geometry section 6-5 1112

Section 6-5Rhombi and Squares

Tuesday, April 24, 2012

Essential Questions

• How do you recognize and apply the properties of rhombi and squares?

• How do you determine whether quadrilaterals are rectangles, rhombi, or squares?


Vocabulary1. Rhombus:

Properties:

2. Square:

Properties:


Vocabulary1. Rhombus: A parallelogram with four congruent sides

Properties:

2. Square:

Properties:



Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals

2. Square:

Properties:




2. Square: A parallelogram with four congruent sides and four right angles

Properties:




2. Square: A parallelogram with four congruent sides and four right angles

Properties: All properties of parallelograms, rectangles, and rhombi


Theorems

Diagonals of a Rhombus

6.15:

6.16:


Theorems


6.15: If a parallelogram is a rhombus, then its diagonals are perpendicular

6.16:


Theorems


6.15: If a parallelogram is a rhombus, then its diagonals are perpendicular

6.16: If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles


Theorems

Conditions for Rhombi and Squares

6.17:

6.18:

6.19:


Theorems


6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus

6.18:

6.19:


Theorems



6.18: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus

6.19:


Theorems



6.18: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus

6.19: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus


Theorems


6.20:


Theorems


6.20: If a quadrilateral is both a rectangle and a rhombus, then it is a square


Example 1The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value.

a. If m∠WZX = 39.5°, find m∠ZYX.




Since m∠WZX = 39.5°, we know that m∠WZY = 79°.





180° − 79°





180° − 79° = 101°





180° − 79° = 101°

m∠ZYX = 101°



b. If WX = 8x − 5 and WZ = 6x + 3, find x.




WX = WZ




WX = WZ

8x − 5 = 6x + 3




WX = WZ

8x − 5 = 6x + 3

2x = 8




WX = WZ

8x − 5 = 6x + 3

2x = 8

x = 4


Example 2

Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus


Example 2


1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6


Example 2



1. Given


Example 2



1. Given

2. LM ∣∣ PN and MN ∣∣ PL


Example 2



1. Given

2. Def. of parallelogram2. LM ∣∣ PN and MN ∣∣ PL


Example 2



1. Given

2. Def. of parallelogram

3. ∠5 ≅ ∠1



Example 2



1. Given


3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅2. LM ∣∣ PN and MN ∣∣ PL


Example 2



1. Given


3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5



Example 2



1. Given


3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive



Example 2



1. Given


3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive5. LMNP is a rhombus



Example 2



1. Given


3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive5. LMNP is a rhombus 5. One diagonal bisects

opposite angles



Example 3Matt Mitarnowski is measuring the boundary of a new garden. He wants the garden to be square. He

has set each of the corner stakes 6 feet apart. What does Matt need to know to make sure that

the garden is a square?


Example 3Matt Mitarnowski is measuring the boundary of a new garden. He wants the garden to be square. He

has set each of the corner stakes 6 feet apart. What does Matt need to know to make sure that

the garden is a square?

Matt knows he at least has a rhombus as all four sides are congruent. To make it a square, it must also be a rectangle, so the diagonals must also be

congruent, as anything that is both a rhombus and a rectangle is also a square.


Example 4Determine whether parallelogram ABCD is a

rhombus, rectangle, or square for A(−2, −1), B(−1, 3), C(3, 2), and D(2, −2). List all that apply and

explain how you know.





AC = (−2 − 3)2 + (−1− 2)2





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34

The diagonals are congruent, so it is a rectangle





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34


m( AC ) =

2 +13 + 2





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34


m( AC ) =

2 +13 + 2

=35





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34


m( AC ) =

2 +13 + 2

=35

m(BD) =−2 − 32 +1





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34


m( AC ) =

2 +13 + 2

=35

m(BD) =−2 − 32 +1

=−53





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34


m( AC ) =

2 +13 + 2

=35

m(BD) =−2 − 32 +1

=−53 AC ⊥ BD





AC = (−2 − 3)2 + (−1− 2)2

= (−5)2 + (3)2 = 25 + 9 = 34

BD = (−1− 2)2 + (3 + 2)2

= (−3)2 + (5)2 = 9 + 25 = 34


m( AC ) =

2 +13 + 2

=35

m(BD) =−2 − 32 +1

=−53 AC ⊥ BD

The diagonals are ⊥, so it is a square, thus also a rhombus


Check Your Understanding

Review #1-6 on p. 431


Problem Set


Problem Set

p. 431 #7-33 odd, 46, 55, 57, 59

"Courage is saying, 'Maybe what I'm doing isn't working; maybe I should try something else.'"

- Anna Lappe


geometry section 6-5 1112

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