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QuestionIn triangle ACEACE, ADAD and EBEB are altitudes. Given ∠CAE=82∘\angle CAE=82^\circ and ∠AEC=47∘\angle AEC=47^\circ, find ∠AFE\angle AFE.

Studdy Solution

STEP 1

Assumptions1. BB and are points on $AC$ and $CE$ respectively. . $EB$ and $AD$ are altitudes of $\triangle ACE$.
3. $AD$ and $EB$ intersect at .
4. ∠CA=82∘\angle CA =82^{\circ} and ∠AEC=47∘\angle AEC =47^{\circ}.
5. We need to find ∠AFE\angle AFE.

STEP 2

First, let's find ∠ACE\angle ACE. We know that the sum of the angles in a triangle is 180∘180^{\circ}.
∠ACE=180∘−∠CA−∠AEC\angle ACE =180^{\circ} - \angle CA - \angle AEC

STEP 3

Now, plug in the given values for ∠CA\angle CA and ∠AEC\angle AEC to calculate ∠ACE\angle ACE.
∠ACE=180∘−82∘−47∘\angle ACE =180^{\circ} -82^{\circ} -47^{\circ}

STEP 4

Calculate the value of ∠ACE\angle ACE.
∠ACE=180∘−82∘−47∘=51∘\angle ACE =180^{\circ} -82^{\circ} -47^{\circ} =51^{\circ}

STEP 5

Since ADAD and EBEB are altitudes, ∠ADE\angle ADE and ∠BEC\angle BEC are right angles. This means ∠AED=90∘−∠ADE=90∘−51∘\angle AED =90^{\circ} - \angle ADE =90^{\circ} -51^{\circ}.

STEP 6

Calculate the value of ∠AED\angle AED.
∠AED=90∘−51∘=39∘\angle AED =90^{\circ} -51^{\circ} =39^{\circ}

STEP 7

In △AEF\triangle AEF, we know that the sum of the angles in a triangle is 180∘180^{\circ}. So, \angle AFE =180^{\circ} - \angle AEF - \angleAF.

STEP 8

We know that ∠AEF=∠AED=39∘\angle AEF = \angle AED =39^{\circ} and \angleAF = \angleAC =82^{\circ}.

STEP 9

Substitute the values of ∠AEF\angle AEF and \angleAF into the equation for ∠AFE\angle AFE.
∠AFE=180∘−39∘−82∘\angle AFE =180^{\circ} -39^{\circ} -82^{\circ}

STEP 10

Calculate the value of ∠AFE\angle AFE.
∠AFE=180∘−39∘−82∘=59∘\angle AFE =180^{\circ} -39^{\circ} -82^{\circ} =59^{\circ}So, ∠AFE=59∘\angle AFE =59^{\circ}.

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