩 = 풎∠푫 Given D C 풎∠푫 = 풓 + 풔, 풎∠푩 = 풕 + 풖 Angle addition

풓 + 풔 = 풕 + 풖 Substitution

풙 + 풓 + 풕 = ퟏퟖퟎ, 풙 + 풔 + 풖 = ퟏퟖퟎ Angles in a triangle add up to ퟏퟖퟎ°.

풓 + 풕 = 풔 + 풖 Subtraction property of equality, substitution

풓 + 풕 − (풓 + 풔) = 풔 + 풖 − (풕 + 풖) Subtraction property of equality

풕 − 풔 = 풔 − 풕 Additive inverse property

풕 − 풔 + (풔 − 풕) = 풔 − 풕 + (풔 − 풕) Addition property of equality

ퟎ = ퟐ(풔 − 풕) Addition and subtraction properties of equality

ퟎ = 풔 − 풕 Division property of equality

풔 = 풕 Addition property of equality

풔 = 풕 ⇒ 풓 = 풖 Substitution and subtraction properties of equality

푨푩̅̅̅̅ ∥ 푪푫̅̅̅̅, 푨푫̅̅̅̅ ∥ 푩푪̅̅̅̅ If two lines are cut by a transversal such that a pair of alternate interior angles are equal in measure, then the lines are parallel.

Quadrilateral 푨푩푪푫 is a parallelogram. Definition of a parallelogram

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: Quadrilateral 푨푩푪푫 with 푨푩 = 푪푫, 푨푫 = 푩푪

Prove: Quadrilateral 푨푩푪푫 is a parallelogram.

Construction: Label the quadrilateral 푨푩푪푫, and mark opposite sides as A B equal. Draw diagonal 푩푫̅̅̅̅̅.

PROOF:

Quadrilateral 푨푩푪푫 with 푨푩 = 푪푫, 푨푫 = 푪푩 Given D C 푩푫 = 푫푩 Reflexive property

△ 푨푩푫 ≅ △ 푪푫푩 SSS

∠푨푩푫 ≅ ∠푪푫푩, ∠푨푫푩 ≅ ∠푪푩푫 Corresponding angles of congruent triangles are congruent.

푨푩̅̅̅̅ ∥ 푪푫̅̅̅̅, 푨푫̅̅̅̅ ∥ 푪푩̅̅̅̅ If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Quadrilateral 푨푩푪푫 is a parallelogram. Definition of a parallelogram

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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: Quadrilateral 푨푩푪푫, diagonals 푨푪̅̅̅̅ and ̅푩푫̅̅̅̅ bisect each other.

Prove: Quadrilateral 푨푩푪푫 is a parallelogram.

Construction: Label the quadrilateral 푨푩푪푫, and mark opposite sides as equal. A B Draw diagonals 푨푪̅̅̅̅ and 푩푫̅̅̅̅̅.

PROOF: E ̅̅̅̅ Quadrilateral 푨푩푪푫, diagonals 푨푪 and 푩푫̅̅̅̅̅ bisect each other. Given D C 푨푬 = 푪푬, 푫푬 = 푩푬 Definition of a segment bisector

풎∠푫푬푪 = 풎∠푩푬푨, 풎∠푨푬푫 = 풎∠푪푬푩 Vertical angles are equal in measure.

△ 푫푬푪 ≅ △ 푩푬푨, △ 푨푬푫 ≅ △ 푪푬푩 SAS

∠푨푩푫 ≅ ∠푪푫푩, ∠푨푫푩 ≅ ∠푪푩푫 Corresponding angles of congruent triangles are congruent.

푨푩̅̅̅̅ ∥ 푪푫̅̅̅̅ , ̅푨푫̅̅̅̅ ∥ 푪푩̅̅̅̅ If two lines are cut by a transversal such that a pair of alternate interior angles are congruent, then the lines are parallel.

Quadrilateral 푨푩푪푫 is a parallelogram. Definition of a parallelogram

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: Parallelogram 푮푯푰푱 with diagonals of equal length, 푮푰 = 푯푱

Prove: 푮푯푰푱 is a rectangle.

Construction: Label the quadrilateral 푮푯푰푱. Draw diagonals 푮푰̅̅̅̅ and 푯푱̅̅̅̅. G H

PROOF:

Parallelogram 푮푯푰푱 with diagonals of equal length, 푮푰 = 푯푱 Given 푮푱 = 푱푮, 푯푰 = 푰푯 Reflexive property J I 푮푯 = 푰푱 Opposite sides of a parallelogram are congruent.

