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    which shape must have opposite sides that are parallel and congruent, and diagonals that are perpendicular bisectors of each other?

    James

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    get which shape must have opposite sides that are parallel and congruent, and diagonals that are perpendicular bisectors of each other? from EN Bilgi.

    Rhombuses_Kites_and_Trapezia

    The Improving Mathematics Education in Schools (TIMES) Project

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    RHOMBUSES, KITES AND TRAPEZIA

    Geometry and measurement : Module 21Years : 9-10

    June 2011

    PDF Version of module

    Assumed Knowledge Motivation Content

    Symmetries of triangles, parallelograms and rectangles

    Rhombuses Squares Kites Trapezia Links Forward

    History and Applications

    Answers to Exercises

    return to top ASSUMED KNOWLEDGE

    The material in this module is a continuation of the module, Parallelograms and Rectangles, which is assumed knowledge for the present module. Thus the present module assumes:

    Confidence in writing logical argument in geometry written as a sequence of steps, each justified by a reason.

    Ruler-and-compasses constructions.

    The four standard congruence tests and their application to:

    proving properties of and tests for isosceles and equilateral triangles,

    proving properties of and tests for parallelograms and rectangles.

    Informal experience with rhombuses, kites, squares and trapezia.

    return to top MOTIVATION

    Logical argument, precise definitions and clear proofs are essential if one is to understand mathematics. These analytic skills can be transferred to many areas in commerce, engineering, science and medicine but most of us first meet them in high school mathematics.

    Apart from some number theory results such as the existence of an infinite number of primes and the Fundamental Theorem of Arithmetic, most of the theorems students meet are in geometry starting with Pythagoras’ theorem.

    Many of the key methods of proof such as proof by contradiction and the difference between a theorem and its converse arise in elementary geometry.

    As in the module, Parallelograms and Rectangles, this module first stresses precise definitions of each special quadrilateral, then develops some of its properties, and then reverses the process, examining whether these properties can be used as tests for that particular special quadrilateral. We have seen that a test for a special quadrilateral is usually the converse of a property. For example, a typical property−test pair from the previous module is the pair of converse statements:

    If a quadrilateral is a parallelogram, then its diagonals bisect each other.

    If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

    Congruence is again the basis of most arguments concerning rhombuses, squares, kites and trapezia, because the diagonals dissect each figure into triangles.

    A number of the theorems proved in this module rely on one or more of the previous theorems in the module. This means that the reader must understand a whole ‘sequence of theorems’ to achieve some results. This is typical of more advanced mathematics.

    In addition, two other matters are covered in these notes.

    The reflection and rotation symmetries of triangles and special quadrilaterals

    are identified and related to congruence.

    The tests for the kite also allow several important standard constructions to be explained very simply as constructions of a kite.

    return to top CONTENT return to top

    SYMMETRIES OF TRIANGLES, PARALLELOGRAMS AND RECTANGLES

    We begin by relating the reflection and rotation symmetries of isosceles triangles, parallelograms and rectangles to the results that we proved in the previous module, Paralleograms and Rectangles.

    The axis of symmetry of an isosceles triangle

    In the module, Congruence, congruence was used to prove that the base angles of an isosceles triangle are equal. To prove that B = C in the diagram opposite, we constructed the angle-bisector AM of the apex A, then used the SAS congruence test to prove that

    ABM ACM

    This congruence result, however, establishes much more than the equality of the base angles. It also establishes that the angle bisector AM is the perpendicular bisector of the base BC. Moreover, this fact means that AM is an axis of symmetry of the isosceles triangle.

    These basic facts of about isosceles triangles will be used later in this module and in the module, Circle Geometry:

    Theorem

    In an isosceles triangle, the following four lines coincide:

    The angle bisector of the apex angle.

    The line joining the apex and the midpoint of the base.

    The line through the apex perpendicular to the base.

    The perpendicular bisector of the base.

