1. Chapter 3 Geometry and Measurement

2. What You Will Learn: To identify, describe, and draw:  Parallel line segments  Perpendicular line segments To draw:  Perpendicular bisectors  Angle bisectors Generalize rules for finding the area of:  Parallelograms  Triangles Explain how the area of a rectangle can be used to find the area of:  Parallelograms  Triangles

3. 3.1 – Parallel and Perpendicular Line Segments What you will learn:  To identify, describe, and draw: Parallel line segments Perpendicular line segments

4. Parallel Describes lines in the same plane that never cross, or intersect The perpendicular distance btw parallel line segments must be the same at each end of the line segments. They are always marked using “arrows” http://www.mathopenref.com/parallel.html

5. Some ways to create parallel line segments: Using paper folding Using a ruler and a right triangle Example: Draw a line segment, AB. Draw another line segment, CD, parallel to AB.

6. Example: Draw a line segment, AB. Draw another line segment, CD, parallel to AB. B A C D B A C D Label the endpoints (A, B, C, D). Mark the lines with arrows to show the lines are parallel. B A Use a ruler to draw a line segment. Slide the triangle, draw a parallel line.

7. Perpendicular Describes lines that intersect at right angles (90°) They are marked using a small square http://www.mathopenref.com/perpendicular.html right angle

8. Using paper folding (p. 85) Using a ruler and protractor (p. 85) http://www.mathopenref.com/constperplinepoint.html Some ways to create perpendicular line segments:

9. Assignment P. 86  #1, 3-5, 7, 9, 11, Math Link  Still Good? #2, 8, 10, 12, 13  ProStar? #14-16 right angle

10. 3.2 – Draw Perpendicular Bisectors Bisect:  Bi means “two.” Sect means “cut.” So, Bisect means to cut in two. Perpendicular bisector  A line that divides a line segment in half and is at right angles (90°) to the line segment.  Equal line segments are marked with “hash” marks

11. Some ways to create a perpendicular bisector: Using a compass (p. 90)  http://www.mathopenref.com/constbisectline.html http://www.mathopenref.com/constbisectline.html Using a ruler and a right triangle (p. 91) Using paper folding (p. 91)

12. Assignment P. 92, # 1-5, 8  Still Good? # 6, 7, 9, MathLink  ProStar? #10

13. 3.3 – Draw Angle Bisectors Terms: Acute angle  An angle that is less than 90° Obtuse angle  An angle that is more than 90° Angle Bisector  A line that divides an angle into two equal parts  Equal angles are marked with the same symbol Less than 90° Greater than 90°

14. Some ways to create an angle bisector include: Using a ruler and compass (p. 95)  http://www.mathopenref.com/constbisectangle.html http://www.mathopenref.com/constbisectangle.html Using a ruler and protractor (p. 95) Using paper folding (p.95)

15. Assignment P. 97, # 1 & 2, 5, 6, 8  Still Good? # 3 & 4, 9, 11, 13, MathLink  ProStar? #12, 14, 15 Greater than 90°: obtuseLess than 90°: acute Angle Bisector

16. 3.4 – Area of a Parallelogram Area of a rectangle: Area = length x width Parallelogram  A four-sided figure with opposite sides parallel and equal in length http://www.mathopenref.com/parallelogramarea.html w l 6 cm 4 cm A = l x w A = 6 cm x 4 cm A = 24 cm 2

17. Making a Parallelogram from a Rectangle cut paste

18. Base  A side of a two-dimensional closed figure  Common symbol is b Height  The perpendicular distance from the base to the opposite side  Common symbol is h Suggest a formula for calculating the area of a parallelogram. b h

19. Area of a Rectangle vs. Area of a Parallelogram Area = length x width = 12 cm x 8 cm = 96 cm Area = base x height = 12 cm x 8 cm = 96 cm 12 cm 8 cm 12 cm 8 cm Are they the same? Try it! 2 2 b h Sometimes it is necessary to extend the line of the base to measure the height

20. Key Ideas The formula for the area of a rectangle can be used to determine the formula for the area of a parallelogram. The formula for the area of a parallelogram is A = b x h, where b is the base and h is the height. The height of a parallelogram is ALWAYS perpendicular to its base. h b

21. Assignment P. 104, # 1-3, 5, 7, 9, 11  Still Good? # 13-18, MathLink  ProStar? # 19, 20 b h A = b x h

22. 3.5 – Area of a Triangle What you will learn:  Develop the formula for the area of a triangle  Calculate the area of a triangle What we know:  The area of a rectangle A = l x w  The area of a parallelogram A = b x h

23. Key Ideas The formula for the area of a rectangle or parallelogram can be used to determine the formula for the area of a triangle The formula for the area of a triangle is A = b x h 2, or A = b x h, 2 where b is the base of the triangle and h is the height of the triangle. The height of the triangle is always measured perpendicular to its base. http://www.mathopenref.com/trianglearea.html h b A = b x h h b Cut the rectangle in half A = b x h 2 Cut the area in half

24. Your Assignment P. 113, #1-3 as a class. Area of a Triangle, Notebook Area of a Triangle Questions, Notebook P. 113, #4a), 5b) No problem? #8, 10, 11 Still good? #13-15 Pro Star? #16-19