Geometry Unit 6 Note Sheets

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Date Name of Lesson

6.2 Parallelograms

6.3 Tests for Parallelograms

6.4 Rectangles

6.5 Rhombi and Squares

6.6 Trapezoids and Kites

Geometry Unit 6 Note Sheets

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Quadrilaterals Properties Property Parallelogram Rectangle Rhombus Square Quadrilateral Trapezoid Isosceles

Trapezoid Kite

Both pairs of opp sides are A A A A S N N N

Exactly 1 pair of opp sides are N N N N S A A N

Diagonals are ^ S S A A S S S A Diagonals are @ S A S A S A A S

Diagonals bisect each other

A A A A S S S S

Interior Angles add up to 360 degrees. A A A A A A A A

Both pairs of opp sides are @

A A A A S S S N

All sides are @ S S A A S S S S

Both pairs of opp angles are @

A A A A S S S S

Exactly 1 pair of opp angles are @

N N N N S S S A

All angles are @ A A S A S S S S

All Ð ’s @ A A S A S S S S

Base Ð ’s @ n/a n/a n/a n/a A S A n/a

A – Always N – Never S – Sometimes n/a – not applicable

Geometry Unit 6 Note Sheets

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6.2 Parallelograms Notes Parallelogram ______________________________________________________________________________

__________________________________________________________________________________________

Properties of Parallelograms Rule

Example Rule

Example Rule

Example Rule

Example Rule

Example Rule

Example

Geometry Unit 6 Note Sheets

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Guided Practice Find the value of each variable in the parallelogram. 1. 2. Your Turn 3. 4. Guided Practice 5. Your Turn 6.

Geometry Unit 6 Note Sheets

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Guided Practice 7. Determine the coordinates of the intersection of the diagonals of parallelogram FGHJ with vertices !(−2, 4), ((3, 5), +(2,−3), and ,(−3,−4). Steps: 1. Name the diagonals. 2. Find the midpoint of the diagonals. Your Turn 8. Determine the coordinates of the intersection of the diagonals of parallelogram RSTU with vertices -(−8,−2), /(−6, 7), 2(6, 7), and 3(4,−2). Steps: 1. Name the diagonals. 2. Find the midpoint of the diagonals. 9.

Geometry Unit 6 Note Sheets

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6.3 Tests for Parallelograms Notes Conditions for Parallelograms

Rule

Example Rule

Example Rule

Example Rule

Example

Guided Practice Determine whether the quadrilateral is a parallelogram. Justify your answer. 1. 2. Your Turn 3. 4.

Geometry Unit 6 Note Sheets

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Guided Practice 5. If !4 = 36 − 1, 4( = 48 + 3, ,4 = 68 − 2, and 4+ = 26 + 3, find 6 and 8 so that the quadrilateral is a parallelogram. Your Turn 6. Find 6 and 8 so that the quadrilateral is a parallelogram. Guided Practice Graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer. 7. 4 2, 3 , : 8, 4 , ;(7,−2), and <(1,−3) Steps: 1. Find the distance of each side. 2. Find the slopes of sides.

Geometry Unit 6 Note Sheets

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Your Turn 8. =(−1, 3), -(3, 1), /(2, −3), and 2(−2,−1) Steps: 1. Find the midpoint of the diagonals. 2. Find the slopes of sides. Guided Practice 9. A student is given the following information and then asked to write a paragraph proof. Determine which statement would correctly complete the student’s proof. Given: Parallelogram >-/2 and Parallelogram >=?3 Prove: ∠? ≅ ∠/

Proof: We are given Parallelogram >-/2 and Parallelogram >=?3. Since opposite angles of a parallelogram are congruent, ∠> ≅ ∠? and ∠> ≅ ∠/. __________________.

A. Therefore, ∠? ≅ ∠/ by the Transitive Property of Congruence.

B. Therefore, ∠? ≅ ∠/ by the Transformative Property of Congruence.

C. Therefore, ∠? ≅ ∠/ by the Reflective Property of Congruence.

D. Therefore, ∠? ≅ ∠/ by the Reflexive Property of Congruence.

Geometry Unit 6 Note Sheets

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6.4 Rectangles Notes

Rectangle _________________________________________________________________________________ Properties of Rectangles

Diagonals of a Rectangle

Rule

Example Rule

Example

Guided Practice A rectangular park has two walking paths as shown. 1. If >/ = 180DEFEGH and >- = 200DEFEGH, find =2. 2. If D∠>-/ = 64°, find D∠/=-. Your Turn 3. Using the rectangular park, if 2/ = 120DEFEGH find >-.

