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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative.
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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org
7th Grade Math 2D Geometry 2012-01-07 www.njctl.org
Setting the PowerPoint View • Use Normal View for the Interactive Elements • To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. • On the View tab, click Slide Master | Page Setup. Select On-screen Show (4:3) under Slide sized for and click Close Master View. • On the Slide Show menu, confirm that Resolution is set to 1024x768. • Use Slide Show View to Administer Assessment Items • To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 11 for an example.)
Table of Contents Determining if a Triangle is Possible Special Pairs of Angles Perimeter & Circumference Click on a topic to go to that section Area of Rectangles Area of Parallelograms Area of Triangles Area of Trapezoids Area of Circles Mixed Review Area of Irregular Figures Area of Shaded Regions Common Core: 7.G.2, 7.G.4-6, 7.EE.3
Determining if a Triangle is Possible Return to table of contents
How many different acute triangles can you draw? How many different right scalene triangles can you draw? Recall that triangles can be classified according to their side lengths and the measure of their angles. Sides: Scalene - no sides are congruent Isosceles - two sides are congruent Equilateral - all three sides are congruent Angles: Acute - all three angles are acute Right - contains one right angle Obtuse - contains one obtuse angle
There is another property that applies to triangles: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. What does this mean? If you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side. If that is not true, then it is not possible to construct a triangle with the given side lengths.
Example: Determine if sides of length 5 cm, 8 cm and 12 cm can form a triangle? Test all three pairs to see if the sum is greater: 5 + 8 > 12 5 + 12 > 8 8 + 12 > 5 13 > 12 17 > 8 20 > 5 Yes, it is possible to construct a triangle with sides of lengths 5 cm, 8 cm and 12 cm.
Example: Determine if sides of length 3 ft, 4 ft and 9 ft can form a triangle? Test all three pairs to see if the sum is greater: 3 + 4 > 9 3 + 9 > 4 4 + 9 > 3 7 > 9 12 > 4 13 > 3 No, it is not possible to construct a triangle with sides of lengths 3 ft, 4 ft and 9 ft.
Try These: Determine if triangles can be formed with the following side lengths: 1. 4 cm, 7 cm, 10 cm 2. 24 mm, 20 mm, 30 mm 4 + 7 > 10 24 + 20 > 30 4 + 10 > 7 24 + 30 > 20 7 + 10 > 4 20 + 30 > 24 YES YES 3. 7 ft, 9 ft, 16 ft 4. 9 in, 13 in, 24 in 7 + 9 = 16 9 + 13 < 24 7 + 16 > 9 9 + 24 > 13 16 + 9 > 7 13 + 24 > 9 NO NO
1 Determine if sides of length 5 mm, 14 mm and 19 mm can form a triangle. Be prepared to show your work! A Yes B No
2 Determine if sides of length 6 in, 9 in and 14 in can form a triangle. Be prepared to show your work! A Yes B No
3 Determine if sides of length 5 yd, 13 yd and 21 yd can form a triangle. Be prepared to show your work! A Yes B No
4 Determine if sides of length 3 ft, 8 ft and 15 ft can form a triangle. Be prepared to show your work! A Yes B No
5 Determine if sides of length 5 in, 5 in and 9 in can form a triangle. Be prepared to show your work! A Yes B No
6 A triangle could have which of the following sets of angles? A B C D
7 A triangle could have which of the following sets of angles? A B C D
Example: Predict the length of the third side of a triangle with sides of length 12 ft and 16 ft. Side 1 = 12 ft Side 2 = 16 ft The 3rd side must be less than: 12 + 16 > 3rd side 28 ft > 3rd side The 3rd side must be greater than: 12 + 3rd side > 16 3rd side > 4 The 3rd side must be greater than 4 ft and less than 28 ft.
Example: Predict the length of the third side of a triangle with sides of length 9 cm and 15 cm. Side 1 = 9 cm Side 2 = 15 cm The 3rd side must be less than: 9 + 15 > 3rd side 24 cm > 3rd side The 3rd side must be greater than: 9 + 3rd side > 15 3rd side > 6 The 3rd side must be greater than 6 cm and less than 24 cm.
Try These: Predict the length of the third side of a triangle whose known sides are lengths: 1. 13 mm, 20 mm 2. 7 in, 19 in 13 + 20 > Side 3 7 + 19 > Side 3 33 > Side 3 26 > Side 3 13 + Side 3 > 20 7 + Side 3 > 19 Side 3 > 7 Side 3 > 12 7 < side 3 < 33 12 < side 3 < 26
Try These: Predict the length of the third side of a triangle whose known sides are lengths: 3. 4 ft, 11 ft 4. 23 cm, 34 cm 4 + 11 > Side 3 23 + 34 > Side 3 15 > Side 3 57 > Side 3 4 + Side 3 > 11 23 + Side 3 > 34 Side 3 > 7 Side 3 > 11 7 < side 3 < 15 11 < side 3 < 57
8 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
9 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
10 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
11 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
12 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
13 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
Special Pairs of Angles Return to table of contents
14 Are the two angles congruent? A Yes B No 75o 110o
15 Are the two angles congruent angles? A Yes B No 40o 40o
16 Are the two angles congruent? A Yes B No 75o 105o
Complementary Angles are two angles with a sum of 90 degrees. These two angles are complementary angles because their sum is 90. Notice that they form a right angle when placed together.
Complementary Angles are two angles with a sum of 90 degrees. These two angles are complementary angles because their sum is 90. Although they aren't placed together, they can still be complementary.
17 What is the measure of A? A 50
What is the measure of A? 18 A 575757 57 57 57 57
19 Tell whether the two angles are complementary? A Yes B No Angle 1 = 63 degrees Angle 2 = 27 degrees
20 Tell whether the angles are complementary. A Yes B No Angle 1 = 146 degrees Angle 2 = 44 degrees
Supplementary Angles are two angles with a sum of 180 degrees. These two angles are supplementary angles because their sum is 180. Notice that they form a straight angle when placed together.
Supplementary Angles are two angles with a sum of 180 degrees. These two angles are supplementary angles because their sum is 180. Although they aren't placed together, they can still be supplementary.
21 What is the measurement of angle A? Angle A 125o
22 What is the measurement of angle A? Angle A 40o
23 Tell whether the two angles are supplementary. A Yes B No Angle 1 = 115 degrees Angle 2 = 65 degrees
24 Find the supplement of 51
25 Find the complement of 51
28 Find the supplement of 102
29 Find the complement of 102
Vertical Angles are two angles that are opposite each other when two lines intersect. b a c d In this example, the vertical angles are: Vertical angles have the same measurement. So:
