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Warm Up/ Activator Using scissors and a ruler, cut out a triangle from a piece of colored paper. Clean up your area afterwards. We will be using the triangle during our lesson today. Constructing Triangles Common Core 7.G.2 Vocabulary • Uniquely defined • Ambiguously defined • Nonexistent Triangle Inequality Theorem Let’s review what we learned yesterday with this video. http://www.youtube.com/watch?v=OoLb_NnnKSQ Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. z x y + > + > + > Practice Can these measures be the sides of a triangle? 1. 2. 3. 4. 7, 5, 4 2, 1, 5 9, 6, 3 7, 8, 4 Practice Can these measures be the sides of a triangle? 1. 2. 3. 4. 7, 5, 4 2, 1, 5 9, 6, 3 7, 8, 4 yes no no yes Example 1 Using the measurements 6 in and 8 in, what is the smallest possible length of the third side? What is the largest possible length of the third side? Example 1 If you assume that 6 and 8 are the shorter sides, then their sum is greater than the third side. Therefore, the third side has to be less than 14. 6+8> 14 > < 14 Example 1 If you assume that the larger of these values, 8, is the largest side of the triangle, then 6 plus the missing value must be greater than 8. Therefore, the third side has to be more than 2. 6+ >8 −6 −6 >2 Example 1 If you put these two inequalities together, then you get the range of values that can be the length of the third side: 2 < < 14 Therefore, any value between 2 and 14 (but not equal to 2 or 14) can be the length of the third side. Practice Solve for the range of values that could be the length of the third side for triangles with these 2 sides: 1. 2 and 6 2. 9 and 11 3. 10 and 18 (Be sure to look for patterns!) Practice Solve for the range of values that could be the length of the third side for triangles with these 2 sides: 1. 2 and 6 2. 9 and 11 3. 10 and 18 << < < < < What patterns do you see? Types of Triangles If three side lengths do not make a triangle, you would say that the triangle is nonexistent. If three side lengths do make a triangle, you would say that the triangle is uniquely defined because it creates one, specific triangle. Angles of Triangles Tear off the corners of the triangle that you created in the warm up/activator. Lay them on your paper with all of the vertices pointing inwards and the edges touching. Like this: What do they create? Angles of Triangles What do they create? A straight line, which is equal to 180 degrees; therefore, the sum of the angles in a triangle always equal 180 degrees. This is called the Triangle Angle Sum Theorem. Glue your triangle corners in your math notebook and explain this in your own words. Practice Given the following angle measurements, determine the third angle measurement. 1. 60°, 80° 2. 110°, 20° Do these measurements create triangles? 3. 55°, 75°, 50° 4. 80°, 90°, 80° Practice Given the following angle measurements, determine the third angle measurement. 1. 60°, 80° ° 2. 110°, 20° ° Do these measurements create triangles? 3. 55°, 75°, 50° yes 4. 80°, 90°, 80° no Constructing Triangles from Angles Look at these triangles. They have the same angle measurements, which is why they are similar in shape. However, do they have the same side lengths? Constructing Triangles from Angles Look at these triangles. They have the same angle measurements, which is why they are similar in shape. However, do they have the same side lengths? No. Since they aren’t the same size, will angle measurements construct unique triangles? Constructing Triangles from Angles Look at these triangles. They have the same angle measurements, which is why they are similar in shape. However, do they have the same side lengths? No. Since they aren’t the same size, will angle measurements construct unique triangles? No. Constructing Triangles from Angles Conditions, such as angle measurements, that can create more than one triangle are called ambiguously defined. Summary Take turns with your partner explaining the Triangle Angle Sum Theorem and the Triangle Inequality Theorem in your own words.