Understanding the Problem: Regular Pentagon and Extended Sides

The question asks for the measure of the angle formed when two sides of a regular pentagon, specifically sides BA and DE, are extended until they meet at a point F. We need to find the measure of angle $\angle$EFA.

Let's visualize a regular pentagon ABCDE. Side BA is extended past A to meet side DE extended past E at point F. This forms a triangle, $\triangle$AFE.

Properties of a Regular Pentagon

A regular pentagon has 5 equal sides and 5 equal interior angles. We need to know the measure of the interior and exterior angles of a regular pentagon.

• The sum of the interior angles of a polygon with $n$ sides is given by the formula $(n-2) \times 180^\circ$. For a regular pentagon, $n=5$.
• Sum of interior angles = $(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$.
• Each interior angle of a regular pentagon = $\frac{540^\circ}{5} = 108^\circ$.
• The sum of the exterior angles of any convex polygon is $360^\circ$.
• Each exterior angle of a regular pentagon = $\frac{360^\circ}{5} = 72^\circ$.

Calculating Angles in $\triangle$AFE

The triangle formed is $\triangle$AFE. We need to find the measures of the angles within this triangle at vertices A and E to calculate the angle at vertex F ($\angle$EFA).

• Angle at vertex A ($\angle$FAE): The side BA is produced (extended) to meet at F. This means F lies on the line extending BA past A. The line segment FA is part of the extended line of BA. The angle $\angle$FAE is formed by the extended side BA (segment AF) and the side AE of the pentagon. This angle is the exterior angle of the pentagon at vertex A, formed by extending side BA. The interior angle at A is $\angle$BAE = $108^\circ$. The exterior angle at A, formed by extending BA, is adjacent to the interior angle on a straight line (the extended line of BA). Thus, $\angle$FAE is equal to the exterior angle of the regular pentagon at A.

So, $\angle$FAE = Exterior angle of regular pentagon = $72^\circ$.

• Angle at vertex E ($\angle$FEA): Similarly, the side DE is produced (extended) to meet at F. This means F lies on the line extending DE past E. The line segment FE is part of the extended line of DE. The angle $\angle$FEA is formed by the extended side DE (segment FE) and the side AE of the pentagon. This angle is the exterior angle of the pentagon at vertex E, formed by extending side DE. The interior angle at E is $\angle$AED = $108^\circ$. The exterior angle at E, formed by extending DE, is adjacent to the interior angle on a straight line (the extended line of DE). Thus, $\angle$FEA is equal to the exterior angle of the regular pentagon at E.

So, $\angle$FEA = Exterior angle of regular pentagon = $72^\circ$.

Now we have two angles in $\triangle$AFE: $\angle$FAE = $72^\circ$ and $\angle$FEA = $72^\circ$. The sum of the angles in any triangle is $180^\circ$.

In $\triangle$AFE:

$\angle$EFA + $\angle$FAE + $\angle$FEA = $180^\circ$

$\angle$EFA + $72^\circ$ + $72^\circ$ = $180^\circ$

$\angle$EFA + $144^\circ$ = $180^\circ$

$\angle$EFA = $180^\circ - 144^\circ$

$\angle$EFA = $36^\circ$

Final Answer for the Angle

The measure of $\angle$EFA is $36^\circ$. This is the angle formed where the extended sides BA and DE of the regular pentagon meet.

Revision Table: Key Geometry Concepts

Additional Information: Related Polygon Geometry

Understanding the properties of regular polygons is fundamental in geometry. Here are a few more points:

• All exterior angles of any convex polygon, one at each vertex, always sum up to $360^\circ$.
• The number of sides of a regular polygon can be found if you know the measure of an interior or exterior angle. If the exterior angle is $\theta_{ext}$, $n = \frac{360^\circ}{\theta_{ext}}$. If the interior angle is $\theta_{int}$, first find the exterior angle $\theta_{ext} = 180^\circ - \theta_{int}$, then use the formula for $n$.
• Isosceles Triangles: Since $\angle$FAE = $\angle$FEA = $72^\circ$, $\triangle$AFE is an isosceles triangle with AF = FE.
• The intersection point F lies outside the pentagon, as expected when extending non-adjacent sides.