Lesson Properties of the sides of a parallelogram

Source code of 'Properties of the sides of a parallelogram'

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<H2>Properties of the sides of a parallelogram</H2>

Let me remind you that a parallelogram is a quadrilateral which has both pairs of the opposite sides parallel.

Theorem 1

In a parallelogram, &nbsp;the opposite sides are of equal length in pairs.

<TABLE>
 <TR>
 <TD>
Proof

Let &nbsp;ABCD&nbsp; be a parallelogram &nbsp;(Figure 1)&nbsp; with the sides &nbsp;AB, &nbsp;BC, DC&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

and &nbsp;AD. &nbsp;We need to prove that the opposite sides &nbsp;AB&nbsp; and &nbsp;DC&nbsp; are

of equal length, &nbsp;as well as the sides &nbsp;BC&nbsp; and &nbsp;AD.

Let us draw the diagonal &nbsp;BD&nbsp; in the parallelogram &nbsp;ABCD&nbsp; and consider

the triangles &nbsp;ABD&nbsp; and &nbsp;DCB&nbsp; (Figure 2).

The diagonal &nbsp;BD&nbsp; is the common side of these triangles. &nbsp;The angles

ABD&nbsp; and &nbsp;BDC&nbsp; are congruent as the alternate interior angles at

the parallel lines &nbsp;AB&nbsp; and &nbsp;DC&nbsp; and the transverse &nbsp;BD. &nbsp;The angles

</TD>
 <TD>
{{{drawing( 520, 150, 0, 10.4, 0, 3, 
 line( 0.3, 0.5, 3.7, 0.5), 
 line( 0.3, 0.5, 1.65, 2.5),
 line( 1.65, 2.5, 5.15, 2.5),
 line( 3.7, 0.5, 5.15, 2.5),

locate(0.3, 0.5,  A),
            locate(3.7, 0.5,  B),
            locate(5.1, 2.85, C),
            locate(1.4, 2.85, D),

line( 2.0,  0.6,  2.0,  0.4),

line( 2.9,  2.6,  2.9,  2.4),

line( 0.88,  1.5,  1.08,  1.5),
            line( 0.88,  1.58, 1.08,  1.58),

line( 4.35,  1.5,  4.55,  1.5),
            line( 4.35,  1.58, 4.55,  1.58),

line( 5.3,  0.5, 8.7,    0.5), 
            line( 5.3,  0.5, 6.65,   2.5),
            line( 6.65, 2.5, 10.15,  2.5),
            line( 8.7,  0.5, 10.15,  2.5),

locate(5.3,  0.5,  A),
            locate(8.7,  0.5,  B),
            locate(10.1, 2.85, C),
            locate(6.4,  2.85, D),

line( 6.65,  2.5, 8.7,  0.5),

line( 7.0,  0.6,  7.0,  0.4),

line( 7.9,  2.6,  7.9,  2.4),

line( 5.9,  1.5,  6.13,  1.5),
            line( 5.9,  1.58, 6.13,  1.58),

line( 9.3,  1.5,  9.55,  1.5),
            line( 9.3,  1.58, 9.55,  1.58),

line( 7.55,  1.5, 7.75,  1.5),

arc( 6.65, 2.5, 0.6, 0.6, 50,  120),

arc( 8.7,  0.5, 0.6, 0.6, 230, 300),

arc( 6.65, 2.5, 0.7, 0.7, 0, 45),
            arc( 6.65, 2.5, 0.8, 0.8, 0, 45),

arc( 8.7, 0.5, 0.7, 0.7, 180, 225),
 arc( 8.7, 0.5, 0.8, 0.8, 180, 225)
)}}}
&nbsp;&nbsp;Figure 1. To the Theorem 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. To the proof of the Theorem 1
 </TD>
 </TR>
</TABLE>ADB&nbsp; and &nbsp;DBC&nbsp; are congruent as the alternate interior angles at the parallel lines &nbsp;AD&nbsp; and &nbsp;BC&nbsp; and the transverse &nbsp;BD.

Hence, &nbsp;the triangles &nbsp;ABD&nbsp; and &nbsp;DCB&nbsp; are congruent in accordance to the &nbsp;postulate P2 (ASA)&nbsp; of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A>, &nbsp;which is under the

topic &nbsp;Triangles&nbsp; of the section &nbsp;Geometry&nbsp; in this site.

This implies that the sides &nbsp;AB&nbsp; and &nbsp;DC&nbsp; are of equal length as the corresponding sides the triangles &nbsp;ABD&nbsp; and &nbsp;DCB.

Similarly, &nbsp;the sides &nbsp;BC&nbsp; and &nbsp;AD&nbsp; are of equal length by the same reason.

The proof is completed.

The converse statement is valid too.

Theorem 2

If in a convex quadrilateral the opposite sides are of equal length by pairs, &nbsp;then the quadrilateral is a parallelogram.

<TABLE>
 <TR>
 <TD>
Proof

The &nbsp;Figure 3&nbsp; shows a convex quadrilateral &nbsp;ABCD&nbsp; with the sides &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

AB, &nbsp;BC, &nbsp;DC&nbsp; and &nbsp;AD.