    This line is an axis of symmetry of the isosceles triangle. It has, as a consequence, the interesting property that the centroid, the incentre, the circumcentre and the orthocentre of ABC all tie on the line AM. In general, they are four different points. See the module, Construction for details of this.

    Extension − Some further tests for a triangle to be isosceles

    The theorem above suggests three possible tests for a triangle to be isosceles. The first two are easy to prove, but the third is rather difficult because simple congruence cannot be used in this ‘non-included angle’ situation.

    click for screencast

    EXERCISE 1

    Use congruence to prove that ABC is isosceles with AB = AC if:

    Source : amsi.org.au

    What shapes are diagonals congruent? – Hollows.info

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    What shapes are diagonals congruent?

    Table of Contents

    What shapes are diagonals congruent?

    Quadrilaterals A B

    in these quadrilaterals, the diagonals bisect each other paralellogram, rectangle, rhombus, square

    in these quadrilaterals, the diagonals are congruent rectangle, square, isosceles trapezoid

    in these quadrilaterals, each of the diagonals bisects a pair of opposite angles rhombus, square

    What are congruent diagonals?

    Properties of a square The diagonals are congruent. The diagonals are perpendicular to and bisect each other. A square is a special type of parallelogram whose all angles and sides are equal. Also, a parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

    What shapes are diagonals equal?

    A parallelogram with one right angle is a rectangle. A quadrilateral whose diagonals are equal and bisect each other is a rectangle.

    Are diagonals congruent in a rhombus?

    A rhombus is a type of parallelogram, and what distinguishes its shape is that all four of its sides are congruent. All 4 sides are congruent. Diagonals bisect vertex angles. Diagonals are perpendicular.

    Can two triangles with different perimeters are congruent?

    For instance, if one triangle has all three sides the same length as another triangle, then the triangles must be congruent; but this isn’t true for any polygons with more than three sides. So maybe, if two triangles have the same perimeter and the same area, they must be congruent.

    How do you know if a diagonal is congruent?

    The first way to prove that the diagonals of a rectangle are congruent is to show that triangle ABC is congruent to triangle DCB. Since ABCD is a rectangle, it is also a parallelogram. Since ABCD is a parallelogram, segment AB ≅ segment DC because opposite sides of a parallelogram are congruent.

    How do you prove a rhombus is congruent?

    If two consecutive sides of a parallelogram are congruent, then it’s a rhombus (neither the reverse of the definition nor the converse of a property). If either diagonal of a parallelogram bisects two angles, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

    Does a rhombus have all 4 sides equal?

    A rhombus has all sides equal, while a rectangle has all angles equal. A rhombus has opposite angles equal, while a rectangle has opposite sides equal. The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.

    What shapes have perpendicular diagonals?

    A rhombus is a parallelogram with four equal sides. The diagonals of a rhombus bisect each other and are perpendicular. A rectangle is a parallelogram with four 90° angles. The rectangle of a rhombus bisect each other and have equal length.

    Which quadrilaterals always have congruent diagonals?

    A Rectangle is a quadrilateral that must have congruent diagonals. An Isosceles trapezoid is a quadrilateral that must have congruent diagonals. A square, because it is a rectangle, is a quadrilateral that must congruent diagonals.

    Are diagonals perpendicular?

    The diagonals are perpendicular bisectors of each other. The rectangle has the following properties: All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals bisect each other). All angles are right angles by definition.

    Does rectangle have congruent diagonals?

    Diagonals of Rectangle. As you can hopefully see, both diagonals equal 13, and the diagonals will always be congruent because the opposite sides of a rectangle are congruent allowing any rectangle to be divided along the diagonals into two triangles that have a congruent hypotenuse.

