Lesson OVERVIEW of Lessons on Parallel lines cutting off transverse lines in congruent or proportional segments

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<H2>OVERVIEW of Lessons on Parallel lines cutting off transverse lines in congruent or proportional segments</H2>

For your convenience, this file contains the list of my lessons on parallel lines that cut off congruent or proportional segments in transverse lines. &nbsp;The lessons are listed in the logical order, &nbsp;which means that every given lesson refers to the preceding ones and does not refer to that follow.

My lessons in this site on Parallel lines cutting off transverse lines in congruent or proportional segments are

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Property-of-parallel-lines-cutting-congruent-segments-in-a-transverse-line.lesson>Three parallel lines cutting off congruent segments in a transverse line</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/n-parallel-lines-cutting-congruent-segments-in-a-transverse-line.lesson>n parallel lines cutting off congruent segments in a transverse line</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/HOW-TO-divide-a-given-straight-segment-into-n-congruent-parts.lesson>HOW TO divide a given straight segment into n congruent parts</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Three-parallel-lines-cut-off-proportional-segments-in-any-two-transverse-lines.lesson>Three parallel lines cut off proportional segments in any two transverse lines</A>, &nbsp;&nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/HOW-TO-construct-the-segment-whose-length-is-an-unknown-term-of-a-proportion.lesson>HOW TO construct the segment whose length is an unknown term of a proportion</A>

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Straight-line-in-a-triangle-parallel-to-its-side-cuts-off-proportional-segments-in-two-other-sides.lesson>Straight line in a triangle parallel to its side cuts off proportional segments in two other sides</A>

under the current topic, &nbsp;&nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-Parallel-lines-cutting-off-congruent-segments-in-transverse-lines.lesson>Solved problems on Parallel lines cutting off congruent segments in transverse lines</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-Parallel-lines-cutting-off-proport-segments-in-transverse-lines.lesson>Solved problems on Parallel lines cutting off proportional segments in transverse lines</A>

under the topic &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

Below the same lessons are listed &nbsp;with short annotations:

1.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Property-of-parallel-lines-cutting-congruent-segments-in-a-transverse-line.lesson>Three parallel lines cutting off congruent segments in a transverse line</A>

&nbsp;&nbsp;&nbsp;&nbsp;Theorem 1. &nbsp;If three parallel lines cut off two congruent segments in one transverse line, &nbsp;then they cut off two congruent segments in any other transverse line.

&nbsp;&nbsp;&nbsp;&nbsp;Important &nbsp;particular cases:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;If a straight line bisects the lateral side of a trapezoid and is parallel to the trapezoid's bases &nbsp;then this straight line bisects the other lateral side of the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;trapezoid too.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;If a straight line bisects one side of a triangle and is parallel to the other side of the triangle &nbsp;then this straight line bisects the third side of the triangle too.

&nbsp;&nbsp;&nbsp;&nbsp;Solved problem (example). &nbsp;In a trapezoid, &nbsp;any straight line segment connecting a point at the shorter base with a point at the larger base is bisected by the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;mid-line of the trapezoid.

2.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/n-parallel-lines-cutting-congruent-segments-in-a-transverse-line.lesson>n parallel lines cutting off congruent segments in a transverse line</A>

&nbsp;&nbsp;&nbsp;&nbsp;Theorem 1. &nbsp;If &nbsp;n&nbsp; parallel lines cut off &nbsp;n-1&nbsp; congruent segments in one transverse line, &nbsp;then they cut off &nbsp;n-1&nbsp; congruent segments in any other transverse line.

3.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/HOW-TO-divide-a-given-straight-segment-into-n-congruent-parts.lesson>HOW TO divide a given straight segment into n congruent parts</A>

&nbsp;&nbsp;&nbsp;&nbsp;A procedure is described on how to divide a given segment in a plane into &nbsp;n&nbsp; congruent parts using a ruler and a compass, &nbsp;where &nbsp;n&nbsp; is a given integer number &nbsp;n >= 2.

4.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Three-parallel-lines-cut-off-proportional-segments-in-any-two-transverse-lines.lesson>Three parallel lines cut off proportional segments in any two transverse lines</A>

&nbsp;&nbsp;&nbsp;&nbsp;Theorem 1. &nbsp;If three parallel lines are intersected by two transverse lines, &nbsp;then the ratio of the segment lengths they cut off in one transverse line is equal to the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ratio of the segment lengths they cut off in the second line.

&nbsp;&nbsp;&nbsp;&nbsp;The Theorem is proved for the case when the ratio is a rational number &nbsp;{{{alpha}}} = {{{m/n}}}, &nbsp;where &nbsp;{{{m}}}&nbsp; and &nbsp;{{{n}}}&nbsp; are arbitrary integer numbers.

&nbsp;&nbsp;&nbsp;&nbsp;Important &nbsp;particular cases:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;Let a straight line connects two points in opposite legs of a trapezoid. &nbsp;This line is parallel to the bases of the trapezoid if and only if the ratio of the segment

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;lengths it divides one leg is equal to the ratio of the segment lengths it divides the other leg.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;Let a straight line connects two sides of a triangle. &nbsp;This line is parallel to the third side of the triangle if and only if the ratio of the segment lengths it divides

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the first side is equal to the ratio of the segment lengths it divides the second side.

