Prentice Hall Geometry

Chapter 6 Polygons and Quadrilaterals > 6-2 Properties of Parallelograms

You will use parallelograms to prove the following theorem.

Take note Theorem 6-7

Theorem

If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

If …

A B ↔ ∥ C D ↔ ∥ E F ↔ modified eh b with left right arrow above , parallel to , modified c d with left right arrow above , parallel to , modified e f with left right arrow above and A C ¯ ≅ C E ¯ eh c bar , approximately equal to , c e bar

Two transversals intersect three parallel horizontal lines, at A and B on the top line, C and D on the middle line, and E and F on the bottom line. On the left transversal, segments AC and CE are congruent.

Then …

B D ¯ ≅ D F ¯ b d bar , approximately equal to , d f bar

Two transversals intersect three parallel horizontal lines, at A and B on top, C and D in the middle, and E and F on bottom. On the left transversal, segments AC and CE are congruent. On the right transversal, segments BD and DF are congruent.

You will prove Theorem 6-7 in Exercise 43.

Problem 4 Using Parallel Lines and Transversals

In the figure, A E ↔ ∥ B F ↔ ∥ C G ↔ ∥ D H ↔ , modified eh e with left right arrow above , parallel to , modified b f with left right arrow above , parallel to , modified c g with left right arrow above , parallel to , modified d h with left right arrow above , comma AB = BC = CD = 2, and e f equals , 2.25 , . What is EH?

Plan

What information do you need?

You know the length of E F ¯ . e f bar , . To find EH, you need the lengths of F G ¯ f g bar and G H ¯ . g h bar , .

Two transversals, a horizontal above a diagonal, intersect four parallel vertical lines, the horizontal at A, B, C, and D, from left to right, and the diagonal at E, F, G, and H, from left to right.

E F = F G = G H Since ∥ lines divide A D ¯ into equal parts, they also divide E H ¯ into equal parts. E H = E F + F G + G H Segment Addition Postulate E H = 2.25 + 2.25 + 2.25 = 6.75 Substitute. table with 3 rows and 2 columns , row1 column 1 , e f equals f g equals g h , column 2 cap since , parallel to . lines divide . eh d bar . into equal partscomma they also divide . e h bar . into equal parts. , row2 column 1 , e h equals e f plus f g plus g h , column 2 cap segment cap addition cap postulate , row3 column 1 , e h equals , 2.25 , plus , 2.25 , plus , 2.25 , equals , 6.75 , column 2 cap substitute. , end table

✓ Got It?

Use the figure in Problem 4. If EF = FG = GH = 6 and AD = 15, what is CD?

✓ Lesson Check

Do you know HOW?

Use the diagram of white parallelogram ABCD to find each value.

Parallelogram ABCD has side AB, measuring x + 2, parallel to side CD, measuring 2x minus 3. Interior angle B measures 127 degrees.

m angle eh
m angle d
x
AB
What are ED and FD in the figure below?

Do you UNDERSTAND?

Reasoning If you know one angle measure of a parallelogram, how do you find the other three angle measures? Explain.
Compare and Contrast What is the difference between a quadrilateral and a parallelogram?
Error Analysis Your classmate says that QV = 10. Explain why the statement may not be correct.

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