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Rhombi and Squares
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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?
Tuesday, April 24, 2012
Vocabulary1. Rhombus:
Properties:
2. Square:
Properties:
Tuesday, April 24, 2012
Vocabulary1. Rhombus: A parallelogram with four congruent sides
Properties:
2. Square:
Properties:
Tuesday, April 24, 2012
Vocabulary1. Rhombus: A parallelogram with four congruent sides
Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals
2. Square:
Properties:
Tuesday, April 24, 2012
Vocabulary1. Rhombus: A parallelogram with four congruent sides
Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals
2. Square: A parallelogram with four congruent sides and four right angles
Properties:
Tuesday, April 24, 2012
Vocabulary1. Rhombus: A parallelogram with four congruent sides
Properties: All properties of parallelograms plus four congruent sides, perpendicular diagonals, and angles that are bisected by the diagonals
2. Square: A parallelogram with four congruent sides and four right angles
Properties: All properties of parallelograms, rectangles, and rhombi
Tuesday, April 24, 2012
Theorems
Diagonals of a Rhombus
6.15:
6.16:
Tuesday, April 24, 2012
Theorems
Diagonals of a Rhombus
6.15: If a parallelogram is a rhombus, then its diagonals are perpendicular
6.16:
Tuesday, April 24, 2012
Theorems
Diagonals of a Rhombus
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
Tuesday, April 24, 2012
Theorems
Conditions for Rhombi and Squares
6.17:
6.18:
6.19:
Tuesday, April 24, 2012
Theorems
Conditions for Rhombi and Squares
6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
6.18:
6.19:
Tuesday, April 24, 2012
Theorems
Conditions for Rhombi and Squares
6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
6.18: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus
6.19:
Tuesday, April 24, 2012
Theorems
Conditions for Rhombi and Squares
6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
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
Tuesday, April 24, 2012
Theorems
Conditions for Rhombi and Squares
6.20:
Tuesday, April 24, 2012
Theorems
Conditions for Rhombi and Squares
6.20: If a quadrilateral is both a rectangle and a rhombus, then it is a square
Tuesday, April 24, 2012
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.
Tuesday, April 24, 2012
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°.
Tuesday, April 24, 2012
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°
Tuesday, April 24, 2012
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° = 101°
Tuesday, April 24, 2012
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° = 101°
m∠ZYX = 101°
Tuesday, April 24, 2012
Example 1The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3, find x.
Tuesday, April 24, 2012
Example 1The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3, find x.
WX = WZ
Tuesday, April 24, 2012
Example 1The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3, find x.
WX = WZ
8x − 5 = 6x + 3
Tuesday, April 24, 2012
Example 1The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3, find x.
WX = WZ
8x − 5 = 6x + 3
2x = 8
Tuesday, April 24, 2012
Example 1The diagonals of a rhombus WXYZ intersect at V. Use the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3, find x.
WX = WZ
8x − 5 = 6x + 3
2x = 8
x = 4
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive5. LMNP is a rhombus
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6Prove: LMNP is a rhombus
1. LMNP is a parallelogram, ∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
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
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 24, 2012
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?
Tuesday, April 24, 2012
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.
Tuesday, April 24, 2012
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.
Tuesday, April 24, 2012
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
Tuesday, April 24, 2012
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
= (−5)2 + (3)2
Tuesday, April 24, 2012
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
= (−5)2 + (3)2 = 25 + 9
Tuesday, April 24, 2012
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
= (−5)2 + (3)2 = 25 + 9 = 34
Tuesday, April 24, 2012
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
= (−5)2 + (3)2 = 25 + 9 = 34
BD = (−1− 2)2 + (3 + 2)2
Tuesday, April 24, 2012
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
= (−5)2 + (3)2 = 25 + 9 = 34
BD = (−1− 2)2 + (3 + 2)2
= (−3)2 + (5)2
Tuesday, April 24, 2012
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
= (−5)2 + (3)2 = 25 + 9 = 34
BD = (−1− 2)2 + (3 + 2)2
= (−3)2 + (5)2 = 9 + 25
Tuesday, April 24, 2012
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
= (−5)2 + (3)2 = 25 + 9 = 34
BD = (−1− 2)2 + (3 + 2)2
= (−3)2 + (5)2 = 9 + 25 = 34
Tuesday, April 24, 2012
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
= (−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
Tuesday, April 24, 2012
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
= (−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
m( AC ) =
2 +13 + 2
Tuesday, April 24, 2012
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
= (−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
m( AC ) =
2 +13 + 2
=35
Tuesday, April 24, 2012
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
= (−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
m( AC ) =
2 +13 + 2
=35
m(BD) =−2 − 32 +1
Tuesday, April 24, 2012
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
= (−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
m( AC ) =
2 +13 + 2
=35
m(BD) =−2 − 32 +1
=−53
Tuesday, April 24, 2012
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
= (−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
m( AC ) =
2 +13 + 2
=35
m(BD) =−2 − 32 +1
=−53 AC ⊥ BD
Tuesday, April 24, 2012
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
= (−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
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
Tuesday, April 24, 2012
Check Your Understanding
Review #1-6 on p. 431
Tuesday, April 24, 2012
Problem Set
Tuesday, April 24, 2012
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
Tuesday, April 24, 2012
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