Question

in a right triangle, one unknown angle is five times larger than the other unknown angle. determine the measure of the missing angles.

90^\circ and 75^\circ
45^\circ and 45^\circ
x 15^\circ and 75^\circ
15^\circ and 90^\circ

Explanation:

Identify the given properties

The problem describes a right triangle. By Triangle Classification, a right triangle contains one angle measuring \(90^\circ\). Let the two unknown acute angles be \(x\) and \(y\).

Set up the equations

The sum of angles in any triangle is \(180^\circ\). Therefore, the two unknown acute angles must sum to \(90^\circ\):

$$x + y = 90^\circ$$

We are given that one unknown angle is five times larger than the other:

$$y = 5x$$

Solve for the unknown angles

Substitute \(y = 5x\) into the sum equation:

$$x + 5x = 90^\circ$$

$$6x = 90^\circ$$

$$x = 15^\circ$$

Calculate the second angle

Use the value of \(x\) to find \(y\):

$$y = 5 \times 15^\circ = 75^\circ$$

The two missing angles are \(15^\circ\) and \(75^\circ\).

Answer:

• (A) \(90^\circ\) and \(75^\circ\)
• (B) \(45^\circ\) and \(45^\circ\)
• (C) \(15^\circ\) and \(75^\circ\) (Correct answer)
• (D) \(15^\circ\) and \(90^\circ\)