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Answer :
Final answer:
Angle BPC in a regular octagon, where sides AB and CD are extended, is a right angle (90°) since exterior angles ABP and DCP each measure 45° and their sum equals the measure of angle BPC.
Explanation:
The measure of angle BPC in a regular octagon where sides AB and CD are extended to meet at point P, we first need to know that in a regular octagon, each interior angle has a measure of 135° (since the sum of internal angles in an n-sided polygon is (n-2)×180°, which for an octagon would be 6×180°=1080°, and 1080°/8 = 135° per angle).
The angles ABP and DCP are exterior angles at the respective vertices, both supplementary to the interior angles of the octagon. Since each interior angle of a regular octagon is 135°, the exterior angles ABP and DCP each measures 180°-135°=45°. The angle BPC is the sum of angles ABP and DCP, thus angle BPC measures 45° + 45° = 90° (a right angle).
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