assignment p. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 p. 723-724: 1, 3, 4,...
TRANSCRIPT
![Page 1: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/1.jpg)
Assignment
• P. 518-521: 1, 2, 4-16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48
• P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36
• Challenge Problems
![Page 2: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/2.jpg)
Parallelograms
What makes a polygon a parallelogram?
![Page 3: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/3.jpg)
8.2 Use Properties of Parallelograms
Objectives:
1. To discover and use properties of parallelograms
2. To find side, angle, and diagonal measures of parallelograms
3. To find the area of parallelograms
![Page 4: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/4.jpg)
The Story of Parallelograms
For this lesson, we’ll be writing The Story of Parallelograms. While short on plot, the story is definitive and is illustrated. In order to write this book, we’ll need to make a small, 8-page booklet with no staples. That part is magic.
The Story of Parallelograms: Page Layout
1. Title 2. Definition 3. Picture 4. Theorem 1
5. Theorem 2 6. Theorem 3 7. Theorem 4 8. Area
![Page 5: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/5.jpg)
Parallelogram
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
• Written PQRS • PQ||RS and QR||PS
![Page 6: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/6.jpg)
Investigation 1
In this Investigation, we will be using Geometer’s Sketchpad to construct a perfect parallelogram, and then we will discover four useful properties about parallelograms.
![Page 7: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/7.jpg)
Theorem 1
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
If PQRS is a parallelogram, then and . RSPQ PSQR
![Page 8: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/8.jpg)
Theorem 2
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
If PQRS is a parallelogram, then and .
RP SQ
![Page 9: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/9.jpg)
Theorem 3
If a quadrilateral is a parallelogram, then consecutive angles are supplementary.
If PQRS is a parallelogram, then x + y = 180°.
![Page 10: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/10.jpg)
Theorem 4
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
![Page 11: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/11.jpg)
Example 1
Find each indicated measure.
1. NM
2. KM
3. mJKL4. mLKM
![Page 12: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/12.jpg)
Example 2
The diagonals of parallelogram LMNO intersect at point P. What are the coordinates of P?
![Page 13: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/13.jpg)
Example 3
Find the values of c and d.
![Page 14: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/14.jpg)
Example 4
For the parallelogram below, find the values of t and v.
![Page 15: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/15.jpg)
Example 5: SAT
For parallelogram ABCD, if AB > BD, which of the following statements must be true?
I. CD < BD
.II ADB > C.III CBD > A
C
A D
B
![Page 16: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/16.jpg)
Bases and Heights
Any one of the sides of a parallelogram can be considered a base. But the height of a parallelogram is not necessarily the length of a side.
![Page 17: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/17.jpg)
Bases and Heights
The altitude is any segment from one side of the parallelogram perpendicular to a line through the opposite side. The length of the altitude is the height.
![Page 18: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/18.jpg)
Bases and Heights 2
The altitude is any segment from one side of the parallelogram perpendicular to a line through the opposite side. The length of the altitude is the height.
![Page 19: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/19.jpg)
Investigation 2
Now you will discover a formula for computing the area of a parallelogram.
![Page 20: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/20.jpg)
Area of a Parallelogram Theorem
The area of a parallelogram is the product of a base and its corresponding height.
Base (b)
Height (h)
Base (b)
Height (h)
A = bh
![Page 21: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/21.jpg)
Area of a Parallelogram Theorem
The area of a parallelogram is the product of a base and its corresponding height.
A = bh
![Page 22: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/22.jpg)
Example 6
Find the area of parallelogram PQRS.
![Page 23: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/23.jpg)
Example 7
What is the height of a parallelogram that has an area of 7.13 m2 and a base 2.3 m long?
![Page 24: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/24.jpg)
Example 8
Find the area of each triangle or parallelogram.1. 2. 3.
![Page 25: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/25.jpg)
Example 9
Find the area of the parallelogram.
![Page 26: Assignment P. 518-521: 1, 2, 4- 16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48 P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36 Challenge Problems](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d965503460f94a7fa69/html5/thumbnails/26.jpg)
Assignment
• P. 518-521: 1, 2, 4-16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48
• P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36
• Challenge Problems