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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Warm Up
1.
Find AB for A (–3, 5) and B (1, 2).
2.
Find the slope of JK for J(–4, 4) and K(3, –3).
ABCD is a parallelogram. Justify each statement.
3.
∠ABC ≅ ∠CDA
4. ∠AEB ≅ ∠CED
5
–1
Vert. ∠s Thm.
opp. ∠s ≅
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
When you are given a parallelogram with certainproperties, you can use the theorems below to determine whether the parallelogram is a rectangle.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 1: Carpentry Application
A manufacture builds a mold for a desktop so that
, , and m∠ABC = 90°. Why must ABCD be a rectangle?
Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since m∠ABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem 6-5-1.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Check It Out! Example 1
A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle?
Both pairs of opp. sides of WXYZ are ≅, so WXYZ is a parallelogram. The contractor can use the carpenter’s square to see if one ∠
of WXYZ is a
right ∠. If one angle is a right ∠, then by Theorem 6-5-1 the frame is a rectangle.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Below are some conditions you can use to determine whether a parallelogram is a rhombus.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram.
Caution
To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
You can also prove that a given quadrilateral is arectangle, rhombus, or square by using the definitions of the special quadrilaterals.
Remember!
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 2A: Applying Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given:Conclusion: EFGH is a rhombus.
The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. By Theorem 6-5-4, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply either theorem, you must first know that ABCD is a parallelogram.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 2B: Applying Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given:
Conclusion: EFGH is a square.
Step 1
Determine if EFGH is a parallelogram.
Given
EFGH is a parallelogram. Quad. with diags. bisecting each other
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Angle Relationships in TrianglesConditions for Special Parallelograms
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Example 2B Continued
Step 2
Determine if EFGH is a rectangle.
Given.
EFGH is a rectangle.
Step 3
Determine if EFGH is a rhombus.
EFGH is a rhombus.
with diags. ≅ rect.
with one pair of cons. sides ≅ rhombus
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 2B Continued
Step 4
Determine is EFGH is a square.
Since EFGH is a rectangle and a rhombus, it has four right angles and four congruent sides. So EFGH is a square by definition.
The conclusion is valid.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Check It Out! Example 2
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given:
∠ABC is a right angle.
Conclusion:
ABCD is a rectangle.
The conclusion is not valid. By Theorem 6-5-1, if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. To apply this theorem, you need to know that ABCD is a parallelogram .
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 3A: Identifying Special Parallelograms in the Coordinate Plane
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 3A Continued
Step 1
Graph PQRS.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 2
Find PR and QS to determine is PQRS is a rectangle.
Example 3A Continued
Since , the diagonals are congruent. PQRS is a rectangle.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 3
Determine if PQRS is a rhombus.
Step 4
Determine if PQRS is a square.
Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.
Example 3A Continued
Since , PQRS is a rhombus.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Example 3B: Identifying Special Parallelograms in the Coordinate Plane
W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3)
Step 1
Graph WXYZ.
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 2
Find WY and XZ to determine is WXYZ is a rectangle.
Thus WXYZ is not a square.
Example 3B Continued
Since , WXYZ is not a rectangle.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 3
Determine if WXYZ is a rhombus.
Example 3B Continued
Since (–1)(1) = –1, , PQRS is a rhombus.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Check It Out! Example 3A
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
K(–5, –1), L(–2, 4), M(3, 1), N(0, –4)
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 1
Graph KLMN.
Check It Out! Example 3A Continued
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Check It Out! Example 3A Continued
Step 2
Find KM and LN to determine is KLMN is a rectangle.
Since , KMLN is a rectangle.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 3
Determine if KLMN is a rhombus.
Since the product of the slopes is –1, the two lines are perpendicular. KLMN is a rhombus.
Check It Out! Example 3A Continued
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 4
Determine if PQRS is a square.
Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.
Check It Out! Example 3A Continued
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Check It Out! Example 3B
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Check It Out! Example 3B Continued
Step 1
Graph PQRS.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 2
Find PR and QS to determine is PQRS is a rectangle.
Check It Out! Example 3B Continued
Since , PQRS is not a rectangle. Thus PQRS is not a square.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Step 3
Determine if KLMN is a rhombus.
Check It Out! Example 3B Continued
Since (–1)(1) = –1, are perpendicular and congruent. KLMN is a rhombus.
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Lesson Quiz: Part I
1.
Given that AB = BC = CD = DA, what additional information is needed to conclude that ABCD is a square?
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Lesson Quiz: Part II
2.
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.Given:
PQRS and PQNM are parallelograms.
Conclusion:
MNRS is a rhombus.
valid
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Angle Relationships in TrianglesConditions for Special Parallelograms
Holt Geometry
Lesson Quiz: Part III
3.
Use the diagonals to determine whether a parallelogram with vertices A(2, 7), B(7, 9), C(5, 4), and D(0, 2) is a rectangle, rhombus, or square. Give all the names that apply.
AC ≠ BD, so ABCD is not a rect. or a square. The slope of AC = –1, and the slope of BD= 1, so AC ⊥
BD. ABCD is a rhombus.