Hinge Thm

Triangle Inequalities and Construction

  • Triangle Formation

    • A triangle can be formed if the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem).
    • Example of three side lengths that form a triangle:
    • Example 1: 5 m, 7 m, 9 m (5 + 7 > 9, 5 + 9 > 7, 7 + 9 > 5)
    • Example of three side lengths that do not form a triangle:
    • Example 2: 2 m, 3 m, 6 m (2 + 3 < 6)

  • Perimeter Constraint

    • Example with a perimeter of 15:
    • Example 3: 5 m, 5 m, 5 m (5 + 5 + 5 = 15)

  • Wooden Frame Problem

    • Given lengths: 2 ft and 5 ft
    • Range of possible lengths for third side (let x be the third side):
    • 2 + 5 > x → x < 7
    • 2 + x > 5 → x > 3
    • 5 + x > 2 → x > -3 (not a constraint)
    • Thus, the possible length for the third side is 3 < x < 7.

Clock Hand Angle Problem

  • Changing lengths of x
    • Distance between hour and minute hands (x) varies with the angle formed between them.
    • Time interpretation: Longer angle results in greater length of x.
    • Order of times based on length of x from least to greatest is:
    • 1:00 (30°) < 8:30 (75°) < 3:00 (90°) < 12:20 (110°) < 1:30 (135°) < 5:00 (150°)

Hinge Theorems

  • Theorem 5-13 (Hinge Theorem)

    • If two sides of triangle 1 are equal to two sides of triangle 2, and the included angles are different, then:
    • The longer side is opposite the larger angle.
    • Example condition: For triangles ABC and XYZ, if m∠A > m∠X, then BC > YZ.

  • Theorem 5-14 (Converse of Hinge Theorem)

    • If two sides of triangle 1 are equal to two sides of triangle 2, but the third sides are different, then:
    • The larger third side is opposite the larger included angle.

Triangle Comparison Problems

  • Comparing Perimeters

    • Problem: Compare perimeter of right isosceles triangle to an isosceles triangle with an 80° vertex.
    • Right isosceles has legs of equal lengths, while the second triangle will have shorter legs due to the 80° angle, resulting in a smaller perimeter.

  • Generating Inequalities

    • Write inequality for segments based on angles or side lengths:
    • Example: If m∠ABC > m∠DCB then AB > BC.

Additional Notes

  • Ensure all angles in a triangle must sum to 180°.
  • When solving for ranges and values, always consider the constraints dictated by the triangle inequality.
  • Include all values when comparing lengths, angles, or other triangle properties, noting their relationships clearly.