Solving Triangle Angles Problem

The question asks us to find the measures of the other two angles of a triangle, given that one angle is 55 degrees and the ratio of the other two angles is 9:16.

Understanding Triangle Angle Sum Property

A fundamental property of any triangle is that the sum of its interior angles is always 180 degrees.

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

According to the property, \(A + B + C = 180^\circ\).

Setting up the Equation

We are given that one angle is 55 degrees. Let's say \(A = 55^\circ\).

The ratio of the other two angles is given as 9:16. Let the other two angles be \(B\) and \(C\).

Since the ratio is 9:16, we can represent these angles as \(9x\) and \(16x\), where \(x\) is a common multiplier.

So, \(B = 9x\) and \(C = 16x\).

Now, we can substitute these values into the angle sum equation:

\(55^\circ + 9x + 16x = 180^\circ\)

Solving for x

Combine the terms with \(x\):

\(55^\circ + 25x = 180^\circ\)

Subtract 55 degrees from both sides of the equation:

\(25x = 180^\circ - 55^\circ\)

\(25x = 125^\circ\)

Now, divide by 25 to find the value of \(x\):

\(x = \frac{125^\circ}{25}\)

\(x = 5^\circ\)

Calculating the Other Two Angles

Now that we have the value of \(x\), we can calculate the measures of the other two angles:

• First angle: \(9x = 9 \times 5^\circ = 45^\circ\)
• Second angle: \(16x = 16 \times 5^\circ = 80^\circ\)

So, the other two angles are 45 degrees and 80 degrees.

Verifying the Solution

Let's check if these angles satisfy the conditions:

• Sum of all three angles: \(55^\circ + 45^\circ + 80^\circ = 100^\circ + 80^\circ = 180^\circ\). The sum is 180 degrees, which is correct.
• Ratio of the two angles: The ratio of 45 degrees to 80 degrees is \(\frac{45}{80}\). Dividing both numerator and denominator by their greatest common divisor (which is 5), we get \(\frac{45 \div 5}{80 \div 5} = \frac{9}{16}\). The ratio is 9:16, which is also correct.

The calculated angles 45 degrees and 80 degrees match the conditions given in the problem.

Matching with Options

Let's compare our calculated angles with the given options:

The angles 45 degrees and 80 degrees correspond to Option D.

Conclusion

Based on the calculations, the other two angles of the triangle are 45 degrees and 80 degrees.

Revision Table: Key Concepts

Additional Information: Types of Triangles

Understanding angle properties helps classify triangles:

• Equilateral Triangle: All three angles are equal (60 degrees each).
• Isosceles Triangle: Two angles are equal.
• Scalene Triangle: All three angles are different.
• Acute Triangle: All three angles are less than 90 degrees. (Our triangle with angles 55°, 45°, 80° is an acute triangle).
• Right Triangle: One angle is exactly 90 degrees.
• Obtuse Triangle: One angle is greater than 90 degrees.

The problem helps illustrate how knowing some information about a triangle's angles allows us to determine the others using fundamental geometric principles and algebraic methods.