Question

in δcde, de = 15, ec = 9, and cd = 13. what is the order of the angles smallest to largest. smallest choose choose largest choose

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the side opposite an angle, the larger the angle. So we need to first identify the sides and their opposite angles in \(\triangle CDE\).

Let's label the triangle: In \(\triangle CDE\), side \(DE = 15\), side \(EC=9\), side \(CD = 13\).

• The angle opposite \(DE\) is \(\angle C\) (because in \(\triangle CDE\), side \(DE\) is opposite vertex \(C\)).
• The angle opposite \(EC\) is \(\angle D\) (side \(EC\) is opposite vertex \(D\)).
• The angle opposite \(CD\) is \(\angle E\) (side \(CD\) is opposite vertex \(E\)).

Step2: Order the sides from smallest to largest

First, order the lengths of the sides: \(EC = 9\), \(CD=13\), \(DE = 15\). So \(9<13<15\), which means \(EC < CD

Step3: Order the angles based on the side - angle relationship

Since the angle opposite a smaller side is smaller, we have:

• The side opposite \(\angle D\) is \(EC = 9\) (the smallest side), so \(\angle D\) is the smallest angle.
• The side opposite \(\angle E\) is \(CD = 13\) (the middle - length side), so \(\angle E\) is the middle - sized angle.
• The side opposite \(\angle C\) is \(DE=15\) (the largest side), so \(\angle C\) is the largest angle.

So the order of the angles from smallest to largest is \(\angle D\), \(\angle E\), \(\angle C\).

Answer:

Smallest: \(\angle D\)
Middle: \(\angle E\)
Largest: \(\angle C\)