Geometry Unit 6 Note Sheets

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Guided Practice 4. Quadrilateral ,4:; is a rectangle. If D∠4,: = 26 + 4 and D∠,:4 = 76 + 5, find 6. Your Turn 5. Quadrilateral -/23 is a rectangle. If D∠-23 = 86 + 4 and D∠/3- = 36 − 2, find 6. Guided Practice 6. Quadrilateral >=-/ has verticies >(−5, 3), =(1,−1), -(−1,−4), and /(−7, 0). Determine whether >=-/ is a rectangle. Find the area. Steps: 1. Find the distance of each side. 2. Find the distance of the diagonals. 3. Find the area.

Geometry Unit 6 Note Sheets

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Your Turn 7. Quadrilateral ,4:; has verticies ,(−2, 3), 4(1, 4), :(3, −2), and ;(0,−3). Determine whether ,4:; is a rectangle. Steps: 1. Find the slopes. Guided Practice 8. What is the value of x in the rectangle?

Your Turn 9. What is the value of x in the rectangle?

Geometry Unit 6 Note Sheets

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6.5 Rhombi and Squares Notes

Rhombus _________________________________________________________________________________

Square ___________________________________________________________________________________

Diagonals of a Rhombus Rule

Example Rule

Example

Guided Practice The diagonals of rhombus !(+, intersect at 4. Use the given information to find the each measure or value. 1. If D∠!,+ = 82, find D∠4+,. 2. If (+ = 6 + 9 and ,+ = 56 − 2, find 6. Your Turn The diagonals of rhombus KLMN intersect at ?. Use the given information to find the each measure or value. 3. If D∠KNL = 39.5, find D∠NML. 4. If KL = 86 − 5 and KN = 66 + 3, find 6.

Geometry Unit 6 Note Sheets

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Guided Practice 5. ,4:; is a rhombus. If D∠,;: = 84and D∠,4; = (56 + 16)°, find the value of 6.

Your Turn 6. ,4:; is a rhombus. If D∠,;: = 100and D∠,4; = (36 + 20)°, find the value of 6.

Geometry Unit 6 Note Sheets

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Conditions for Rhombi and Squares Rule

Example Rule

Example Rule

Example Rule

Guided Practice 5. Determine whether parallelogram JKLM with vertices ,(−7,−2), 4(0, 4), :(9, 2), and ;(2,−4) is a rhombus, a rectangle, or a square. List all that apply. Explain (Remember to find distances, slopes, midpoints to help explain.) Your Turn 6. Determine whether parallelogram ABCD with vertices P(−2,−1), Q(−1, 3), R(3, 2), and S(2,−2) is a rhombus, a rectangle, or a square. List all that apply. Explain

Geometry Unit 6 Note Sheets

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6.6 Trapezoids and Kites Notes Trapezoid _______________________________________________________________

________________________________________________________________________

Isosceles Trapezoid ________________________________________________________

Isosceles Trapezoids Rule

Example Rule

Example Rule

Example

Guided Practice The speaker shown is an isosceles trapezoid. If D∠!,+ = 85, !4 = 8TUVℎEH, and ,( = 19TUVℎEH, find each measure. 1. D∠!(+ 2. 4+ Your Turn Each side of the basket shown is an isosceles trapezoid. If D∠,;: = 130, 4< = 6.7XEEF, and ;< =3.6XEEF, find each measure. 3. D∠;,4 4. ,:

Geometry Unit 6 Note Sheets

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Guided Practice 5. Quadrilateral ABCD has vertices P(−3, 4), Q(2, 5), R(3, 3), and S(−1, 0). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. Your Turn 6. Quadrilateral ABCD has vertices P(5, 1), Q(−3,−1), R(−2, 3), and S(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.

Geometry Unit 6 Note Sheets

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midsegment of a trapezoid ____________________________________________________________________

Trapezoid Midsegment Theorem Rule

Example

Guided Practice 7. In the figure :+ is the midsegment of trapezoid !(,4. What is the value of x? Your Turn 8. In the figure ;< is the midsegment of trapezoid !(,4. What is the value of x?

Geometry Unit 6 Note Sheets

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Kite ______________________________________________________________________________________

Kites Rule

Example Rule

Example

Guided Practice 9. If !(+, is a kite, find D∠(!,. 10. If KLMN is a kite, find NM. Your Turn 11. If KLMN is a kite, find D∠LMN. 12. If ;<>= is a kite, find <>.