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Sam is drawing triangles on banners using a ruler and protractor.

Inline Dropdown Question
On the second banner Sam wants to create a variety of different sizes of triangles.

For each triangle condition listed, determine whether the condition will produce one and only one triangle, more than one triangle, or no triangle.

Use the drop-down menu to select the appropriate answer.

A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches
Query 1
A triangle with side lengths of 5 inches, 7 inches, and 10 inches
Query 2
A triangle with each angle measuring 60 degrees
Query 3
A triangle with angle measures of , , and
Query 4

1 answer

Let's evaluate each condition for creating triangles:

  1. A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches:

    • The sum of the lengths of the two shorter sides (4 + 5 = 9) is equal to the third side (9). According to the triangle inequality theorem, this does not satisfy the condition for forming a triangle.
    • Answer: No triangle

  2. A triangle with side lengths of 5 inches, 7 inches, and 10 inches:

    • The sum of the lengths of the two shorter sides (5 + 7 = 12) is greater than the length of the third side (10). Thus, these side lengths can form a triangle.
    • Answer: One and only one triangle

  3. A triangle with each angle measuring 60 degrees:

    • Since the angles add up to 180 degrees and each angle is the same, this condition will form an equilateral triangle. There is only one unique equilateral triangle that can be formed for a given side length.
    • Answer: One and only one triangle

  4. A triangle with angle measures of X degrees, Y degrees, and Z degrees:

    • Without specific values for X, Y, and Z, we can't determine if they will form a triangle. However, if it is assumed that the measures sum up to 180 degrees and all are positive, then it will form one triangle. If the angles do not satisfy these conditions, it could be no triangle or a valid triangle with infinite solutions based on varying side lengths.
    • Assuming they are valid angle measures that add up to 180 degrees, the answer would be:
    • Answer: One and only one triangle

If you have specific values for Query 4, please provide them for a more accurate assessment.