# Parallel Sides: Definition & Concept

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• 0:05 Definition of Parallel
• 0:34 Properties of Parallel Sides
• 1:07 The Parallel Symbol
• 1:45 Shapes with Parallel Sides
• 2:35 Lesson Summary
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Lesson Transcript
Instructor
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

In this lesson, you will learn the special properties of parallel sides. Only certain shapes have the distinguished honor of having parallel sides. Learn what these are and discover how to mathematically mark two sides that are parallel.

## Definition of Parallel

Two sides or lines are parallel if they are lines that are always the same distance from each other and will never intersect or touch.

The two lines above are parallel. Can you see how they are always the same distance from each other no matter where you go on either line? If you and a buddy each walked on a line, the two of you would never cross paths, and if you kept pace with each other, you would always be the same distance apart. That is what it means to be parallel.

## Properties of Parallel Sides

When a shape has a pair of parallel sides, it gives the shape certain unique properties.

Look at this trapezoid. Usually when a shape has a pair of parallel sides, they make up the bases of the shape. This trapezoid has two bases with each base being one of the two parallel sides.

Also, when a shape has a pair of parallel sides, the shape's height will be the same regardless of how long you stretch it out. Take this trapezoid, for instance. If I were to stretch the shape out, it would still have the same height.

## The Parallel Symbol

There are two ways to mathematically notate that two sides are parallel to each other. One way is by writing it with a math symbol. Another way is to note it directly on the shape using matching arrow marks on the parallel sides.

If a shape had more than one pair of parallel sides, then you would mark the difference by adding one more arrow mark to the second pair of parallel sides.

Note how this rectangle has two pairs of parallel sides and how I've marked each pair differently. The second pair has two matching arrow marks instead of just one. This lets me know that while each pair is parallel, the two pairs are not parallel to each other.

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## Exploring Relationships Between Equal and Equivalent Sets Using Real-World Examples

### Reminders:

• Two sets are equal if they contain the same elements.
• Two sets are equivalent if they have the same cardinality or the same number of elements.

### Questions:

1. Jamie and Grace go shopping. Jamie buys 2 shirts, 1 pair of shoes, and 3 pairs of pants. Grace buys 1 pair of shoes, 1 shirt, 1 pair of pants, 1 necklace, 1 ring, and 1 bottle of perfume. Is the set of things that Jamie bought equal to the set of things that Grace bought? Why or why not? Are the two sets equivalent? Why or why not?
2. Laura and Tristyn both have \$4. Laura has 4 one-dollar bills. Tristyn has 1 two-dollar bill, 1 one-dollar bill, and 2 fifty-cent coins. Is Laura's set of currency equal to Tristyn's set of currency? Why or why not? Are the two sets equivalent? Why or why not?
3. Erica and Tessa are both given a swag bag at a fashion show they are attending. Each bag contains a tube of mascara, a candy cane, a tube of lip gloss, and a bottle of hand lotion. Is the set of contents of Erica's bag equal to the set of contents of Tessa's bag? Why or why not? Are they equivalent sets? Why or why not?
4. Based on your answers to these questions, do you think all equal sets are also equivalent sets? Explain.
5. Based on your answers to these questions, do you think that all equivalent sets are also equal sets? Explain.

### Solutions:

1. The set of things that Jamie bought is not equal to the set of things that Grace bought, because they bought different things, so the sets contain different elements. However, the set of things that Jamie bought is equivalent to the set of things that Grace bought, because they each bought 6 things total, so the two sets each have the same number of things in them.
2. The set of Laura's currency is not equal to the set of Tristyn's currency, because they have different types of currency that add up to \$4. However, Laura's set of currency consists of 4 pieces of currency, and Tristyn's set of currency also consists of 4 pieces of currency, so the two sets are equivalent since they contain the same number of elements.
3. Since the swag bags have the exact same contents, the set of contents of Erica's bag is equal to the set of contents of Tessa's bag, because they contain the exact same elements. They are also equivalent sets because they both contain 4 items, so they have the same number of elements.
4. Yes, all equal sets are also equivalent sets. Equal sets have the exact same elements, so they must have the same number of elements. Therefore, equal sets must also be equivalent.
5. No, not all equivalent sets are also equal sets. We saw that this is the case with the first two questions because we had sets that are equivalent, but not equal.

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