Determining the Longest Side in a Triangle Using Angle Measures

In the study of geometry, particularly in the context of triangles, understanding how to determine the longest side based on the given angles is a fundamental concept. This article aims to provide a clear explanation of how to find the longest side when the measures of two angles in a triangle are given. We will use the triangle ABC and explore the process step-by-step.

Given Angles and Finding the Third Angle

In triangle ABC, given that ∠A∠A∠A 40° and ∠B∠B∠B 60°. The sum of the angles in any triangle is 180°, so we can calculate the measure of angle C

∠C180|∠A?∠B∠C180|∠A-∠B∠C180||∠A?∠B

Substituting the values of ∠B∠B∠B , we get:

Solving this, we find:

∠C80°?∠C80^°∠C80°

Thus, angle C 80°.

Which Side is the Longest?

According to the angle sum property of a triangle, the side opposite to the largest angle is the longest side. In this case, angle C is the largest angle. Therefore, the side opposite to angle C is the longest side.

Side AB is the longest.

Sine Rule and Longest Side

Using the sine rule, we can further confirm the longest side. The sine rule states that:

( frac{a}{sin A} frac{b}{sin B} frac{c}{sin C} )

Substituting the values, we have:

( frac{a}{sin 40°} frac{b}{sin 60°} frac{c}{sin 80°} )

Since ∠C∠C∠C 80° is the largest angle, the corresponding side c (opposite to angle ∠C∠C∠C ) is the longest.

Conclusion

Using the angle sum property and the sine rule, we can determine that in triangle ABC, the longest side is AB as angle C is the largest.

Final Answer: The longest side is AB.

For further study and practice, exploring the sine rule and angle properties in more complex geometric shapes and scenarios can be beneficial. Understanding these concepts will not only enhance problem-solving skills but also enrich one's understanding of geometry.