△ 푯푱푮 ≅ △ 푰푮푱, △ 푮푯푰 ≅ △ 푱푰푯 SSS

풎∠푮 = 풎∠푱, 풎∠푯 = 풎∠푰 Corresponding angles of congruent triangles are equal in measure.

풎∠푮 + 풎∠푱 = ퟏퟖퟎ°, 풎∠푯 + 풎∠푰 = ퟏퟖퟎ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary.

ퟐ(풎∠푮) = ퟏퟖퟎ°, ퟐ(풎∠푯) = ퟏퟖퟎ° Substitution property of equality

풎∠푮 = ퟗퟎ°, 풎∠푯 = ퟗퟎ° Division property of equality

풎∠푮 = 풎∠푱 = 풎∠푯 = 풎∠푰 = ퟗퟎ° Substitution property of equality

푮푯푰푱 is a rectangle. Definition of a rectangle

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Closing (1 minute) . A parallelogram is a quadrilateral whose opposite sides are parallel. We have proved the following properties of a parallelogram to be true: Opposite sides are congruent, opposite angles are congruent, diagonals bisect each other.

Exit Ticket (5 minutes)

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Name ______Date______

Lesson 28: Properties of Parallelograms

Exit Ticket

Given: Equilateral parallelogram 퐴퐵퐶퐷 (i.e., a rhombus) with diagonals 퐴퐶̅̅̅̅ and ̅퐵퐷̅̅̅ Prove: Diagonals intersect perpendicularly.

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Exit Ticket Sample Solutions

Given: Equilateral parallelogram 푨푩푪푫 (i.e., a rhombus) with diagonals 푨푪̅̅̅̅ and 푩푫̅̅̅̅̅

Prove: Diagonals intersect perpendicularly.

Rhombus 푨푩푪푫 Given

푨푬 = 푪푬, 푫푬 = 푩푬 Diagonals of a parallelogram bisect each other.

푨푩 = 푩푪 = 푪푫 = 푫푨 Definition of equilateral parallelogram

△ 푨푬푫 ≅ △ 푨푬푩 ≅ △ 푪푬푩 ≅ △ 푪푬푫 SSS

풎∠푨푬푫 = 풎∠푨푬푩 = 풎∠푩푬푪 = 풎∠푪푬푫 Corr. angles of congruent triangles are equal in measure.

풎∠푨푬푫 = 풎∠푨푬푩 = 풎∠푩푬푪 = 풎∠푪푬푫 = ퟗퟎ° Angles at a point sum to ퟑퟔퟎ°. Since all four angles are congruent, each angle measures ퟗퟎ°.

Problem Set Sample Solutions

Use the facts you have established to complete exercises involving different types of parallelograms.

1. Given: 푨푩̅̅̅̅ ∥ 푪푫̅̅̅̅, 푨푫 = 푨푩, 푪푫 = 푪푩 Prove: 푨푩푪푫 is a rhombus.

Construction: Draw diagonal 푨푪̅̅̅̅.

푨푫 = 푨푩, 푪푫 = 푪푩 Given

푨푪 = 푪푨 Reflexive property

△ 푨푫푪 ≅ △ 푪푩푨 SSS

푨푫 = 푪푩, 푨푩 = 푪푫 Corresponding sides of congruent triangles are equal in length.

푨푩 = 푩푪 = 푪푫 = 푨푫 Transitive property

푨푩푪푫 is a rhombus. Definition of a rhombus

2. Given: Rectangle 푹푺푻푼, 푴 is the midpoint of 푹푺̅̅̅̅. Prove: △ 푼푴푻 is isosceles.

Rectangle 푹푺푻푼 Given

푹푼 = 푺푻 Opposite sides of a rectangle are congruent.

∠푹, ∠푺 are right angles. Definition of a rectangle

푴 is the midpoint of 푹푺̅̅̅̅. Given

푹푴 = 푺푴 Definition of a midpoint

△ 푹푴푼 ≅ △ 푺푴푻 SAS

푼푴̅̅̅̅̅ ≅ 푻푴̅̅̅̅̅ Corresponding sides of congruent triangles are congruent.

