Prove that.
(i) diagonals of a rectangle are equal.
(ii) diagonals of a square are equal.
Answers
Step-by-step explanation:
parallelogram is a quadrilateral whose opposite sides are parallel. Thus the quadrilateral ABCD shown opposite is a parallelogram because AB || DC and DA || CB.
The word ‘parallelogram’ comes from Greek
words meaning ‘parallel lines’.
Constructing a parallelogram using the definition
To construct a parallelogram using the definition, we can use the copy-an-angle construction to form parallel lines. For example, suppose that we are given the intervals AB and AD in the diagram below. We extend AD and AB and copy the angle at A to corresponding angles at B and D to determine C and complete the parallelogram ABCD. (See the module, Construction.)
This is not the easiest way to construct a parallelogram.
First property of a parallelogram − The opposite angles are equal
The three properties of a parallelogram developed below concern first, the interior angles, secondly, the sides, and thirdly the diagonals. The first property is most easily proven using angle-chasing, but it can also be proven using congruence.
Theorem
The opposite angles of a parallelogram are equal.
Proof
Let ABCD be a parallelogram, with angleA = α and angleB = β.
Prove that angleC = α and angleD = β.
α + β = 180° (co-interior angles, AD || BC),
so angleC = α (co-interior angles, AB || DC)
and angleD = β (co-interior angles, AB || DC).
Second property of a parallelogram − The opposite sides are equal
As an example, this proof has been set out in full, with the congruence test fully developed. Most of the remaining proofs however, are presented as exercises, with an abbreviated version given as an answer.
Theorem
The opposite sides of a parallelogram are equal.
Proof
ABCD is a parallelogram.
To prove that AB = CD and AD = BC.
Join the diagonal AC.
In the triangles ABC and CDA:
angleBAC = angleDCA (alternate angles, AB || DC)
angleBCA = angleDAC (alternate angles, AD || BC)
AC = CA (common)
so triangleABC ≡ triangleCDA (AAS)
Hence AB = CD and BC = AD (matching sides of congruent triangles).
Third property of a parallelogram − The diagonals bisect each other
Theorem
The diagonals of a parallelogram bisect each other.
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EXERCISE 1
a Prove that triangleABM ≡ triangleCDM.
b Hence prove that the diagonals bisect each other.
As a consequence of this property, the intersection of the diagonals is the centre of two concentric circles, one through each pair of opposite vertices.
Notice that, in general, a parallelogram does not have a circumcircle through all four vertices.
First test for a parallelogram − The opposite angles are equal
Besides the definition itself, there are four useful tests for a parallelogram. Our first test is the converse of our first property, that the opposite angles of a quadrilateral are equal.
Theorem
If the opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram.
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EXERCISE 2
Prove this result using the figure below.
2 In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted {\displaystyle \square }\square ABCD.[1][2]
Square
Regular polygon 4 annotated.svg
A regular quadrilateral
Type
Regular polygon
Edges and vertices
4
Schläfli symbol
{4}
Coxeter diagram
CDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.png
Symmetry group
Dihedral (D4), order 2×4
Internal angle (degrees)
90°
Dual polygon
Self
Properties
Convex, cyclic, equilateral, isogonal, isotoxal
Answer
Let ABCD be a square. Let the diagonals AC and BD intersect each other at a point O. To prove
that the diagonals of a square are equal and bisect each other at right angles, we have to
prove AC = BD, OA = OC, OB = OD.
In ΔABC and ΔDCB,
AB = DC (Sides of a square are equal to each other)
∠ABC = ∠DCB (All interior angles are of 90)
BC = CB (Common side)
So, ΔABC ≅ ΔDCB (By SAS congruency)
Hence, AC = DB (By CPCT)
Hence, the diagonals of a square are equal in length.
For Rectangle--->
Answer
Given ABCD is a rectangle
Given ABCD is a rectanglethen AC and BD are diagonals
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angle
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common side
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CD
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CDso by SAS congruency,
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CDso by SAS congruency,triangle ABC is congruent to BCD
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CDso by SAS congruency,triangle ABC is congruent to BCDso by cpct,
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CDso by SAS congruency,triangle ABC is congruent to BCDso by cpct,AC = BD
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CDso by SAS congruency,triangle ABC is congruent to BCDso by cpct,AC = BDso diagonals are equal
Given ABCD is a rectanglethen AC and BD are diagonalsthen in triangle ABC and BCD,angle b is common angleBC is common sideAB = CDso by SAS congruency,triangle ABC is congruent to BCDso by cpct,AC = BDso diagonals are equalhence proved
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