Question

What are the measures of DBE and BEF?

Answer 1

Answer: DBE - 71 degrees, BEF = 161 degrees

Explanation:

Angle ABD and Angle DBE are supplementary, which means they add up to 180 degrees.

Therefore, 3x + 4 + 2x + 1 = 180

5x + 5 = 180

5x = 175

x = 35 degrees

So, angle DBE = 2*35 + 1

= 71 degrees

Angle BEF is supplementary to Angle BED = x -16 = 19

19 + angle BEF = 180

Angle BEF = 161 degrees

Answer 2

• ∠DBE = 71°
• ∠BEF = 161°

Explanation:

You want to know the measures of interior angle DBE and exterior angle BEF in the right triangle figure shown.

Complementary angles

Angles DBE and DEB in the right triangle are complementary, so we have ...

(2x +1) +(x -16) = 90

3x = 105 . . . . . . . . . . . . add 15

x = 35

DBE

The measure of angle DBE is ...

(2x +1)° = (2·35 +1)° = 71°

The measure of ∠DBE is 71°.

Exterior angle theorem

The exterior angle theorem tells you that exterior angle BEF is equal to the sum of remote interior angles EBD and EDB.

∠BEF = ∠EBD +∠EDB

∠BEF = 71° +90° = 161°

The measure of ∠BEF is 161°.

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Additional comment

As in a lot of angle problems there are a number of different relations that could be used to find the solution. Angles DBE and DBA are supplementary, so their sum could be used to find the value of x. This would also tell you ∠DBA = 109°. Having found x, we know ∠DEB = 19°, so the fact that ∠DEB and ∠BEF are supplementary could also be used to find ∠BEF.

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