Triangle Unit: Day 6 10/24 & 10/25

Agenda: EQ: What is the Triangle Inequality Theorem? Warm Up (questioning to find range of missing side) Go over Answers to Homework Go over Triangle Inequality Activity Answers Worksheet on Triangle Congruence Homework: Finish Worksheet “Congruence and Triangle”

Please collect the following items and place them in this order on your desk: Homework (1/2 worksheet) Triangle Inequality Activity (spaghetti) Inequality of Sides and Angles Activity (other activity)

Warm Up Are 10cm and 7cm the small and medium sides? What would the triangle inequality look like? Could 10 cm and 7 cm be the small and large sides? What would the triangle inequality look like? Could they be the medium and large sides? What would the triangle inequality look like? Pick 3 numbers you know CANNOT be the 3rd side. Pick 5 numbers you know COULD be the 3rd side. Would 3.000001 be a possibility for the 3rd side? What about 16.9999999? What is the range of options for the 3rd side?

Mathematically how can we figure out the range for missing side length? The upper bound for the missing side? (the largest it could be) The lower bound for the missing side? (the smallest it could be) Range notation:

Homework Answers

Triangle Inequality Activity Answers Answers will vary as long as “Small + Medium” is always bigger than “Large” Answers will vary as long as “Large” is always bigger than “Small + Medium” The “Small + Medium” column was always bigger then the “Large” column The “Large” column was always bigger then the “Small + Medium” column To make a triangle the sum of the small and medium sides of the figure must be bigger than the large side. This held true for every real triangle that I made.

Triangle Inequality Activity Answers Yes, my conjecture holds. If the conjecture works for the three different triangles I completed and the three totally different triangles someone else completed it is very likely that it will hold for any triangle. No, the sum of the small and medium sides cannot be equal to the larger side because it would make two lines rather than a triangle. (Using the spaghetti the sum of the small and medium sides made a line the exact same length as the large side.) This conclusion is possible because the larger side alone is bigger than the small and medium sides therefore no matter what you add to it, it will still be bigger than the 3rd side. Example: Sides: 5, 6, 10 5 + 6 > 10 10 + ___ > ____ Because 10 is already bigger than 5 or 6. It doesn’t matter which one you add to 10, the left side is already bigger. 𝑠+𝑚>𝑙 𝑠+𝑙>𝑚 m+𝑙>𝑠

Inequalities for sides and angles Activity Answers The longest side is across from the largest angle The shortest side is across from the smallest angle The triangle would not be a closed figure/would not be a triangle. A scalene triangle has no congruent sides and thus no congruent angles. A B

Triangle Inequality Theorem Inequalities of Sides and Angles The sum of the lengths of any two sides of a triangle is greater than the lengths of the third side. Inequalities of Sides and Angles Theorem: If two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal, and the greater angle is opposite the greater side. Theorem: If two angles of a triangle are unequal, then the measures of the sides opposite these angles are also unequal, and the longer side is opposite the greater angle. i.e. the largest side is always across from the largest angle, and the smallest side is always across from the smallest angle. Ex: If 𝒎𝑨<𝒎𝑩<𝒎𝑪 then 𝑩𝑪 < 𝑨𝑪 < 𝑨𝑩 A C B

Practice Problems: On a ½ sheet of paper answer the following. Which of the following could be triangles? Show your work. 7, 10, 12 6, 5, 14 17, 9, 8 What is the range of values for the third side. Show your work. 13, 8 6, 12 4, 20 Bring me your paper when you are finished

How to grade Practice Problems -5 for every answer that was not correct -5 for any problem in which they did not show work

Congruence and Triangles Worksheet What you don’t finish is for homework!