Lesson Opposite angles of a parallelogram

Source code of 'Opposite angles of a parallelogram'

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<H2>Opposite angles of a parallelogram</H2>

Theorem 1
In a parallelogram, &nbsp;the opposite angles are congruent.

<TABLE>
 <TR>
 <TD>
Proof

We have the parallelogram &nbsp;ABCD&nbsp; (Figure 1).&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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
We need to prove that the opposite angles are congruent: &nbsp;L&nbsp;A = L&nbsp;C 
and &nbsp;L&nbsp;B = L&nbsp;D.

Let us draw the straight line &nbsp;AE&nbsp; as the continuation of the side &nbsp;AB&nbsp;

of the parallelogram &nbsp;ABCD&nbsp; (Figure 2).

Then the angles &nbsp;DAB&nbsp; and &nbsp;CBE&nbsp; are congruent as the corresponding

angles at the parallel lines &nbsp;AD&nbsp; and &nbsp;BC&nbsp; and the transverse &nbsp;AE&nbsp; (see 
the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A>&nbsp; under the topic &nbsp;Angles, complementary, 
supplementary angles&nbsp; of the section &nbsp;Geometry&nbsp; in this site).

</TD>
 <TD>
{{{drawing( 280, 160, 0.5, 7.5, 0.5, 4.5, 
 line( 1.0, 1.0, 6.0, 1.0), 
 line( 2.0, 4.0, 7.0, 4.0),
 line( 1.0, 1.0, 2.0, 4.0),
 line( 6.0, 1.0, 7.0, 4.0),

locate(1.0, 1.0, A),
            locate(6.0, 1.0, B),
            locate(7.0, 4.4, C),
            locate(2.0, 4.4, D),

arc(1.0, 1.0, 1.12, 1.12, 290, 360),

arc(7.0, 4.0, 1.12, 1.12, 110, 180),

arc(6.0, 1.0, 1.0, 1.0, 180, 287),
            arc(6.0, 1.0, 1.2, 1.2, 180, 287),

arc(2.0, 4.0, 1.0, 1.0, 0, 112),
 arc(2.0, 4.0, 1.2, 1.2, 0, 112)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. To the Theorem 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 <TD>
{{{drawing( 320, 160, 0.5, 8.5, 0.5, 4.5, 
 line( 1.0, 1.0, 6.0, 1.0), 
 line( 2.0, 4.0, 7.0, 4.0),
 line( 1.0, 1.0, 2.0, 4.0),
 line( 6.0, 1.0, 7.0, 4.0),

locate(1.0, 1.0, A),
            locate(6.0, 1.0, B),
            locate(7.0, 4.4, C),
            locate(2.0, 4.4, D),

arc(1.0, 1.0, 1.12, 1.12, 290, 360),

arc(7.0, 4.0, 1.12, 1.12, 110, 180),

arc(6.0, 1.0, 1.0, 1.0, 180, 287),
            arc(6.0, 1.0, 1.2, 1.2, 180, 287),

arc(2.0, 4.0, 1.0, 1.0, 0, 112),
            arc(2.0, 4.0, 1.2, 1.2, 0, 112),

green(line( 6.0, 1.0, 7.3, 1.0)), 
            locate(7.2, 1.0, E),
            arc(6.0, 1.0, 1.12, 1.12, 290, 360),

red(line( 7.0, 4.0, 8.3, 4.0)), 
 locate(8.2, 4.4, F),
 arc(7.0, 4.0, 1.0, 1.0, 0, 112),
 arc(7.0, 4.0, 1.2, 1.2, 0, 112)
)}}}
Figure 2. To the proof of the Theorem 1
 </TD>
 </TR>
</TABLE>The angles &nbsp;CBE&nbsp; and &nbsp;BCD&nbsp; are congruent as the alternate interior angles at the parallel lines &nbsp;DC&nbsp; and &nbsp;AE&nbsp; and the transverse &nbsp;BC.

Hence, &nbsp;the angles &nbsp;DAB&nbsp; and &nbsp;BCD&nbsp; are congruent, &nbsp;or &nbsp;L&nbsp;A = L&nbsp;C.

Regarding the angles &nbsp;L&nbsp;B&nbsp; and &nbsp;L&nbsp;D, &nbsp;let us draw the straight line &nbsp;DF&nbsp; as the continuation of the side &nbsp;DC&nbsp; of the parallelogram &nbsp;ABCD&nbsp; (Figure 2).

Then the angles &nbsp;ADC&nbsp; and &nbsp;BCF&nbsp; are congruent as the corresponding angles at the parallel lines &nbsp;AD&nbsp; and &nbsp;BC&nbsp; and the transverse &nbsp;DF.

The angles &nbsp;BCF&nbsp; and &nbsp;ABC&nbsp; are congruent as the alternate interior angles at the parallel lines &nbsp;DF&nbsp; and &nbsp;AB&nbsp; and the transverse &nbsp;BC.

Hence, &nbsp;the angles &nbsp;ADC&nbsp; and &nbsp;ABC&nbsp; are congruent, &nbsp;or &nbsp;L&nbsp;B = L&nbsp;D.

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-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-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/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>.