Lesson Opposite angles of a parallelogram
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<H2>Opposite angles of a parallelogram</H2> <B>Theorem 1</B> In a parallelogram, the opposite angles are congruent. <TABLE> <TR> <TD> <B>Proof</B> We have the parallelogram <B>ABCD</B> (<B>Figure 1</B>). We need to prove that the opposite angles are congruent: <I>L</I> <B>A</B> = <I>L</I> <B>C</B> and <I>L</I> <B>B</B> = <I>L</I> <B>D</B>. Let us draw the straight line <B>AE</B> as the continuation of the side <B>AB</B> of the parallelogram <B>ABCD</B> (<B>Figure 2</B>). Then the angles <B>DAB</B> and <B>CBE</B> are congruent as the corresponding angles at the parallel lines <B>AD</B> and <B>BC</B> and the transverse <B>AE</B> (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A> under the topic <B>Angles, complementary, supplementary angles</B> of the section <B>Geometry</B> 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) )}}} <B>Figure 1</B>. To the <B>Theorem 1</B> </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) )}}} <B>Figure 2</B>. To the proof of the <B>Theorem 1</B> </TD> </TR> </TABLE>The angles <B>CBE</B> and <B>BCD</B> are congruent as the alternate interior angles at the parallel lines <B>DC</B> and <B>AE</B> and the transverse <B>BC</B>. Hence, the angles <B>DAB</B> and <B>BCD</B> are congruent, or <I>L</I> <B>A</B> = <I>L</I> <B>C</B>. Regarding the angles <I>L</I> <B>B</B> and <I>L</I> <B>D</B>, let us draw the straight line <B>DF</B> as the continuation of the side <B>DC</B> of the parallelogram <B>ABCD</B> (<B>Figure 2</B>). Then the angles <B>ADC</B> and <B>BCF</B> are congruent as the corresponding angles at the parallel lines <B>AD</B> and <B>BC</B> and the transverse <B>DF</B>. The angles <B>BCF</B> and <B>ABC</B> are congruent as the alternate interior angles at the parallel lines <B>DF</B> and <B>AB</B> and the transverse <B>BC</B>. Hence, the angles <B>ADC</B> and <B>ABC</B> are congruent, or <I>L</I> <B>B</B> = <I>L</I> <B>D</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-a-parallelogram.lesson>Properties of the sides of a parallelogram</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/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>.