Lesson Alternate interior angles and Alternate exterior angles
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<b>Alternate interior angles</b> Alternate interior angles are two <A HREF=Congruent_angles.wikipedia>congruent</A> (equal in measurement) interior angles that lie on different <A HREF=Parallel_lines.wikipedia>parallel lines</A> and on opposite sides of a <A HREF=Transversal_line.wikipedia>transversal</A>. {{{drawing( 160, 160, -3.5, 4.5, -0.2, 5.4, line(-3, 1, 4, 1 ), line( -3, 4, 4, 4 ),line( -2, 0, 2, 5 ), locate( -3, 1, A ), locate( 4, 1, B ),locate( 4, 4, D ),locate(-3, 4, C ), locate( -1.6, 1.7, w ), locate( 1.3, 4.1, x ), locate( .3, 4.1, y ), locate( -.6, 1.7, z), locate( -1.5, .2, P), locate( 1.3, 5.3, Q ))}}} In the figure shown, <b>PQ</b> is the transversal that cut the parallel lines AB and CD. Angles <b>w</b> and <b>x</b> and angles <b>y</b> and <b>z</b> are alternate interior angles. {{{Angle}}}{{{w}}} = {{{Angle}}}{{{x}}} {{{Angle}}}{{{y}}} = {{{Angle}}}{{{z}}} <b>proof</b> for the proof we will modify the figure as follows {{{drawing( 160, 160, -3.5, 4.5, -0.2, 5.4, line(-3, 1, 4, 1 ), line( -3, 4, 4, 4 ),line( -2, 0, 2, 5 ), locate( -3, 1, A ), locate( 4, 1, B ),locate( 4, 4, D ),locate(-3, 4, C ), locate( -1.6, 1.7, w ), locate( 1.3, 4.1, x ), locate( .3, 4.1, y ), locate( -.6, 1.7, z), locate( -1.5, .2, P), locate( 1.3, 5.3, Q ), locate( -2, 1.1, y ), locate( 1.8, 4.7, z ), locate( .3, 4.7, w ), locate( -.8, 1.1, x))}}} In the proof we will use the property of the corresponding angles that they are congruent. When two lines are crossed by another line (Transversal), the angles in matching corners are called corresponding angles. in the figure, corresponding angles are shown by same name i.e w, x, y and z. Hence by using this property we will proof that alternate interior angles are equal. At the intersection point of straight lines AB and PQ {{{Angle}}}{{{w}}} + {{{Angle}}}{{{z}}} = {{{180}}}..................(1) (AB is a straight line) and {{{Angle}}}{{{x}}} + {{{Angle}}}{{{z}}} = {{{180}}}..................(2) (PQ is a straight line) from equation (1) and (2) {{{Angle}}}{{{w}}} = {{{Angle}}}{{{x}}} Again applying the same concept at the intersection point of straight lines AB and PQ {{{Angle}}}{{{w}}} + {{{Angle}}}{{{z}}} = {{{180}}}..................(1) (AB is a straight line) and {{{Angle}}}{{{w}}} + {{{Angle}}}{{{y}}} = {{{180}}}..................(2) (PQ is a straight line) from equation (1) and (2) {{{Angle}}}{{{y}}} = {{{Angle}}}{{{z}}} Hence Proved that alternate interior angles are congruent. <b>Alternate exterior angles</b> Alternate exterior angles are two <A HREF=Congruent_angles.wikipedia>congruent</A> exterior angles that lie on different <A HREF=Parallel_lines.wikipedia>parallel lines</A> and on opposite sides of a <A HREF=Transversal_line.wikipedia>transversal</A>. {{{drawing( 160, 160, -3.5, 4.5, -0.2, 5.4, line(-3, 1, 4, 1 ), line( -3, 4, 4, 4 ),line( -2, 0, 2, 5 ), locate( -3, 1, A ), locate( 4, 1, B ),locate( 4, 4, D ),locate(-3, 4, C ), locate( -2, 1.1, w ), locate( 1.8, 4.7, x ), locate( 0, 4.7, y ), locate( -.8, 1.1, z), locate( -1.5, .2, P), locate( 1.3, 5.3, Q ))}}} In the figure shown, <b>PQ</b> is the transversal that cut the parallel lines AB and CD. Angles <b>w</b> and <b>x</b> and angles <b>y</b> and <b>z</b> are alternate exterior angles. {{{Angle}}}{{{w}}} = {{{Angle}}}{{{x}}} {{{Angle}}}{{{y}}} = {{{Angle}}}{{{z}}} Proof is same as "Alternate interior angles" For more information refer to <A HREF=Interior_angles.wikipedia>Interior Angles</A>.
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