Lesson Properties of the sides of parallelograms

Source code of 'Properties of the sides of parallelograms'

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<H2>Properties of the sides of parallelograms</H2>

It is better to read this lesson after the lesson <A HREF = http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A>, which is under the topic Triangles in the section Geometry in this site.
Here I apply congruence tests for triangles to prove geometry facts on on some special quadrilaterals.

Theorem 1

If a quadrilateral has two opposite sides parallel and of equal length, then two other opposite sides of the quadrilateral are parallel and of equal length too.
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 <TR>
 <TD>
Proof

The proof is actually very simple. The Figure 1 will help us.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
This Figure (the left part) shows a quadrilateral ABFE with 
parallel sides AE and BF of equal length. We are going to prove 
that two other sides AB and EF are parallel and have equal length.

Draw the diagonal BE in the quadrilateral ABFE and consider 
the triangles ABE and BEF (Figure 1, the right part). They 
have the common side BE, the congruent sides AE and BF 
(by the condition) and the congruent angles AEB and EBF (these 
angles are the alternate interior angles at the parallel straight 
lines AE and BF and the transverse BE). 
 </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(1.4, 2.85, E),
            locate(5.1, 2.85, F),

line( 0.85,  1.5,  1.05,  1.5),
            line( 0.85,  1.56, 1.05,  1.56),

line( 4.35,  1.5,  4.55,  1.5),
            line( 4.35,  1.56, 4.55,  1.56),

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(6.4,  2.85, E),
            locate(10.1, 2.85, F),

line( 6.65,  2.5, 8.7,  0.5),

line( 5.9,  1.5,  6.1,  1.5),
            line( 5.9,  1.56, 6.1,  1.56),

line( 9.3,  1.5,  9.5,  1.5),
            line( 9.3,  1.56, 9.5,  1.56),

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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. To the Problem 1
 </TD>
 </TR>
</TABLE>Hence, these triangles are congruent in accordance to the postulate P1 (SAS) (see the lesson <A HREF=> Congruence tests for triangles</A> under the topic Triangles 
in the section Geometry in this site).
This means that the corresponding angles ABE and BEF are congruent. It implies that the sides AB and EF are parallel, because the angles ABE and BEF are 
alternate interior angle for the straight lines AB and EF and the transverse BE. 
This means also that the sides AB and EF are of equal length as the corresponding sides of congruent triangles.
The proof is completed.

Theorem 2

If in a quadrilateral two opposite sides are parallel and two other opposite sides are parallel too, then the opposite sides of each pair are of equal length.
<TABLE>
 <TR>
 <TD>
Proof

The Figure 2 (its left part) shows a quadrilateral ADEF with&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
parallel sides AD and FE and parallel sides FA and ED. 
We are going to prove that the sides AD and FE are of equal 
length, and the sides FA and ED are of equal length, too.

Draw the diagonal AE in the quadrilateral ADEF and consider 
the triangles FAE and DEA (Figure 2, the right part). They 
have the congruent angles FAE and DEA as the alternate 
interior angles at the parallel straight lines FA and ED 
and the transverse AE. They also have the congruent angles 
FEA and DAE as the alternate interior angles at 
the parallel straight lines AD and FE and the transverse AE.
 </TD>
 <TD>
{{{drawing( 450, 150, -2, 7, 0, 3,

locate(0.3, 0.5, A),

line (0.3, 0.5, 2.85, 2.5),

locate(2.9, 2.7, D),
            locate(0.9, 2.7, E),

line (1.35, 1.35, 1.55, 1.35),
            line (1.35, 1.42, 1.55, 1.42),

line (-1.4,  0.5, 1.15, 2.5),
            locate(-1.4, 0.5, F),
            line (-1.4,  0.5, 0.3, 0.5),
            line (1.15,  2.5, 2.85, 2.5),

line (-0.35, 1.35, -0.15, 1.35),
            line (-0.35, 1.42, -0.15, 1.42),

line (-0.55, 0.6, -0.55, 0.4),

line ( 1.9,  2.6,  1.9,  2.4),

locate(3.3, 0.5, A),

line (3.3, 0.5, 5.85, 2.5),

locate(5.9, 2.7, D),
            locate(3.9, 2.7, E),

line (4.35, 1.35, 4.55, 1.35),
            line (4.35, 1.42, 4.55, 1.42),

line ( 1.6,  0.5, 4.15, 2.5),
            locate(1.6, 0.5, F),
            line ( 1.6,  0.5, 3.3, 0.5),
            line (4.15,  2.5, 5.85, 2.5),

line (2.65, 1.35, 2.85, 1.35),
            line (2.65, 1.42, 2.85, 1.42),

line (2.45, 0.6, 2.45, 0.4),

line ( 4.9,  2.6,  4.9,  2.4),

line ( 3.3, 0.5, 4.15, 2.5),

line ( 3.62, 1.5, 3.83, 1.5),

arc ( 3.3,  0.5, 0.6, 0.6, 180, 290),

arc ( 4.15, 2.5, 0.6, 0.6,   0, 110),

arc ( 3.3,  0.5, 0.8, 0.8, 290, 320),
            arc ( 3.3,  0.5, 1.0, 1.0, 290, 320),

arc ( 4.15, 2.5, 0.8, 0.8, 115, 140),
 arc ( 4.15, 2.5, 1.0, 1.0, 115, 140)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 3. To the Problem 2
 </TD>
 </TR>
</TABLE>The listed congruent angles include the common side AE.
Therefore, the triangles FAE and DEA are congruent in accordance to the postulate P2 (ASA) of the lesson <A HREF = http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A> (under the topic 
Triangles in the section Geometry in this site).
Hence, the sides AD and FE are of equal length as the corresponding sides of the congruent triangles. The sides FA and ED are of equal length by the same reason.
The proof is completed.

Quadrilaterals that are considered in this lesson are called parallelograms. They are quadrilaterals that have both pairs of opposite sides parallel. 
There is a special topic named Parallelograms in the section Geometry in this site. 
Parallelograms have a number of properties, and they are studied intensively in numerous lessons under that topic.
We placed the current lesson here under the topic Triangles because of two reasons.
First, this lesson gives the examples of applications of the congruence tests for triangles. 
Second, the properties of parallelograms that are proved in this lesson, are used in other lessons under the topic Triangles, so we need to have them ready to use.

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-a-parallelogram.lesson>Properties of the sides of a parallelogram</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>

For navigation over the lessons on Properties of Triangles use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/Compendium-of-properties-of-triangles.lesson>Properties of Trianles</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>.