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Decide whether it is possible for a triangle to have the three angle measures or three side lengths given.

If it is possible, then decide whether all such triangles are congruent.

Let's work through the four parts of this problem step by step:

### (a) 133°, 124°, 51°
To determine if a triangle is possible with these angles, we apply the **Triangle Angle Sum Theorem**, which states that the sum of the interior angles of a triangle must equal 180°. Let's calculate:

\[
133° + 124° + 51° = 308°
\]

Since the sum is greater than 180°, a triangle is **not possible** with these angles.

---

### (b) 56°, 32°, 44°
Again, using the Triangle Angle Sum Theorem:

\[
56° + 32° + 44° = 132°
\]

The sum is less than 180°, so this does **not** form a valid triangle. Hence, a triangle is **not possible**.

---

### (c) 12 m, 5 m, 5 m
For side lengths, we use the **Triangle Inequality Theorem**, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's check the combinations:

1. \(12 + 5 > 5\) (True)
2. \(12 + 5 > 5\) (True)
3. \(5 + 5 > 12\) (False)

Since the third inequality fails, a triangle is **not possible** with these side lengths.

---

### (d) 60 cm, 40 cm, 30 cm
We'll apply the Triangle Inequality Theorem again:

1. \(60 + 40 > 30\) (True)
2. \(60 + 30 > 40\) (True)
3. \(40 + 30 > 60\) (True)

All the inequalities hold, so a triangle **is possible** with these side lengths. Since the side lengths are different, the triangles will be **unique** but **not congruent** to one another.

---

### Summary of Answers:
- (a) No triangle possible.
- (b) No triangle possible.
- (c) No triangle possible.
- (d) Triangle possible but **not congruent**.

Would you like more details or further clarification on any of these points?

### Related Questions:
1. How does the Triangle Inequality Theorem apply to obtuse triangles?
2. Can a triangle have side lengths that are exactly equal to the sum of the other two sides?
3. How would changing one of the angle measurements in part (a) affect the result?
4. How can we check for triangle congruency using the SAS, SSS, and ASA theorems?
5. What are the conditions for forming a right triangle with given sides?

### Tip:
Always remember to check both angle and side conditions using their respective theorems to ensure accurate results when determining if a triangle is possible.

Decide whether it is possible for a triangle to have the three angle measures or three side lengths given. If it is possible, then decide whether all such triangles are congruent.

Determine if a triangle can be formed with the given angle measurements of 133°, 124°, and 51°, or the side lengths 12 m, 5 m, 5 m, and others. Learn how to apply the Triangle Angle Sum Theorem and Triangle Inequality Theorem to check for congruent triangles.

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