The opposite sides &nbsp;AB&nbsp; and &nbsp;DC&nbsp; are of equal length: &nbsp;AB = DC.

The opposite sides &nbsp;AD&nbsp; and &nbsp;BC&nbsp; are of equal length too: &nbsp;AD = BC.

We need to prove that the quadrilateral &nbsp;ABCD&nbsp; is a parallelogram.

Draw the diagonal &nbsp;BD&nbsp; in the parallelogram &nbsp;ABCD&nbsp; and consider 
the triangles &nbsp;ABD&nbsp; and &nbsp;DCB&nbsp; (Figure 4).

These triangles have the congruent sides &nbsp;AB&nbsp; and &nbsp;DC&nbsp; by the condition.

The sides &nbsp;AD&nbsp; and &nbsp;BC&nbsp; are congruent by the condition, &nbsp;too.

The side &nbsp;BD&nbsp; is the common side.

locate(0.3, 0.5,  A),
            locate(3.7, 0.5,  B),
            locate(5.1, 2.85, C),
            locate(1.4, 2.85, D),

line( 2.0,  0.6,  2.0,  0.4),

line( 2.9,  2.6,  2.9,  2.4),

line( 0.88,  1.5,  1.08,  1.5),
            line( 0.88,  1.58, 1.08,  1.58),

line( 4.35,  1.5,  4.55,  1.5),
            line( 4.35,  1.58, 4.55,  1.58),

line( 5.3,  0.5, 8.7,    0.5), 
            line( 5.3,  0.5, 6.65,   2.5),
            line( 6.65, 2.5, 10.15,  2.5),
            line( 8.7,  0.5, 10.15,  2.5),

locate(5.3,  0.5,  A),
            locate(8.7,  0.5,  B),
            locate(10.1, 2.85, C),
            locate(6.4,  2.85, D),

line( 6.65,  2.5, 8.7,  0.5),

line( 7.0,  0.6,  7.0,  0.4),

line( 7.9,  2.6,  7.9,  2.4),

line( 5.9,  1.5,  6.13,  1.5),
            line( 5.9,  1.58, 6.13,  1.58),

line( 9.3,  1.5,  9.55,  1.5),
            line( 9.3,  1.58, 9.55,  1.58),

line( 7.55,  1.5, 7.75,  1.5),

arc( 6.65, 2.5, 0.6, 0.6, 50,  120),

arc( 8.7,  0.5, 0.6, 0.6, 230, 300),

arc( 6.65, 2.5, 0.7, 0.7, 0, 45),
            arc( 6.65, 2.5, 0.8, 0.8, 0, 45),

arc( 8.7, 0.5, 0.7, 0.7, 180, 225),
 arc( 8.7, 0.5, 0.8, 0.8, 180, 225)
)}}}
&nbsp;&nbsp;Figure 3. To the Theorem 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 4. To the proof of the Theorem 2
 </TD>
 </TR>
</TABLE>

Hence, &nbsp;the triangles &nbsp;ABD&nbsp; and &nbsp;DCB&nbsp; are congruent in accordance to the &nbsp;postulate P3 (SSS)&nbsp; of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A>, &nbsp;which is under the

topic &nbsp;Triangles&nbsp; in the section &nbsp;Geometry&nbsp; in this site.

This means that the angles &nbsp;ABD&nbsp; and &nbsp;BDC&nbsp; are congruent as the corresponding angles of the congruent triangles &nbsp;ABD&nbsp; and &nbsp;DCB.

Hence, &nbsp;the straight lines &nbsp;AD&nbsp; and &nbsp;BC&nbsp; are parallel as they have congruent alternate interior angles &nbsp;ABD&nbsp; and &nbsp;BDC.

This also means that the angles &nbsp;ABD&nbsp; and &nbsp;DBC&nbsp; are congruent as the corresponding angles of the congruent triangles &nbsp;ABD&nbsp; and &nbsp;DCB.

Hence, &nbsp;the straight lines &nbsp;AB&nbsp; and &nbsp;BC&nbsp; are parallel as they have congruent alternate interior angles &nbsp;ADB&nbsp; and &nbsp;DBC.

The proof is completed.

My other lessons on parallelograms in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/In-a-parallelogram-each-diagonal-divides-it-in-two-congruent-triangles.lesson>In a parallelogram, each diagonal divides it in two congruent triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-the-sides-of-parallelograms.lesson>Properties of the sides of parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Opposite-angles-of-a-parallelogram-are-congruent.lesson>Opposite angles of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Consecutive-angles-of-a-parallelogram.lesson>Consecutive angles of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Midpoints-of-a-quadrilateral-are-vertices-of-the-parallelogram.lesson>Midpoints of a quadrilateral are vertices of the parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-length-of-diagonals-of-a-parallelogram.lesson>The length of diagonals of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Parallelograms/Remarcable-advanced-problems-on-parallelograms.lesson>Remarcable advanced problems on parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-parallelogram-sides-measures-Examples.lesson>HOW TO solve problems on the parallelogram sides measures - Examples</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-angles-of-parallelograms.lesson>HOW TO solve problems on the angles of parallelograms - Examples</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Parallelograms/PROPERTIES-OF-PARALLELOGRAMS.lesson>PROPERTIES OF PARALLELOGRAMS</A>

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.