Detailed Solution for Angle Problems MCQ

This solution breaks down each statement in the question to find the values of $x$, the complementary angle, and the smallest angle based on the given conditions.

Statement 1: Solving for $x$ with Supplementary Angles

Two angles are supplementary if their sum is 180 degrees. The given angles are $4x + 40$ and $10x$.

To find $x$, we set up the equation:

$ (4x + 40) + 10x = 180 $

Combine the terms with $x$:

$ 14x + 40 = 180 $

Subtract 40 from both sides:

$ 14x = 180 - 40 $

$ 14x = 140 $

Divide by 14:

$ x = \frac{140}{14} $

$ x = 10 $

So, the value of $x$ is 10.

Statement 2: Finding the Complementary Angle

Two angles are complementary if their sum is 90 degrees. We need to find the complementary angle of $(4/5) * 45$.

First, calculate the value of the given angle:

$ \text{Angle} = \frac{4}{5} \times 45 $

$ \text{Angle} = 4 \times \frac{45}{5} $

$ \text{Angle} = 4 \times 9 $

$ \text{Angle} = 36 \text{ degrees} $

Now, find the complementary angle by subtracting this value from 90 degrees:

$ \text{Complementary Angle} = 90 - 36 $

$ \text{Complementary Angle} = 54 \text{ degrees} $

The complementary angle is 54 degrees.

Statement 3: Calculating the Smallest Angle from a Ratio

Two angles are complementary, meaning they add up to 90 degrees. They are in the ratio of $4 : 5$.

Let the angles be $4y$ and $5y$.

Their sum is 90 degrees:

$ 4y + 5y = 90 $

Combine the terms with $y$:

$ 9y = 90 $

Divide by 9:

$ y = \frac{90}{9} $

$ y = 10 $

The two angles are:

• First angle = $4y$ = $4 * 10$ = 40 degrees
• Second angle = $5y$ = $5 * 10$ = 50 degrees

The smallest angle is 40 degrees.

Conclusion

Based on the calculations for each statement:

• Statement 1 yields $x = 10$.
• Statement 2 yields the complementary angle = 54 degrees.
• Statement 3 yields the smallest angle = 40 degrees.

These values correspond to the sequence 10, 54, 40.

Comparing these results with the options, option (C) matches the calculated values: 10, 54, 40.