Question

A city park can be represented by trapezoid DEFG, where DE¯¯¯¯¯ || FG¯¯¯¯¯, and DF¯¯¯¯¯ and EG¯¯¯¯¯ represent hiking paths that span the diagonals of the park.

A planning team is interested in planting additional trees throughout the park and has measured some of the angles between the edges of the park and the sidewalks as shown in the diagram below to determine the best location for the trees.



Using the information in the diagram, find m∠DFG.

Answer 1

H. 26°

Explanation:

Two facts from geometry come into play here:

• alternate interior angles are congruent where a transversal crosses parallel lines
• the exterior angle of a triangle is equal to the sum of the remote interior angles

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The second of these relations tells us the exterior angle at X of triangle XDE (where X is the point the diagonals cross) is equal to the sum of angles D and E of that triangle. That is, ...

50° = ∠FDE +∠GED

∠FDE = 50° -24° = 26°

The second of the relations described above tells us that angles FDE and DFG are congruent.

∠DFG = ∠FDE = 26°