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Honors Geometry Textbook24 chapters | 200 lessons

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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.

In this lesson, you'll learn that there are five ways you can go about proving that a certain kind of quadrilateral is a parallelogram. You'll also get a demonstration of how knowing the ways can help you when you need to prove something.

When it comes to math, you have to be able to prove that what you're doing is correct. When it comes to geometry, it's the same. In geometry, you'll often be asked to prove that a certain shape is, indeed, that certain shape. For example, you might be shown a quadrilateral and be asked to prove that it is a parallelogram. Remember that a **quadrilateral** is a four-sided flat shape. A **parallelogram** is a quadrilateral with two pairs of opposite, parallel sides.

Looking at this shape, you might think that it is a parallelogram, but unless the problem specifically tells you and/or you can prove that it is, you can't say for sure that it's a parallelogram.

This is where mathematical proofs are very important. You can only say for sure that this is a parallelogram with a mathematical proof. Most times, when you're asked to prove that a certain quadrilateral is a parallelogram, you'll be given information about just a few sides. It's then your job to prove that these sides have the right properties of a parallelogram. We'll be taking a closer look at this in a little bit.

But first, let's go over five ways you can use to prove that a quadrilateral is a parallelogram. Depending on what information you have to work with, you'll be using one of these five ways.

This one is simply the reverse of the definition of a parallelogram. If you can prove that the quadrilateral fits the definition of a parallelogram, then it is a parallelogram.

If both pairs of opposite sides of a quadrilateral are congruent, then you'll always have two opposite pairs of parallel sides. **Congruent** means that they measure the same. Think about it: two congruent sides separating the other pair of opposite sides must always keep those opposite lines the same distance apart. This means, then, that the opposite sides are also parallel.

This one is kind of similar to the method before. It just goes about proving the case in another way. You can actually try this out with four toothpicks. Try it by placing two of the toothpicks opposite and parallel to each other. Now connect these two toothpicks at both ends with the other two toothpicks. You'll notice that no matter how you place your first pair of toothpicks, your second pair of toothpicks will always be parallel.

This one is a bit harder to visualize. But you can play around with it by taking two different-sized sticks and crossing them in the middle of both sticks. These two sticks are the diagonals inside your parallelogram. You'll see that no matter how you cross your sticks, as long as they cross in the middle, you'll always get a parallelogram. Your two sticks are the blue and green lines. You can see that these are the diagonals inside the parallelogram.

If both pairs of opposite angles are congruent, then your opposite pairs of sides will always be the same distance apart, thus making sure that they remain parallel and congruent. You can try this out by making two identical angles and then placing the two angles opposite each other so that the other pair of opposite angles are also congruent. Then you'll see that you'll always get a parallelogram.

Now let's look at an example:

Prove that quadrilateral PORK is a parallelogram if triangle PRK is isosceles with base KR and triangle POR is also isosceles with base OP. Angle PRK is also congruent to angle RPO.

We begin by making our necessary marks to show our given information. Place tick marks on sides KP, PR, and OR to show that they are all the same, since both triangles are isosceles and they share a common side. An **isosceles triangle** is a triangle with two equal sides and a third side called the base. The two angles next to the base are also congruent. We also place congruent marks on angles K, PRK, O, and OPR.

Now we can go ahead with our proof.

Statement | Reason |
---|---|

Triangle PRK is isosceles with base KR | Given |

Triangle ROP is isosceles with base OP | Given |

Side PK = Side PR | Triangle PRK is isosceles, therefore sides are congruent |

Side PR = Side PR | Reflexive |

Side PR = Side OR | Triangle POR is isosceles, therefore sides are congruent |

Angle PRK = Angle RPO | Given |

Angle K = KRP | Triangle PRK is isosceles, therefore base angles are congruent |

Angle O = Angle OPR | Triangle POR is isosceles, therefore base angles are congruent |

Angle K = Angle O | Transitive |

Triangle PRK = Triangle ROP | AAS (Angle Angle Side Theorem) |

Angle KPR = Angle ORP | Congruent angle of congruent triangles |

Angle KRO = Angle KPO | Angle KRP + Angle PRO = Angle KPR + Angle OPR because Angle KRP = Angle OPR and Angle KPR = Angle PRO |

Quadrilateral PORK is a parallelogram | Both pairs of opposite angles are congruent |

With this proof, we prove that the quadrilateral is a parallelogram by proving that both pairs of opposite angles are congruent.

Making your proof can take a while, and there is definitely more than one way to go about writing this proof. The most important thing for you to remember is that your proof needs to prove one of the five ways mentioned.

Let's review. A **quadrilateral** is a four-sided flat shape. A **parallelogram** is a quadrilateral with two pairs of opposite and parallel sides.

To prove a quadrilateral is a parallelogram, you must use one of these five ways.

- Prove that both pairs of opposite sides are parallel.
- Prove that both pairs of opposite sides are congruent.
- Prove that one pair of opposite sides is both congruent and parallel.
- Prove that the diagonals bisect each other.
- Prove that both pairs of opposite angles are congruent.

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Honors Geometry Textbook24 chapters | 200 lessons

- Measuring the Area of a Parallelogram: Formula & Examples 4:02
- What Is a Rhombus? - Definition and Properties 4:24
- Measuring the Area of a Rhombus: Formula & Examples 6:30
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- Squares: Definition and Properties 6:52
- Trapezoids: Definition and Properties 4:24
- Measuring the Area of a Trapezoid 4:38
- Using Heron's Formula in Geometry 5:54
- Solving Problems using the Quadratic Formula 8:32
- How to Measure the Angles of a Polygon & Find the Sum 6:00
- Proving That a Quadrilateral is a Parallelogram 6:59
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