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

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

For your convenience, this file contains the list of my lessons on parallel lines that cut off congruent or proportional segments in transverse lines. The lessons are listed in the logical order, 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

    -Three parallel lines cutting off congruent segments in a transverse line,

    -n parallel lines cutting off congruent segments in a transverse line,

    -HOW TO divide a given straight segment into n congruent parts,

    -Three parallel lines cut off proportional segments in any two transverse lines,   and

    -HOW TO construct the segment whose length is an unknown term of a proportion

    -Straight line in a triangle parallel to its side cuts off proportional segments in two other sides

under the current topic,   and

    -Solved problems on Parallel lines cutting off congruent segments in transverse lines,

    -Solved problems on Parallel lines cutting off proportional segments in transverse lines

under the topic Geometry of the section Word problems.

Below the same lessons are listed with short annotations:

1. Three parallel lines cutting off congruent segments in a transverse line

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

    Important particular cases:

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

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

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

2. n parallel lines cutting off congruent segments in a transverse line

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

3. HOW TO divide a given straight segment into n congruent parts

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

4. Three parallel lines cut off proportional segments in any two transverse lines

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

    The Theorem is proved for the case when the ratio is a rational number alpha

, where

and

are arbitrary integer numbers.

    Important particular cases:

        a) Let a straight line connects two points in opposite legs of a trapezoid. This line is parallel to the bases of the trapezoid if and only if the ratio of the segment
            lengths it divides one leg is equal to the ratio of the segment lengths it divides the other leg.

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

5. HOW TO construct the segment whose length is an unknown term of a proportion

    A procedure is described on how to construct a segment in a plane, whose length satisfy the proportion a%2Fb

, where

and

are the lengths of the three
    given segments a, b and d, using a ruler and a compass.

6. Straight line in a triangle parallel to its side cuts off proportional segments in two other sides

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

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

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

    Solved problems (samples)

7. Solved problems on Parallel lines cutting off congruent segments in transverse lines

    Problem 1. In a trapezoid ABCD the lateral side AD is divided in 3 congruent parts by the points E and F. The straight line EG is drawn parallel to the
                       trapezoid bases through the first subdivision point E which is closest subdivision point to the trapezoid vertex D. The point G is the intersection
                       point of the straight line EG and the lateral side BC. The length of the segment GC is 7 cm. Find the length of the side BC of the trapezoid.

    Problem 2. In a triangle ABC the side AC is divided in 137 congruent parts by the points C1, C2, C3, . . . , C136. The straight line C1B1 is drawn parallel to
                       the side CB through the first subdivision point C1 which is closest subdivision point to the triangle vertex A. The point B1 is the intersection point
                       of the straight line C1B1 and the side AB. The length of the segment AB1 is equal to 3 cm. Find the length of the side AB of the triangle.

    Problem 3. In a parallelogram ABCD the segments DE and BF connect the opposite vertices D and B with the midpoints E and F of the parallel sides AB.
                       and DC. Prove that the straight lines DE and BF divide the diagonal AC in three congruent parts.

8. Solved problems on Parallel lines cutting off proportional segments in transverse lines

    Problem 1. In a trapezoid ABCD the lateral side AD is divided in two segments by the point E. The straight line EF is drawn parallel to the trapezoid bases
                       through the subdivision point E and intersects the lateral side BC at the point F. The segments DE and AE are of the length 5 cm and 10 cm
                       respectively. The length of the segment CF is 4 cm. Find the length of the side BC of the trapezoid.

    Problem 2. In a triangle ABC the side AC is divided in two parts by the point E. The straight line EF is drawn parallel to the triangle side AB through the
                       subdivision point E and intersects the side BC at the point F. The segments CE and AE are of the length 5 cm and 10 cm respectively. The
                       length of the side BC is 21 cm. Find the length of the segments BF and CF.

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