Rectangles, rhombuses, and also squares space parallelograms identified by their diagonals, angles, and sides.

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Classifying Parallelograms

Rectangles, rhombuses (also dubbed rhombi) and squares room all more specific execution of parallelograms, also called distinct parallelograms.

A square is a rectangle if and also only if it has four right (congruent) angles.
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Figure (PageIndex1)

(ABCD) is a rectangle if and also only if (angle Acong angle Bcong angle Ccong angle D).

A quadrilateral is a rhombus if and also only if it has four congruent sides.
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(ABCD) is a rhombus if and also only if (overlineABcong overlineBC cong overlineCD cong overlineAD).

A quadrilateral is a square if and only if that has four right angles and four congruent sides. Through definition, a square is a rectangle and also a rhombus.
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Figure (PageIndex3)

(ABCD) is a square if and only if (angle Acong angle Bcong angle Ccong angle D) and (overlineABcong overlineBC cong overlineCD cong overlineAD).

You can always show that a parallelogram is a rectangle, rhombus, or square by utilizing the meanings of these shapes. There space some added ways to prove parallelograms space rectangles and also rhombuses, shown below:

1. A parallel is a rectangle if the diagonals room congruent.

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(ABCD) is parallelogram. If (overlineACcong overlineBD), climate (ABCD) is likewise a rectangle.

2. A parallelogram is a rhombus if the diagonals space perpendicular.

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Solution

For the an initial figure, every sides space congruent and one angle is (135^circ), therefore the angles are not congruent. This is a rhombus.

For the second figure, all four angles are congruent but the sides are not. This is a rectangle.


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Is a rhombus SOMETIMES, ALWAYS, or never ever a square? define why.

Solution

A rhombus has four congruent sides and a square has four congruent sides and angles. Therefore, a rhombus is a square once it has congruent angles. This method a rhombus is occasionally a square.


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List everything you know about the square (SQRE).

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Solution

A square has actually all the nature of a parallelogram, rectangle and rhombus.

Properties that a ParallelogramProperties the a RhombusProperties that a Rectangle
(overlineSQparallel overlineER) (overlineSQcong overlineERcong overlineSEcong overlineQR) (mangle SER=mangle SQR=mangle QSE=mangle QRE=90^circ)
(overlineSEparallel overlineQR) (overlineSRperp overlineQE)
(angle SEQcong angle QERcong angle SQEcong angle EQR) (overlineSRcong overlineQE)
(angle QSRcong angle RSEcong angle QRScong angle SRE) (overlineSAcong overlineARcong overlineQAcong overlineAE)

All the bisected angles room (45^circ).


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figure (PageIndex24)

For concerns 16-19 identify if the adhering to are ALWAYS, SOMETIME, or never true. Define your reasoning.

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A rectangle is a rhombus. A square is a parallelogram. A parallel is regular. A square is a rectangle.