Lesson 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. <B>Theorem 1</B> In a parallelogram, the opposite sides are of equal length in pairs. <TABLE> <TR> <TD> <B>Proof</B> Let <B>ABCD</B> be a parallelogram (<B>Figure 1</B>) with the sides <B>AB</B>, <B>BC</B>, <B>DC</B> and <B>AD</B>. We need to prove that the opposite sides <B>AB</B> and <B>DC</B> are of equal length, as well as the sides <B>BC</B> and <B>AD</B>. Let us draw the diagonal <B>BD</B> in the parallelogram <B>ABCD</B> and consider the triangles <B>ABD</B> and <B>DCB</B> (<B>Figure 2</B>). The diagonal <B>BD</B> is the common side of these triangles. The angles <B>ABD</B> and <B>BDC</B> are congruent as the alternate interior angles at the parallel lines <B>AB</B> and <B>DC</B> and the transverse <B>BD</B>. 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) )}}} <B>Figure 1</B>. To the <B>Theorem 1</B> <B>Figure 2</B>. To the proof of the <B>Theorem 1</B> </TD> </TR> </TABLE><B>ADB</B> and <B>DBC</B> are congruent as the alternate interior angles at the parallel lines <B>AD</B> and <B>BC</B> and the transverse <B>BD</B>. Hence, the triangles <B>ABD</B> and <B>DCB</B> are congruent in accordance to the <B>postulate P2 (ASA)</B> of 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 <B>Triangles</B> of the section <B>Geometry</B> in this site. This implies that the sides <B>AB</B> and <B>DC</B> are of equal length as the corresponding sides the triangles <B>ABD</B> and <B>DCB</B>. Similarly, the sides <B>BC</B> and <B>AD</B> are of equal length by the same reason. The proof is completed. The converse statement is valid too. <B>Theorem 2</B> If in a convex quadrilateral the opposite sides are of equal length by pairs, then the quadrilateral is a parallelogram. <TABLE> <TR> <TD> <B>Proof</B> The <B>Figure 3</B> shows a convex quadrilateral <B>ABCD</B> with the sides <B>AB</B>, <B>BC</B>, <B>DC</B> and <B>AD</B>. The opposite sides <B>AB</B> and <B>DC</B> are of equal length: <B>AB</B> = <B>DC</B>. The opposite sides <B>AD</B> and <B>BC</B> are of equal length too: <B>AD</B> = <B>BC</B>. We need to prove that the quadrilateral <B>ABCD</B> is a parallelogram. Draw the diagonal <B>BD</B> in the parallelogram <B>ABCD</B> and consider the triangles <B>ABD</B> and <B>DCB</B> (<B>Figure 4</B>). These triangles have the congruent sides <B>AB</B> and <B>DC</B> by the condition. The sides <B>AD</B> and <B>BC</B> are congruent by the condition, too. The side <B>BD</B> is the common side. </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) )}}} <B>Figure 3</B>. To the <B>Theorem 2</B> <B>Figure 4</B>. To the proof of the <B>Theorem 2</B> </TD> </TR> </TABLE> Hence, the triangles <B>ABD</B> and <B>DCB</B> are congruent in accordance to the <B>postulate P3 (SSS)</B> of 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 <B>Triangles</B> in the section <B>Geometry</B> in this site. This means that the angles <B>ABD</B> and <B>BDC</B> are congruent as the corresponding angles of the congruent triangles <B>ABD</B> and <B>DCB</B>. Hence, the straight lines <B>AD</B> and <B>BC</B> are parallel as they have congruent alternate interior angles <B>ABD</B> and <B>BDC</B>. This also means that the angles <B>ABD</B> and <B>DBC</B> are congruent as the corresponding angles of the congruent triangles <B>ABD</B> and <B>DCB</B>. Hence, the straight lines <B>AB</B> and <B>BC</B> are parallel as they have congruent alternate interior angles <B>ADB</B> and <B>DBC</B>. The proof is completed. My other lessons on parallelograms in this site are - <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> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-the-sides-of-parallelograms.lesson>Properties of the sides of parallelograms</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Opposite-angles-of-a-parallelogram-are-congruent.lesson>Opposite angles of a parallelogram</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Consecutive-angles-of-a-parallelogram.lesson>Consecutive angles of a parallelogram</A> - <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> - <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> - <A HREF=https://www.algebra.com/algebra/homework/Parallelograms/Remarcable-advanced-problems-on-parallelograms.lesson>Remarcable advanced problems on parallelograms</A> - <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> - <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> - <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 <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.