5.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/HOW-TO-construct-the-segment-whose-length-is-an-unknown-term-of-a-proportion.lesson>HOW TO construct the segment whose length is an unknown term of a proportion</A>

&nbsp;&nbsp;&nbsp;&nbsp;A procedure is described on how to construct a segment in a plane, &nbsp;whose length satisfy the proportion &nbsp;{{{a/b}}} = {{{x/d}}}, &nbsp;where &nbsp;{{{a}}}, &nbsp;{{{b}}} and &nbsp;{{{d}}}&nbsp; are the lengths of the three

&nbsp;&nbsp;&nbsp;&nbsp;given segments &nbsp;a, &nbsp;b&nbsp; and &nbsp;d, &nbsp;using a ruler and a compass.

6.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Straight-line-in-a-triangle-parallel-to-its-side-cuts-off-proportional-segments-in-two-other-sides.lesson>Straight line in a triangle parallel to its side cuts off proportional segments in two other sides</A>

&nbsp;&nbsp;&nbsp;&nbsp;Theorem 1. &nbsp;If a straight line connecting two sides of a triangle is parallel to its third side &nbsp;then the straight line divides these sides proportionally.

&nbsp;&nbsp;&nbsp;&nbsp;Theorem 2. &nbsp;If a straight line connects two sides of a triangle and divides these sides proportionally, &nbsp;then this straight line is parallel to the third triangle's side.

&nbsp;&nbsp;&nbsp;&nbsp;Theorems of this lesson are proved for any real value of the proportion ratios.

&nbsp;&nbsp;&nbsp;&nbsp;Solved problems (samples)

7.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-Parallel-lines-cutting-off-congruent-segments-in-transverse-lines.lesson>Solved problems on Parallel lines cutting off congruent segments in transverse lines</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;In a trapezoid &nbsp;ABCD&nbsp; the lateral side &nbsp;AD&nbsp; is divided in 3 congruent parts by the points &nbsp;E and &nbsp;F. &nbsp;The straight line &nbsp;EG&nbsp; is drawn parallel to the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;trapezoid bases through the first subdivision point &nbsp;E&nbsp; which is closest subdivision point to the trapezoid vertex &nbsp;D. &nbsp;The point &nbsp;G&nbsp; is the intersection

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsppoint of the straight line &nbsp;EG&nbsp; and the lateral side &nbsp;BC. &nbsp;The length of the segment &nbsp;GC&nbsp; is &nbsp;7 cm. &nbsp;Find the length of the side &nbsp;BC&nbsp; of the trapezoid.

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;In a triangle &nbsp;ABC&nbsp; the side &nbsp;AC&nbsp; is divided in 137 congruent parts by the points &nbsp;C1, &nbsp;C2, &nbsp;C3, . . . , &nbsp;C136. &nbsp;The straight line &nbsp;C1B1&nbsp; is drawn parallel to

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side &nbsp;CB through the first subdivision point &nbsp;C1&nbsp; which is closest subdivision point to the triangle vertex &nbsp;A. &nbsp;The point &nbsp;B1&nbsp; is the intersection point

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the straight line &nbsp;C1B1&nbsp; and the side &nbsp;AB. &nbsp;The length of the segment &nbsp;AB1&nbsp; is equal to 3 cm. &nbsp;Find the length of the side &nbsp;AB&nbsp; of the triangle.

&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;In a parallelogram &nbsp;ABCD&nbsp; the segments &nbsp;DE&nbsp; and &nbsp;BF&nbsp; connect the opposite vertices &nbsp;D&nbsp; and &nbsp;B&nbsp; with the midpoints &nbsp;E&nbsp; and &nbsp;F&nbsp; of the parallel sides &nbsp;AB.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and &nbsp;DC. &nbsp;Prove that the straight lines &nbsp;DE&nbsp; and &nbsp;BF&nbsp; divide the diagonal &nbsp;AC&nbsp; in three congruent parts.

8.&nbsp; <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-Parallel-lines-cutting-off-proport-segments-in-transverse-lines.lesson>Solved problems on Parallel lines cutting off proportional segments in transverse lines</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;In a trapezoid &nbsp;ABCD&nbsp; the lateral side &nbsp;AD&nbsp; is divided in two segments by the point &nbsp;E. &nbsp;The straight line &nbsp;EF&nbsp; is drawn parallel to the trapezoid bases

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;through the subdivision point &nbsp;E&nbsp; and intersects the lateral side &nbsp;BC&nbsp; at the point &nbsp;F. &nbsp;The segments &nbsp;DE&nbsp; and &nbsp;AE&nbsp; are of the length &nbsp;5 cm&nbsp; and &nbsp;10 cm

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;respectively. &nbsp;The length of the segment &nbsp;CF&nbsp; is &nbsp;4 cm. &nbsp;Find the length of the side &nbsp;BC&nbsp; of the trapezoid.

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;In a triangle &nbsp;ABC&nbsp; the side &nbsp;AC&nbsp; is divided in two parts by the point &nbsp;E. &nbsp;The straight line &nbsp;EF&nbsp; is drawn parallel to the triangle side &nbsp;AB&nbsp; through the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;subdivision point &nbsp;E&nbsp; and intersects the side &nbsp;BC&nbsp; at the point &nbsp;F. &nbsp;The segments &nbsp;CE&nbsp; and &nbsp;AE&nbsp; are of the length &nbsp;5 cm&nbsp; and &nbsp;10 cm&nbsp; respectively. &nbsp;The

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;length of the side &nbsp;BC&nbsp; is &nbsp;21 cm. &nbsp;Find the length of the segments &nbsp;BF&nbsp; and &nbsp;CF.

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>.