Triangle Inequality Theorem Worksheet with multiple-choice and diagram-based questions.

I see that the image for problem 4 is cut off and doesn't show the actual triangles or side lengths needed to solve the problem. Since I have already examined problems 1, 2, and 3, I will now solve each of them step by step.

Let me start with problem 1: Decide whether each set of numbers can form a triangle.

The Triangle Inequality Theorem states that for any three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

For problem 1:
a) 5, 10, 15: 5 + 10 = 15, which is not greater than 15, so these cannot form a triangle.
b) 6, 9, 15: 6 + 9 = 15, which is not greater than 15, so these cannot form a triangle.
c) 9, 40, 41: 9 + 40 = 49 > 41, 9 + 41 = 50 > 40, 40 + 41 = 81 > 9, so these can form a triangle.
d) 7.5, 8.5, 14.5: 7.5 + 8.5 = 16 > 14.5, 7.5 + 14.5 = 22 > 8.5, 8.5 + 14.5 = 23 > 7.5, so these can form a triangle.
e) 11, 12, 9: 11 + 12 = 23 > 9, 11 + 9 = 20 > 12, 12 + 9 = 21 > 11, so these can form a triangle.
f) 1, 16, 16: 1 + 16 = 17 > 16, 1 + 16 = 17 > 16, 16 + 16 = 32 > 1, so these can form a triangle.
g) 9, 5, 7: 9 + 5 = 14 > 7, 9 + 7 = 16 > 5, 5 + 7 = 12 > 9, so these can form a triangle.
h) 2, 15, 16: 2 + 15 = 17 > 16, 2 + 16 = 18 > 15, 15 + 16 = 31 > 2, so these can form a triangle.

Now let's solve problem 2: Two sides of a triangle have the following measures. Find the range of possible measures for the third side.

The range for the third side is given by |a - b| < c < a + b, where a and b are the given sides and c is the third side.

a) 14, 11: |14 - 11| < c < 14 + 11, so 3 < c < 25
b) 6, 10: |6 - 10| < c < 6 + 10, so 4 < c < 16
c) 15, 18: |15 - 18| < c < 15 + 18, so 3 < c < 33
d) 11, 20: |11 - 20| < c < 11 + 20, so 9 < c < 31
e) 47, 21: |47 - 21| < c < 47 + 21, so 26 < c < 68
f) 5, 8: |5 - 8| < c < 5 + 8, so 3 < c < 13
g) 6, 9: |6 - 9| < c < 6 + 9, so 3 < c < 15
h) 17, 19: |17 - 19| < c < 17 + 19, so 2 < c < 36

Finally, let's solve problem 3: Name the largest and smallest angle.

In a triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. However, since we're given the angles directly, we can simply identify the largest and smallest angles.

a) Triangle ABC: angles are 40°, 90°, 30°. Largest angle is 90° (angle B), smallest angle is 30° (angle C).
b) Triangle PQR: angles are 125°, 30°, 25°. Largest angle is 125° (angle P), smallest angle is 25° (angle R).
c) Triangle STU: angles are 80°, 60°, 40°. Largest angle is 80° (angle S), smallest angle is 40° (angle U).

Problem 4 cannot be solved as the image is cut off and doesn't provide the necessary information.

Therefore, the solutions are:

Problem 1:
a) No
b) No
c) Yes
d) Yes
e) Yes
f) Yes
g) Yes
h) Yes

Problem 2:
a) 3 < c < 25
b) 4 < c < 16
c) 3 < c < 33
d) 9 < c < 31
e) 26 < c < 68
f) 3 < c < 13
g) 3 < c < 15
h) 2 < c < 36

Problem 3:
a) Largest: ∠B (90°), Smallest: ∠C (30°)
b) Largest: ∠P (125°), Smallest: ∠R (25°)
c) Largest: ∠S (80°), Smallest: ∠U (40°)

Problem 4: Cannot be solved due to incomplete information in the image.