Question

△RST has angle measures of ∠R=45° and ∠T=105°. Using the angle measures, which list indicates the side lengths in decreasing order?(1 point)ResponsesTR<ST<RSupper T upper R is less than upper S upper T is less than upper R upper STS>TR>SRupper T upper S is greater than upper T upper R is greater than upper S upper RRS<TR<TSupper R upper S is less than upper T upper R is less than upper T upper SRS>TS>TR

Answer 1

To determine the order of the side lengths of triangle \( \triangle RST \) based on the given angle measures, we can use the fact that in a triangle, the side opposite the largest angle is the longest and the side opposite the smallest angle is the shortest.

Given the angle measures:

• \( \angle R = 45^\circ \)
• \( \angle T = 105^\circ \)

To find \( \angle S \): \[ \angle S = 180^\circ - \angle R - \angle T = 180^\circ - 45^\circ - 105^\circ = 30^\circ \]

Now we have the angle measures:

• \( \angle R = 45^\circ \) (opposite side \( ST \))
• \( \angle T = 105^\circ \) (opposite side \( RS \))
• \( \angle S = 30^\circ \) (opposite side \( TR \))

Based on these angles:

• \( \angle T \) is the largest angle, so \( RS \) is the longest side.
• \( \angle R \) is the second largest angle, so \( ST \) is the second longest side.
• \( \angle S \) is the smallest angle, so \( TR \) is the shortest side.

Thus, the order of the sides in decreasing order of length is: \[ RS > ST > TR \]

The correct response is: RS > ST > TR.