Understanding the Right Triangle Problem

The question asks us to find the measure of the smallest angle in a right triangle. We are given a relationship between the smallest angle and one of the other angles in the triangle.

A right triangle is a triangle that has one angle measuring exactly \(90^\circ\).

Let the three angles of the right triangle be \(A\), \(B\), and \(C\).

We know one angle is \(90^\circ\). Let's assume \(C = 90^\circ\).

The sum of the angles in any triangle is always \(180^\circ\). So, \(A + B + C = 180^\circ\).

Substituting \(C = 90^\circ\), we get \(A + B + 90^\circ = 180^\circ\). This simplifies to \(A + B = 180^\circ - 90^\circ\), which means \(A + B = 90^\circ\). The two acute angles in a right triangle are complementary.

Setting Up the Equation for Triangle Angles

The question states that "the measure of one angle of a right triangle is \(30^\circ\) more than the measure of the smallest angle".

In a right triangle, the \(90^\circ\) angle is the largest angle (or equal to the largest if the other two are \(45^\circ\)). Therefore, the smallest angle must be one of the two acute angles (\(A\) or \(B\)), and it must be less than \(90^\circ\).

Let the measure of the smallest angle be \(x\). Since it's an acute angle in a right triangle, \(x \lt 90^\circ\).

The other acute angle is \(90^\circ - x\).

Since \(x\) is the smallest angle, \(x \le 90^\circ - x\), which implies \(2x \le 90^\circ\), or \(x \le 45^\circ\). So, the smallest angle \(x\) must be less than or equal to \(45^\circ\).

The problem says one angle is \(30^\circ\) more than the smallest angle (\(x\)). This 'one angle' cannot be the smallest angle itself (as \(x \ne x + 30^\circ\)). It also cannot be the \(90^\circ\) angle unless \(90^\circ = x + 30^\circ\), which would mean \(x = 60^\circ\). But \(x\) must be the smallest angle and \(x \le 45^\circ\), so \(x\) cannot be \(60^\circ\).

Therefore, the angle that is \(30^\circ\) more than the smallest angle \(x\) must be the other acute angle, which is \(90^\circ - x\).

So, we can write the equation:

Solving for the Smallest Angle

Now we need to solve the equation for \(x\):

\[90^\circ - x = x + 30^\circ\]

Add \(x\) to both sides of the equation:

\[90^\circ - x + x = x + 30^\circ + x\]

\[90^\circ = 2x + 30^\circ\]

Subtract \(30^\circ\) from both sides:

\[90^\circ - 30^\circ = 2x + 30^\circ - 30^\circ\]

\[60^\circ = 2x\]

Divide both sides by 2:

\[\frac{60^\circ}{2} = \frac{2x}{2}\]

\[x = 30^\circ\]

So, the measure of the smallest angle is \(30^\circ\).

Verifying the Triangle Angles

If the smallest angle is \(30^\circ\), the other acute angle is \(90^\circ - 30^\circ = 60^\circ\).

The three angles of the triangle are \(30^\circ\), \(60^\circ\), and \(90^\circ\).

Let's check the condition from the question: "the measure of one angle is \(30^\circ\) more than the measure of the smallest angle".

The smallest angle is \(30^\circ\).

Is \(60^\circ\) equal to \(30^\circ + 30^\circ\)? Yes, \(60^\circ = 60^\circ\). This condition holds true.

Is \(90^\circ\) equal to \(30^\circ + 30^\circ\)? No, \(90^\circ \ne 60^\circ\).

The angles are \(30^\circ\), \(60^\circ\), and \(90^\circ\). The smallest is \(30^\circ\).

Thus, our calculated value for the smallest angle is correct.

Revision Table: Key Concepts for Right Triangles

Additional Information: Exploring Triangle Properties

Understanding triangle properties is crucial for solving geometry problems involving angles. Here are some additional points:

• An equilateral triangle has all three sides equal in length and all three angles equal in measure (\(60^\circ\) each).
• An isosceles triangle has at least two sides equal in length and the two angles opposite those sides are equal in measure.
• A scalene triangle has all three sides of different lengths and all three angles of different measures.
• The relationship between the sides and angles in any triangle is that the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle. In our right triangle example with angles \(30^\circ\), \(60^\circ\), \(90^\circ\), the side opposite the \(30^\circ\) angle is the shortest, the side opposite the \(60^\circ\) angle is the medium length, and the side opposite the \(90^\circ\) angle (the hypotenuse) is the longest.
• The sum of the exterior angles of any convex polygon, including a triangle, is always \(360^\circ\).