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:

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<reasoning>

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 \(75^\circ\).
</reasoning>

<answer>
<mcq-option>(A) \(90^\circ\) and \(75^\circ\)</mcq-option>
<mcq-option>(B) \(45^\circ\) and \(45^\circ\)</mcq-option>
<mcq-correct>(C) \(15^\circ\) and \(75^\circ\)</mcq-correct>
<mcq-option>(D) \(15^\circ\) and \(90^\circ\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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 \(75^\circ\).
</reasoning>

<answer>
<mcq-option>(A) \(90^\circ\) and \(75^\circ\)</mcq-option>
<mcq-option>(B) \(45^\circ\) and \(45^\circ\)</mcq-option>
<mcq-correct>(C) \(15^\circ\) and \(75^\circ\)</mcq-correct>
<mcq-option>(D) \(15^\circ\) and \(90^\circ\)</mcq-option>
</answer>

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