Lesson Vertical angles_
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<H2>Vertical angles</H3> In <B>Figure 1</B> two intersecting straight lines are shown. They form four angles {{{alpha}}},{{{beta}}}, {{{gamma}}} and {{{delta}}}. Pair of angles {{{alpha}}} and {{{beta}}} are <B>vertical</B>. Pair of angles {{{gamma}}} and {{{delta}}} are <B>vertical</B> as well. <B>Definition</B>. Two angles are called <B>vertical</B> if the sides of one of them are continuations of sides of the other.<TABLE> <TR> <TD> <B>Theorem (Vertical angles theorem)</B> Vertical angles are congruent. <B>Proof</B> Referring to <B>Figure 1</B>, we are going to prove that {{{alpha}}}={{{beta}}} and {{{gamma}}}={{{delta}}}. Note that {{{alpha}}} + {{{gamma}}} = 180° (because angles {{{alpha}}} and {{{gamma}}} make in sum the straight angle), and {{{beta}}} + {{{gamma}}} = 180° (because angles {{{gamma}}} and {{{beta}}} make in sum the straight angle). Hence, {{{alpha}}}={{{beta}}}. Similarly, {{{alpha}}} + {{{delta}}} = 180° (because {{{alpha}}} and {{{delta}}} make in sum the straight angle) and {{{alpha}}} + {{{gamma}}} = 180° (because {{{alpha}}} and {{{gamma}}} make in sum the straight angle). Therefore, {{{gamma}}}={{{delta}}}. The proof is completed. </TD> <TD> {{{drawing( 200, 200, 0, 5, 0, 5, line( 0.0, 2, 5.0, 3), line( 0.5, 0, 5, 5), locate ( 0.5, 2.1, A), locate ( 4.5, 2.9, B), locate ( 0.5, 0.6, C), locate ( 4.5, 4.4, D), locate ( 2.0, 2.4, alpha), locate ( 3.4, 3.2, beta), locate ( 2.7, 3.2, gamma), locate ( 2.7, 2.4, delta) )}}} <B>Figure 1. Vertical angles</B> </TD> </TR> </TABLE><TABLE> <TR> <TD> <B>Example</B> If in <B>Figure 1</B> one of vertical angles{{{alpha}}} = 37°, find three other angles {{{beta}}}, {{{gamma}}} and {{{delta}}}. </TD> <TD> <B>Solution</B> {{{beta}}} = 37° as the vertical angle to {{{alpha}}}; {{{gamma}}} = 180° -37° = 143° as the complementary angle to {{{alpha}}}; {{{delta}}} = 143° as the vertical angle to {{{gamma}}}. </TD> </TR> </TABLE> My other lessons in this site on Angles Basics, Supplementary and Complementary angles, Vertical angles, Parallel lines are - <A HREF=http://www.algebra.com/algebra/homework/Angles/Angles-basics.lesson>Angles basics</A> - <A HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-suppl-compl-or-vertical-angles-Examples.lesson>HOW TO solve problems on supplementary, complementary or vertical angles - Examples</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-parallel-lines-Examples.lesson>HOW TO solve problems on parallel lines - Examples</A> - <A HREF=https://www.algebra.com/algebra/homework/Angles/OVERVIEW-of-lessons-on-Angles-basics-Suppl-Compl-Vertical-angles-Parallel-lines.lesson>OVERIEW of lessons on Angles Basics, Supplementary and Complementary angles, Vertical angles, Parallel lines</A> To navigate over the lessons on Properties of Triangles use this file/link <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 <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
