Solving for the Angles of a Triangle Given Equal Measure Conditions

Triangles are fascinating geometric shapes with their angles following specific rules. One such interesting problem involves a triangle where two angles are of equal measure, and each is one-third the measure of the third angle. This article will explore how to solve for the angles of such a triangle using algebraic methods.

Problem Statement

Given a triangle with angles where two angles are of equal measure, and each is one-third the measure of the third angle. Let each of the equal angles be denoted as (frac{x}{3}), and the third angle as (x). The sum of the angles in any triangle is always (180^circ).

Solution

According to the problem, we can write the following equation for the sum of the angles:

(frac{x}{3} frac{x}{3} x 180^circ)

Combining the terms, we get:

(frac{2x}{3} x 180^circ)

Expressing (x) as (frac{3x}{3}), the equation becomes:

(frac{2x}{3} frac{3x}{3} 180^circ)

Simplifying further:

(frac{5x}{3} 180^circ)

Multiplying both sides by 3:

(5x 540^circ)

Dividing both sides by 5:

(x 108^circ)

Now, substituting the value of (x), we can find the two equal angles:

(frac{x}{3} frac{108^circ}{3} 36^circ)

The three angles of the triangle are:

(36^circ, 36^circ, 108^circ)

Conclusion

Thus, the angles of the triangle are (36^circ, 36^circ), and (108^circ). This solution demonstrates the application of algebraic equations to solve geometric problems. If you ever encounter a similar problem, remember to identify the given conditions and set up the necessary equations to find the unknown angles.

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Solving for the Angles of a Triangle Given Equal Measure Conditions

Solving for the Angles of a Triangle Given Equal Measure Conditions

Problem Statement

Solution

Conclusion

Related Topics

Solving for the Angles of a Triangle Given Equal Measure Conditions

Problem Statement

Solution

Conclusion

Related Topics

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