A triangle is not possible whose angles measure

Options

A. `40^(@), 65^(@), 75^(@)`
B. `50^(@), 56^(@), 74^(@)`
C. `72^(@), 63^(@), 45^(@)`
D. `67^(@), 42^(@),81^(@)`

Answer

To determine which set of angle measures cannot form a triangle, we will use the angle sum property of triangles. This property states that the sum of the angles in a triangle must equal 180 degrees.

### Step-by-step Solution:

1. **Understand the Angle Sum Property**:
The sum of the angles in a triangle is always 180 degrees. Therefore, for any three angles \( A \), \( B \), and \( C \) in a triangle, the equation must hold:
\[ A + B + C = 180^\circ \]

2. **Evaluate Each Option**:
We will check each option provided to see if the sum equals 180 degrees.

**Option 1**: \( 40^\circ + 60^\circ + 75^\circ \)
- Calculate:
\[ 40 + 60 + 75 = 175^\circ \]
- This does not equal 180 degrees, so option 1 is not valid.

**Option 2**: \( 50^\circ + 56^\circ + 74^\circ \)
- Calculate:
\[ 50 + 56 + 74 = 180^\circ \]
- This equals 180 degrees, so option 2 is valid.

**Option 3**: \( 72^\circ + 63^\circ + 45^\circ \)
- Calculate:
\[ 72 + 63 + 45 = 180^\circ \]
- This equals 180 degrees, so option 3 is valid.

**Option 4**: \( 67^\circ + 42^\circ + 81^\circ \)
- Calculate:
\[ 67 + 42 + 81 = 190^\circ \]
- This does not equal 180 degrees, so option 4 is not valid.

3. **Conclusion**:
The angles that do not form a triangle are found in options 1 and 4. However, since we are looking for the option that is explicitly stated as "not possible," we conclude that option 4 is the correct answer.

### Final Answer:
The triangle is not possible with the angle measures of **67 degrees, 42 degrees, and 81 degrees** (Option 4).

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