△ 푼푴푻 is isosceles. Definition of an isosceles triangle

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3. Given: 푨푩푪푫 is a parallelogram, 푹푫̅̅̅̅̅ bisects ∠푨푫푪, 푺푩̅̅̅̅ bisects ∠푪푩푨. Prove: 푫푹푩푺 is a parallelogram.

푨푩푪푫 is a parallelogram; Given

푹푫̅̅̅̅̅ bisects ∠푨푫푪, 푺푩̅̅̅̅ bisects ∠푪푩푨.

푨푫 = 푪푩 Opposite sides of a parallelogram are congruent.

∠푨 ≅ ∠푪, ∠푩 ≅ ∠푫 Opposite angles of a parallelogram are congruent.

∠푹푫푨 ≅ ∠푹푫푺, ∠푺푩푪 ≅ ∠푺푩푹 Definition of angle bisector

풎∠푹푫푨 + 풎∠푹푫푺 = 풎∠푫, 풎∠푺푩푪 + 풎∠푺푩푹 = 풎∠푩 Angle addition

풎∠푹푫푨 + 풎∠푹푫푨 = 풎∠푫, 풎∠푺푩푪 + 풎∠푺푩푪 = 풎∠푩 Substitution

ퟐ(풎∠푹푫푨) = 풎∠푫, ퟐ(풎∠푺푩푪) = 풎∠푩 Addition ퟏ ퟏ 풎∠푹푫푨 = 풎∠푫 풎∠푺푩푪 = 풎∠푩 ퟐ , ퟐ Division ∠푹푫푨 ≅ ∠푺푩푪 Substitution

△ 푫푨푹 ≅ △ 푩푪푺 ASA

∠푫푹푨 ≅ ∠푩푺푪 Corresponding angles of congruent triangles are congruent.

∠푫푹푩 ≅ ∠푩푺푫 Supplements of congruent angles are congruent.

푫푹푩푺 is a parallelogram. Opposite angles of quadrilateral 푫푹푩푺 are congruent.

4. Given: 푫푬푭푮 is a rectangle, 푾푬 = 풀푮, 푾푿 = 풀풁 Prove: 푾푿풀풁 is a parallelogram.

푫푬 = 푭푮, 푫푮 = 푭푬 Opposite sides of a rectangle are congruent.

푫푬푭푮 is a rectangle; Given

푾푬 = 풀푮, 푾푿 = 풀풁

푫푬 = 푫푾 + 푾푬, 푭푮 = 풀푮 + 푭풀 Segment addition

푫푾 + 푾푬 = 풀푮 + 푭풀 Substitution

푫푾 + 풀푮 = 풀푮 + 푭풀 Substitution

푫푾 = 푭풀 Subtraction

풎∠푫 = 풎∠푬 = 풎∠푭 = 풎∠푮 = ퟗퟎ° Definition of a rectangle

△ 풁푮풀, △ 푿푬푾 are right triangles. Definition of right triangle

△ 풁푮풀 ≅ △ 푿푬푾 HL

풁푮 = 푿푬 Corresponding sides of congruent triangles are congruent.

푫푮 = 풁푮 + 푫풁; 푭푬 = 푿푬 + 푭푿 Partition property or segment addition

푫풁 = 푭푿 Subtraction property of equality

△ 푫풁푾 ≅ △ 푭푿풀 SAS

풁푾 = 푿풀 Corresponding sides of congruent triangles are congruent.

푾푿풀풁 is a parallelogram. Both pairs of opposite sides of a parallelogram are congruent.

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5. Given: Parallelogram 푨푩푭푬, 푪푹 = 푫푺, 푨푩푪̅̅̅̅̅̅ and 푫푬푭̅̅̅̅̅̅ are segments. Prove: 푩푹 = 푺푬

푨푩̅̅̅̅ ∥ 푭푬̅̅̅̅ Opposite sides of a parallelogram are parallel.

풎∠푩푪푹 = 풎∠푬푫푺 If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

∠푨푩푭 ≅ ∠푭푬푨 Opposite angles of a parallelogram are congruent.

∠푪푩푹 ≅ ∠푫푬푺 Supplements of congruent angles are congruent.

푪푹 = 푫푺 Given

△ 푪푩푹 ≅ △ 푫푬푺 AAS

푩푹 = 푺푬 Corresponding sides of congruent triangles are equal in length.

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