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Properties of Triangles EQ: How can I determine if a set of attributes will create one unique triangle, many triangles, or no triangle at all? What is the triangle inequality theorem? The Triangle Inequality Theorem states that the sum of any two sides must be greater than the length of the third side. a+b>c Example 1: Side lengths 3, 4, & 5 can create one unique triangle because: 3+4>5 4+5>3 3+5>4 Example 2: Side lengths 3, 4, & 9 cannot create a triangle because: 3+4<9 Sample Problem 1: If the two smaller sides of a triangle have lengths 6 cm and 8 cm, the third side must be less than ________ cm. Sample Problem 2: If two sides of a triangle have lengths 5 cm and 8 cm, the third side must be greater than ____________ and less than ___________. What is the The triangle angle-sum theorem states that all three angles of triangle angle- any triangle must have a sum of 180°. sum theorem? Example 1: 30°, 40°, & 110° angles can form triangles because their sum is 180° Example 2: 45°, 50°, & 90° do not form a triangle because 45 + 50 + 90 > 180. Similar triangles will have congruent angles; which means that any set of angles that add up to 180 degrees can create more than one triangle with those measurements. Example: The triangles above have the same angle measurements because they are similar triangles. Angle-Angle-Angle (AAA) - If you are given 3 angle measurements only, you can create many triangles, as long as the 3 angle measurements total up to 180. Which combinations of known sides and angles will create a unique triangle? If you know the length of two adjacent sides and their included angle, then you can only make one unique triangle. (SAS - Side-Angle-Side) This is because the only way to complete the triangle is to connect the two given sides. Example: Likewise, If you know two angle measurements and the side length that connects them, then you can only make one unique triangle. (ASA - Angle-Side-Angle) This is because the only way to complete the triangle is to extend the two angles until they intersect. Example: If you are given three side lengths that meet the requirements of the triangle inequality theorem (a + b > c), then they will create only one unique triangle (SSS - Side-Side-Side) Example: Given side lengths 3 cm, 4 cm, & 5 cm there is only one way to put those together to form a triangle.