In a right triangle, one of the acute angle is four times the other. Its measure is

Options

A. `68^@`
B. `84^@`
C. `80^@`
D. `72^@`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the triangle
We are given a right triangle where one of the acute angles is four times the other. Let's denote the angles as follows:
- Let angle A be the acute angle that is four times angle C.
- Let angle B be the right angle, which measures 90 degrees.

### Step 2: Set up the equation
Since the sum of the angles in any triangle is 180 degrees, we can write the equation for the angles in our triangle:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]

Substituting the known values:
\[ \text{Angle A} + 90^\circ + \text{Angle C} = 180^\circ \]

### Step 3: Substitute angle A
Since angle A is four times angle C, we can express angle A in terms of angle C:
\[ \text{Angle A} = 4 \times \text{Angle C} \]

Now we can substitute this into our equation:
\[ 4 \times \text{Angle C} + 90^\circ + \text{Angle C} = 180^\circ \]

### Step 4: Combine like terms
Now, combine the terms involving angle C:
\[ 5 \times \text{Angle C} + 90^\circ = 180^\circ \]

### Step 5: Isolate angle C
To isolate angle C, subtract 90 degrees from both sides:
\[ 5 \times \text{Angle C} = 180^\circ - 90^\circ \]
\[ 5 \times \text{Angle C} = 90^\circ \]

### Step 6: Solve for angle C
Now, divide both sides by 5 to find angle C:
\[ \text{Angle C} = \frac{90^\circ}{5} \]
\[ \text{Angle C} = 18^\circ \]

### Step 7: Find angle A
Now that we have angle C, we can find angle A:
\[ \text{Angle A} = 4 \times \text{Angle C} = 4 \times 18^\circ = 72^\circ \]

### Conclusion
The measure of the acute angle that is four times the other acute angle is:
\[ \text{Angle A} = 72^\circ \]

### Final Answer
Thus, the measure of the acute angle is **72 degrees**.

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