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    shswisdom / Brijesh Write up 3

    Brijesh Write up 3

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    last edited by Brijesh Patel 12 years, 2 months ago

    1. The shape you get when you Construct two segments of different length that are perpendicular bisectors of each other you get a figure that can be discribed as a Rhombus and a paralellogram. (It is also a quadralateral)

    2. The shape you get when you repeat the same steps for number one but use congruent segments can be discribed as a square, rhombus, rectangle, and paralellogram. (It is also a quadralateral)

    3. The figure formed below when two segments that are not perpendicular cross can be discribed as a paralellogram.

    4. A figure that is formed with the same conditions as number three but with congruent segments can be discribed as a  square, rectangle, rhombus, and parallelogram. (It is also a quadralateral)

    5. This extremely complicated shape to construct can be discribed as a kite.

    6. I have constructed all shaped except for an isosceles trapezoid and a trapezoid. In an isosceles trapezoid the diagnals are congruent and the bases are paralell.

    7.

    Condition Quadrilateral Reasioning

    Diagonals are perpendicular Square, Kite, Rhombus As you can see with the pictures above the diagonals for these three shapes cross at a 90 degree angle, becuase of this they are congruent.

    Diagonals are perpendicular but only one diagonal is bisected Same shapes as above. Again these shape's diagonals bisect each other at a 90 degree angle making them congruent.

    Digonals are congruent and intersect but are not perpendicular Rectangle As you can see with the rectangle above it has two bisectors but these bisectors unlike the ones with the square, kite do not intercect at 90 degress.

    Diagonals bisect each other Square, Kite, Rhombus, Rectangle, paralellogram. As you can see in the pictures above all of those shapes are made with bisecting diagonals except for the concave quadrilateral.

    Diagonals are perpendicular and bisect each other. Same shapes from the first box. As you can see with the pictures above the places where the diagonals intersect are perpendicular

    Diagonals are congruent and bisect each other Square, rectangle, Trapezoid (Isosceles specifically) The diagonals (bisectors) are congruent for the these two shapes.

    Diagonals are congruent, perpendicular and bisect each other. Square For a square all the diagonals are congruent and all intersect at a 90 degree angle.

    8. As you add more conditions to describe the diagonals, less diagonals can be diecribed as a result. For exaple if you have the condition "diagonals bisect each other" then you are talking about All of the shapes made above along with trapezoids, but with the exception of the concave quadrilateral. If you add the condition "Diagonals are congruent, perpendicular, and bisect each other then you have a square not all of those shapes. JUST  THE  SQUARE.

    9.

    Figure Names Properties

    a Paralellogram/Rectangle

    Opposite sides are parallel, Opposite sides are congruent, Opposite angles are congruent, Each diagonal forms two congruent triangles, Diagonals bisect each other,

    Diagonals are congruent,  All angles are right angles. Two pairs of consecutive sides are congruent.

    b Kite Only one pair of poosite angles is congruent, Diagonals are perpendicular, Diagonals bisect vertex angles

    c Square, Rectangle, Rhombus, Paralellogram Opposite sides are parallel, Opposite sides are congruent, Opposite angles are congruent., each diagonal forms 2 congruent trianlges, Diagonals bisect each other, Diagonals are congruent, All angles are right angles, Two pairs of consecutive sides are congruent.

    d Isosceles Trapezoid Only one pair of oppsite sides is parallel. Only one pair of opposite sides is congruent, Diagonals are congruent, Two pairs of consective sides are congruent.

    e Paralellogram Opposite sides are parallel, Opposite sides are congruent, Each digaonal forms 2 congruent triangles, Diagonals bisect each other

    f Rhombus, Kite oppsite sides are parallel, Opposite sides are congruent, Opposite angles are congruent, Each diagonal forms 2 congruent triangles, Diagonals bisect each other,Diagonals are perpendicular. Diagonals bisect vertex angles, All sides are congruent

    10.

    Property Parallelogram Rectangle Rhombus Square Isos Trapezoid Kite

    oppsite sides are parallel x x x x

    Only one pair of opposite sides is parallel         x

    Opposite sides are congruent x x x x

    Opposite Only one pair of opposite sides is congruent         x

    Opposite angles are congruent x x x x

    Only one pair of opposite angles is congruent           x

    Each diagonal forms 2 congruent triangles x x x x

    Diagonals bisect each other x x x x

    Source : shswisdom.pbworks.com

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