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Triangle Inequality Theorem and Triangle Properties

Learn about the Triangle Inequality Theorem, range of third side, identifying types of triangles by sides and angles, and Triangle Properties. Practice tasks included.

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Triangle Inequality Theorem and Triangle Properties

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  1. Comprehensive Test On Friday Homework: Maintenance Sheet 17 *Due Thursday Bellringer

  2. G.5 Triangle Inequalities Modified by Felicia Hill

  3. Triangle Inequality Theorem: Can you make a triangle? Yes!

  4. Triangle Inequality Theorem: Can you make a triangle? NO because 4 + 5 < 12

  5. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a + b > c a + c > b b + c > a video

  6. Finding the range of the third side: Example Given a triangle with sides of length 3 and 7, find the range of possible values for the third side. Solution Let x be the length of the third side of the triangle. The maximum value: x < 3 + 7 = 10 The minimum value: x > 7 – 3 = 4 So 4 < x < 10(x is between 4 and 10.) x x < 10 x x > 4

  7. Finding the range of the third side: GivenThe lengths of two sides of a triangle Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since the third side and the smallest side given cannot be larger than the other side, we find the minimum value by subtracting the two sides. Difference < Third Side < Sum

  8. Finding the range of the third side: Example Given a triangle with sides of length a and b, find the range of possible values for the third side. Solution Let x be the length of the third side of the triangle. The maximum value: x < a + b The minimum value: x > |a – b| So |a – b|< x < a + b (x is between |a – b| and a + b.) x < a + b x > |a – b|

  9. In a Triangle: • The smallest angle is opposite the smallest side. • The largest angle is opposite the largest side. • The smallest side is opposite the smallest angle. • The largest side is opposite the largest angle.

  10. larger angle B shorter side C longer side A smaller angle Theorem • If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle.

  11. Theorem • If one side of a triangle is larger than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

  12. Corollary #1: The perpendicular segment from a point to a line is the shortest segment from the point to the line. This side is longer because it is opposite the largest angle! This is the shortest segment!

  13. Corollary #2: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. This side is longer because it is opposite the largest angle! This is the shortest segment!

  14. Work Session Task 2: identify triangles by the lengths of their sides (isosceles, equilateral, and scalene) as well as by the measure of their angles (right, obtuse, and acute)

  15. Task 1 • What do notice about the two shorter side lengths of triangle #1 in comparison to the longest side length? The sum of the two shorter sides lengths is greater than the longest side length of the triangle. 1. What do notice about the two shorter side lengths of triangle #2 in comparison to the longest side length? The sum of the two shorter sides lengths is greater than the longest side length of the triangle.

  16. Task 1 • What observation(s) have you made about the side lengths of triangle #3 that DOES NOT meet to form a triangle? Compare the sum of the two shortest sides lengths to the longest side. The sum of the two shorter side lengths is less than the longest side length of the triangle. The sum of the two shorter side lengths is equal to the longest side length of the triangle. • Will any combination of three side lengths create a triangle? Explain. No, the sum of the two shorter side lengths must always be greater than the longest side length to create a triangle. 4. Can you draw a triangle with side lengths of 11 cm, 5 cm, and 6 cm? Not possible. 5cm + 6cm = 11 cm

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