document.write( "Question 748157: In a triangle, a segment is drawn joining the midpoint of two sides. What is true about this segment? \n" ); document.write( "
Algebra.Com's Answer #455347 by MathLover1(20850)\"\" \"About 
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The straight line connecting midpoints of two sides of a triangle is \"parallel\" to the \"third\" side of the triangle. \r
\n" ); document.write( "\n" ); document.write( "The straight line connecting midpoints of two sides of a triangle is parallel to the third side of the triangle. \r
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\n" ); document.write( "Proof\r
\n" ); document.write( "\n" ); document.write( "Figure 1 shows the triangle ABC with the midpoints D and E that are                 
\n" ); document.write( "located in its sides BC and AC respectively. The theorem states that
\n" ); document.write( "the straight line ED, which connects the midpoints D and E (green
\n" ); document.write( "line in the Figure 1), is parallel to the triangle side AB. \r
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\n" ); document.write( "\n" ); document.write( "Continue the straight line segment ED to its own length to the point F
\n" ); document.write( "(Figure 2) and connect the points B and F by the straight line segment BF.
\n" ); document.write( "The triangles EDC and FDB have the congruent vertical angles EDC and
\n" ); document.write( "FDB, congruent sides DC and DB as halves of the side BC, and congruent
\n" ); document.write( "sides ED and FD by the construction. Therefore, these triangles are
\n" ); document.write( "congruent in accordance to the postulate P1 (SAS) of the lesson
\n" ); document.write( " Congruence tests for triangles (which is under the topic Triangles
\n" ); document.write( "in the section Geometry in this site).
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\n" ); document.write( "Figure 1. To the Theorem 1      
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\n" ); document.write( "Figure 2. To the proof of the Theorem 1
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\n" ); document.write( "\n" ); document.write( "This means that the angles ECD and DBF are congruent as the corresponding angles of these triangles.
\n" ); document.write( "Hence, the straight lines AC and BF are parallel, because these angles are the alternate interior angles formed by the transverse line BC
\n" ); document.write( "(see the lesson Parallel lines under the topic Angles, complementary, supplementary angles in the section Geometry in this site).
\n" ); document.write( "This means also that the segments CE and BF are of equal length as the corresponding sides of triangles EDC and FDB.
\n" ); document.write( "Since the point E is the midpoint of the side AB and the segments AE and CE are of equal length, this implies that the segments BF and AE are of equal length.\r
\n" ); document.write( "\n" ); document.write( "Thus, we have proved that in the quadrilateral ADFE the two opposite sides BF and AE are parallel and have equal length.\r
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\n" ); document.write( "\n" ); document.write( "At this point we can refer to the geometry fact proven in the lesson Properties of the sides of parallelograms (see the Theorem 1 of that lesson):
\n" ); document.write( "if a quadrilateral has two opposite sides parallel and of equal length, then two other opposite sides of the quadrilateral are parallel and of equal length too.\r
\n" ); document.write( "\n" ); document.write( "It implies that the straight lines AB and EF are parallel.
\n" ); document.write( "This is exactly what we were going to prove.\r
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