2. Triangles

3. Today’s Learning Goals We will learn why triangles are used in construction. We will understand the triangle inequality – the sum of the lengths of any two sides of a triangle is greater than the length of the third side. We will determine the sum of the interior angles for any triangle.

4. Definitions A triangle is a closed, three-sided 2-D figure with straight sides that do not overlap. Although triangles only have three sides and three angles, they come in many different shapes with very useful properties. A right triangle is a triangle with one right angle.

5. Definitions An acute triangle is a triangle with all angles less than 90 .

6. Definitions An obtuse triangle is a triangle with one angle more than 90 . The terms right, acute, and obtuse are all ways that we can name a triangle according to its angle measurements.

7. Definitions We can also name a triangle according to its side lengths. A scalene triangle is a triangle where none of its sides have the same length. How would you name the following scalene triangles? Scalene right Scalene obtuse Scalene acute

8. Definitions An isosceles triangle is a triangle that has two sides the same length. How would you name the following isosceles triangles? Isosceles obtuse Isosceles acute Isosceles right Often times, the same markings will be used to show the sides with the same length.

9. Definitions What is the name of a triangle when all of its sides are the same length? Yes…an equilateral triangle is a triangle with all sides the same length. What do you notice about the angles for all equilateral triangles? Excellent…they are all equal and all acute.

10. Triangles Are Useful Most of us are aware that triangles are used in the construction of buildings, bridges, and homes.

11. Triangle Construction If you were given three different side lengths, would you always be able to make a triangle? Okay…some people think you would be able to and some think you might not. Today, we are going to try to make triangles using metal polystrips with different side lengths.

12. Triangle Construction Let’s try to make a triangle with our metal polystrips with lengths of 6, 8, and 10 units. How could we name a 6, 8, 10 triangle? Nice…it is a scalene right triangle. Could we make a different-looking triangle with the same three side lengths? No…the 6, 8, 10 scalene right triangle is the only one that you can make. Notice how rigid the triangle is when we push on the metal polystrips. This is why triangles are used in construction…for structural stability! 6 8 10

13. Triangle Construction Now, let’s try to make a triangle with our metal polystrips with lengths of 5, 9, and 16 units by putting the 16 length on the bottom. Yes…5 + 9 < 16 so the two sides will never meet to make a triangle. How come we could not make a triangle with side lengths of 5, 9, and 16?

14. Triangle Construction Some of you thought that a triangle could be made with any side lengths. We just saw an example of three side lengths that did not make a triangle. Try to make more triangles with different side lengths greater than 3 and less than 18.

15. Partner Work You have 20 minutes to work on the following problems with your partner.

16. For those that finish early Which set of side lengths from the four below can make a triangle with all angles the same size? a) 5, 5, 3 b) 8, 8, 8 c) 3, 4, 5 d) 6, 8, 9

17. Big Idea from Today’s Lesson A triangle can only be made if the longest side is shorter than the sum of the other two sides (Triangle Inequality)! A triangle is rigid…there’s only 1 triangle that can be made from three given sides. This is why triangles are used in construction. The sum of the interior angles is 180° for ANY triangle.

18. Homework Pgs. 225 (2 – 5, 12, 13)