Question

In \(\triangle P R S, m \angle P=30, m \angle R=45\), and \(m \angle S=105\). Which side of \(\triangle P R S\) has the greatest measure?

Short answer

The side PR is the longest.

Step-by-step solution

Understanding Triangle Angle-Side Relationship

In any triangle, the side opposite to the largest angle is the longest. This is because larger angles in triangles correspond to longer opposite sides.

Identifying the Angles and Their Sides

In this triangle, the angles are given as \(m \angle P = 30^\circ\), \(m \angle R = 45^\circ\), and \(m \angle S = 105^\circ\). The side opposite \(m \angle P\) is \(RS\), opposite \(m \angle R\) is \(PS\), and opposite \(m \angle S\) is \(PR\).

Comparing the Angles

Compare the magnitudes of the angles: \(30^\circ, 45^\circ,\) and \(105^\circ\). Clearly, \(105^\circ\) is the largest angle.

Determining the Longest Side

Since \(105^\circ\) is the largest angle, the side opposite this angle, \(PR\), is the longest side of \(\triangle P R S\).

Key concepts

Angle-Side Relationship

In the world of triangles, the relationship between angles and their opposite sides is crucial. This relationship states that in any triangle, the side opposite the largest angle will be the longest. The reasoning is straightforward: a larger angle will "open up" the triangle more, allowing the opposite side to stretch out further.
In simpler terms, picture the triangle as a door. If you open the door slightly (like a small angle), the gap (side) at the bottom is short. But if you swing it wide open (like a large angle), that gap grows bigger.
When tackling problems involving the angle-side relationship, always identify which angle is the largest. This will directly tell you which side is the longest. It's all about connecting which side is opposite which angle.

Triangle Angles

A fundamental property of triangles is that their interior angles always add up to 180 degrees. This is true for any type of triangle, be it scalene, isosceles, or equilateral.
Knowing the sum of 180 degrees helps in a multitude of calculations. For instance, if you know two angles in a triangle, you can easily find the third by subtracting the sum of the known angles from 180 degrees.
For example, in \(\triangle PRS\), the angles are 30, 45, and 105 degrees. Adding 30 and 45 gives us 75. Subtract that from 180, and what do we have? Right—105 degrees! It all lines up, ensuring our calculations are correct.
Understanding this property can make solving angle problems in triangles much simpler.

Largest Side

Now that we've explored angles, let's find the largest side of the triangle using their relationship. In triangle \(\triangle PRS\) with angles of 30, 45, and 105 degrees, the side opposite the largest angle is what we're after.
The 105-degree angle, being the largest, indicates that its opposing side, \(PR\), will be the longest. This isn't just a guess; it's a geometric truth stemming from how the angle-side relationship works.
Identifying the largest angle could seem tough at first, but once you grasp which angle is the biggest, pinpointing the longest side becomes a breeze. Always remember the rule: larger angle, longer side. This concept is not just limited to this exercise; it's a universal truth in the world of triangles.