List the angle measures of triangle CDE in order f

Questions: List the angle measures of triangle CDE in order from smallest to largest. Assume that w is a positive number. m angle < m < angle < m angle

Transcript text: List the angle measures of $\triangle C D E$ in order from smallest to largest. Assume that $w$ is a positive number. $m \angle$ $\square$ $

Solution

Solution Steps

Step 1: Find the value of $w$

We are given a triangle CDE. The lengths of the sides are CE = 31w, DE = 11w, and the angle E is 96°. According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side. Therefore, we have: CE + DE > CD CE + CD > DE CD + DE > CE

Since w is a positive number, we can divide by w: 31 + 11 > CD/w 31 + CD/w > 11 11 + CD/w > 31

42 > CD/w, CD/w > -20, and CD/w > 20. Combining the inequalities regarding CD/w, we have CD/w > 20.

Step 2: Compare the sides

We have the lengths of the sides as CE = 31w, ED = 11w, and CD is unknown. Since w is a positive number, we can compare the sides based on their coefficients: 11w < 31w So, DE < CE.

Step 3: Relate side lengths to angles

In a triangle, the smallest angle is opposite the smallest side, and the largest angle is opposite the largest side. In our triangle, DE < CE. Therefore, the angle opposite to DE is smaller than the angle opposite to CE. This means $ m\angle C < m\angle D$.

We also know that $ \angle E = 96^\circ$. The sum of the angles in a triangle is 180°. Therefore, $ m\angle C + m\angle D + m\angle E = 180^\circ$. Since $ m\angle E = 96^\circ$, we have $ m\angle C + m\angle D = 180^\circ - 96^\circ = 84^\circ$. Since both $ m\angle C$ and $ m\angle D $ are less than $96^\circ$, and their sum is $84^\circ$, we can conclude that they are both smaller than $96^\circ$. Thus, $m\angle C < m\angle D < m\angle E$.