# How do you determine the number of possible triangles and find the measure of the three angles given DE=24, EF=18, mangleD=15?

Feb 26, 2018

In both cases, triangle is obtuse

Case (1) : $\hat{D} = 15 , \hat{E} = 144.81 , \hat{F} = 20.19$

Case (2) : hat D = 15, hat E = 5.19![enter image source here](https://useruploads.socratic.org/ksXfMFG0QNOktMneNpEQ_7EE5B0EE-75A0-4F6B-81B0-B6B5700E0A94.png) , hat F = 159.81

#### Explanation:

Two solutions possible with the given SSA measurements. $\hat{F}$ - One acute and another obtuse angle.

Case 1: $\hat{F}$ acute.

$\sin F = \frac{f \sin D}{d} = \frac{24 \cdot \sin 15}{18} = 0.3451$

$\hat{F} = {\sin}^{-} 1 0.3451 = {20.19}^{\circ}$

$\hat{E} = 180 - 15 - 20.19 = {144.81}^{\circ}$

Case 2: $\hat{F}$ obtuse.

$\sin F = \frac{f \sin D}{d} = \frac{24 \cdot \sin 15}{18} = 0.3451$

$\hat{F} = {\sin}^{-} 1 0.3451 = 180 - 20.19 = {159.81}^{\circ}$

$\hat{E} = 180 - 15 - 159.81 = {5.19}^{\circ}$