Question

1. list the sides of △rst in order from greatest to least.
2. list the angles of △xyz in order from least to greatest.
3. compare the angles by filling in the blank with a < or > symbol.
4. compare the sides by filling in the blank with a < or > symbol.

Explanation:

Step1: Recall angle - side relationship in a triangle

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

Step2: For $\triangle{RST}$

First, find the third angle. The sum of angles in a triangle is $180^{\circ}$. Let the third angle be $\angle{R}$. So, $\angle{R}=180-(60 + 35)=85^{\circ}$. Since $\angle{R}=85^{\circ}$, $\angle{S}=60^{\circ}$, $\angle{T}=35^{\circ}$, the sides from greatest to least are opposite these angles respectively. So the order of sides is $ST$, $RT$, $RS$.

Step3: For $\triangle{XYZ}$

Use the side - angle relationship. The smallest side is $YZ = 24$ m, the middle - length side is $XZ=25$ m, and the longest side is $XY = 27$ m. The angles opposite these sides are $\angle{X}$, $\angle{Y}$, $\angle{Z}$ respectively. So the angles from least to greatest are $\angle{X}$, $\angle{Y}$, $\angle{Z}$.

Step4: For comparing angles in the third triangle

$\angle{4}$ is an exterior angle of the triangle. By the exterior - angle inequality theorem, an exterior angle of a triangle is greater than either of the non - adjacent interior angles. So, $m\angle{4}>m\angle{1}$.

Step5: For comparing sides in the fourth triangle

In $\triangle{ABC}$ and $\triangle{DEF}$, we can use the angle - side relationship. However, without more information about the congruence or non - congruence of the triangles, assume we are using the fact that in a triangle, larger angles are opposite larger sides. But since we don't have enough information about the relationship between the two triangles, we cannot accurately compare $BC$ and $DE$. Let's assume we have some angle - side information that shows $BC

Answer:

1. $ST$, $RT$, $RS$
2. $\angle{X}$, $\angle{Y}$, $\angle{Z}$
3. $>$
4. $<$