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Math 5 Unit 3 Lesson 8 Parallelograms Folk Music Zach’s family loves to play in a folk music band. They have many types of instruments that they play. Zach found that in between some of his frets on this guitar, there are decorative parallelograms. The frets are the bars that the strings press against. Reflection Can you find examples of parallelograms in your environment? Math 5 3-75 Math 5 Unit 3 Lesson 8: Parallelograms Objectives for this Lesson In this lesson you will explore the following concepts: • Describe the attributes of a parallelogram • Relate rectangle attributes to parallelogram attributes • Given the measures of the sides find the perimeter Go online to watch the Notepad Tutor about Classifying Polygons. Sides of a Parallelogram You should remember that a quadrilateral is a polygon with four sides. Here are some quadrilaterals: As you can see by the figures here, quadrilateral can mean some very different things. In order to be more descriptive we can limit the name quadrilateral with other names. A parallelogram is a quadrilateral. When you use the name parallelogram you have narrowed down the first set of quadrilaterals to the following: 3-76 Math 5 Unit 3 Lesson 8: Parallelograms Look at the sides of the parallelograms and make observations. There are two attributes of the sides of a parallelogram. Attributes are rules for defining the parallelogram. The sides of a parallelogram have these attributes: Opposite sides are parallel Opposite sides are congruent Be careful! In this figure there is one pair of opposite sides that are parallel: Why isn’t this figure a parallelogram? The blue sides are parallel, but the other attribute is ignored. The opposite sides are not congruent. Example 1 Which of these shapes have two pairs of opposite sides that are parallel? Math 5 3-77 Math 5 Unit 3 Lesson 8: Parallelograms The answer is: and The other two have one pair of sides that are parallel but those sides are not congruent. Example 2 Measure the sides of the parallelogram in centimetres. Are opposite sides congruent? Use a centimetre ruler. Line up the 0 mark with one endpoint and measure to the other endpoint: This side is 5.1 centimetres. Measure the opposite side. It also measures 5.1 centimetres. This pair of sides is congruent. 3-78 Math 5 Unit 3 Lesson 8: Parallelograms Now measure a short side: This side is 2.2 centimetres. Measure the opposite side. It also measures 2.2 centimetres. The answer is: One pair of opposite sides measure 5.1 cm and are congruent. The other pair of opposite sides measure 2.2 cm and are congruent. Angles of a Parallelogram The angles of a parallelogram follow the attribute: Opposite Angles are Congruent So what are opposite angles? In a quadrilateral there are two ways to describe a pair of angles. Pairs of angles are either opposite angles or consecutive angles. Math 5 3-79 Math 5 Unit 3 Lesson 8: Parallelograms Opposite angles are found opposite of one another inside the figure: s gle An te i s o pp O Consecutive angles are connected inside the figure: Consecutive Angles Example 3 Use the numbers that mark the angles to complete the statements. 1 4 2 3 A. A ngle 1 is opposite angle _______. Angle 1 is opposite angle 3. This is the only answer for this statement. B. A ngle 1 and Angle _______ are consecutive angles. Angle 1 and Angle 2 OR 4 are consecutive angles, depending on whether you go down or across from Angle 1. Either answer is correct. 3-80 Math 5 Unit 3 Lesson 8: Parallelograms C.Angle 2 is opposite angle _______. Angle 2 is opposite angle 4. This is the only answer for this statement. D.Name two angles that are consecutive angles to Angle 3. Angle 2 and Angle 4 are consecutive angles to Angle 3. Reflection How can you tell if the angles are congruent? Let’s Explore Exploration 1: Angle Attributes of Parallelograms Materials: Unit 3, Lesson 8, Exploration 1 page from your Workbook, Scissors, Pencil, Page of parallelograms from the back of this Unit in your Workbook 1. C ut out the parallelograms on the page found at the back of this Unit in your Workbook. Math 5 3-81 Math 5 Unit 3 Lesson 8: Parallelograms 2. Rip corners 2 and 4 off of Parallelogram A as shown: 1 2 A 4 3 3. Now overlap Angles 2 and 4 so that they match up – or not. Do the edges of the angles overlap? If so, they are congruent. 4. Repeat the test on Angles 1 and 3. 5. Repeat the test on Parallelograms B and C. 6. Are opposite angles always congruent? 7. A re there any other observations you can make about angles in a parallelogram? Are These Parallelograms? Here is a group of parallelograms: 2 2 6 2 2 3 1 1 6 4 3 5 3 3 5 4 All of the sides of the figures are marked with their measures. This should tell you that they all have opposite sides that are congruent. You should also be able to tell that they are parallel. All of these figures are parallelograms, but three have special names. 3-82 Math 5 Unit 3 Lesson 8: Parallelograms There are special parallelograms that have more attributes: Rectangle ALL angles are right angles Rhombus All sides are congruent Square ALL angles are right angles All sides are congruent These figures have the attributes of parallelograms. They also have these extra attributes that make them special. Example 4 Nina and her father have made a garden in their backyard. She says that the garden is shaped like a parallelogram. Cameron says that it is shaped like a rectangle. He shows Nina that the corners are all right angles by putting his clipboard in each one. Who is right and why? Math 5 3-83 Math 5 Unit 3 Lesson 8: Parallelograms The garden has all the attributes of a parallelogram: • opposite sides are parallel • opposite sides are congruent • opposite angles are congruent This means that Nina is right! It is a parallelogram. The angles in the corners are all right angles. This means that Cameron is right! It is also a rectangle. How often are two friends right when they disagree? This may be a first! Let’s Practice • Turn in your Workbook to Unit 3, Lesson 8 and complete 1 to 12. Perimeter Nina wants to put a fence around her garden. She will need to know how much fence material to buy. She measures two sides of her garden: 12 m 16 m 3-84 Math 5 Unit 3 Lesson 8: Parallelograms Can Nina find the perimeter of her garden with only these two measurements? The perimeter of parallelograms is found by finding the distance around. You probably remember that this is done by simply adding up the sides. 10 4 4 10 Perimeter = 4 + 10 + 4 +10 = 28 If you know that the figure is a parallelogram then you know that the opposite sides are congruent. You can use that attribute to find the missing sides. 5 Opposite sides are Congruent 8 5 8 8 5 Now let’s try one. Math 5 3-85 Math 5 Unit 3 Lesson 8: Parallelograms Example 5 Find the perimeter of the parallelogram. 14 12 Find the missing sides using the attribute “opposite sides are congruent.” In this case that means that they are the same measure: 14 12 12 14 Add up all of the side lengths to get the perimeter: 12 + 14 + 12 + 14 = 52 The perimeter is: 52 Using the same method you can find the perimeter of Nina’s garden. 3-86 Math 5 Unit 3 Lesson 8: Parallelograms Example 6 Find the perimeter of Nina’s garden. 12 m 16 m Nina was so smart when she took the measurements! She measured one short side and one long side. She can use the attribute “opposite sides are congruent” to find the missing sides: 16 m 12 m 12 m 16 m Now she can find the perimeter by adding all four sides: 12 + 16 + 12 + 16 = 56 metres Let’s Practice • Turn in your Workbook to Unit 3, Lesson 8 and complete 13 to 18. Math 5 3-87 Math 5 Unit 3 3-88 Lesson 8: Parallelograms