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In a right triangle, if one acute angle measures 30 degrees, what is the measure of the other acute angle?Choices:- 30 degrees 45 degrees 60 degrees 90 degrees

Question

In a right triangle, if one acute angle measures 30 degrees, what is the measure of the other acute angle?

Choices:

  • 30 degrees
  • 45 degrees
  • 60 degrees
  • 90 degrees
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Solution

1. Break Down the Problem

In a right triangle, the sum of all angles is always 180 degrees. Since one of the angles is a right angle (90 degrees), we can use this information to find the other acute angle.

2. Relevant Concepts

The sum of the angles in a triangle can be expressed as: Angle1+Angle2+Angle3=180 \text{Angle}_1 + \text{Angle}_2 + \text{Angle}_3 = 180^\circ In a right triangle:

  • One angle is 9090^\circ (the right angle).
  • Other two angles (acute angles) must sum up to 9090^\circ.

3. Analysis and Detail

Let's denote the acute angles as:

  • Angle1=30\text{Angle}_1 = 30^\circ
  • Angle2=?\text{Angle}_2 = ?

From the properties of a right triangle: Angle1+Angle2+90=180 \text{Angle}_1 + \text{Angle}_2 + 90^\circ = 180^\circ Thus: 30+Angle2+90=180 30^\circ + \text{Angle}_2 + 90^\circ = 180^\circ

Subtracting 9090^\circ from both sides gives: 30+Angle2=90 30^\circ + \text{Angle}_2 = 90^\circ

Now, subtract 3030^\circ from both sides: Angle2=9030=60 \text{Angle}_2 = 90^\circ - 30^\circ = 60^\circ

4. Verify and Summarize

We found that the other acute angle that complements the 3030^\circ angle in the right triangle is 6060^\circ.

Final Answer

The measure of the other acute angle is 60 degrees.

This problem has been solved

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