Question

What is the order of the side lengths in triangle QRT, from greatest to least, given that angle Q measures 80 degrees and angle T measures 35 degrees?

Answer 1

To determine the order of the side lengths of triangle QRT from greatest to least, we will refer to the relationship between the angles and opposite sides in a triangle.

In a triangle, the side opposite the largest angle is the longest side, and the side opposite the smallest angle is the shortest side.

Given:
- Angle \( Q = 80^\circ \)
- Angle \( T = 35^\circ \)

First, we can calculate the remaining angle \( R \) using the fact that the sum of angles in a triangle equals \( 180^\circ \):

\[
R = 180^\circ - Q - T
\]
\[
R = 180^\circ - 80^\circ - 35^\circ = 65^\circ
\]

Now we have the angles:
- \( \angle Q = 80^\circ \)
- \( \angle R = 65^\circ \)
- \( \angle T = 35^\circ \)

Now, we will sort the angles from greatest to least:
1. \( \angle Q = 80^\circ \) (largest angle)
2. \( \angle R = 65^\circ \)
3. \( \angle T = 35^\circ \) (smallest angle)

Consequently, we note the sides opposite these angles:
- Side \( QT \) is opposite angle \( R \) (65 degrees)
- Side \( RT \) is opposite angle \( Q \) (80 degrees)
- Side \( QR \) is opposite angle \( T \) (35 degrees)

Now we can rank the sides based on the angles:
1. The longest side is \( RT \) (opposite the largest angle \( 80^\circ \)).
2. The middle side is \( QT \) (opposite the middle angle \( 65^\circ \)).
3. The shortest side is \( QR \) (opposite the smallest angle \( 35^\circ \)).

Thus, the order of the side lengths from greatest to least is:
\[
RT > QT > QR
\]