75 degrees

To construct an angle of 75^ using only a compass and a pencil, we can combine a 60^ angle and a 15^ angle. The 15^ angle is obtained by bisecting the 30^ angle between the 60^ and 90^ lines. Step 1: Draw a ray and construct a 60^ angle. • Draw a ray OA. • Place the compass at point O (the vertex) and draw a large arc that intersects OA at point P. • With the same compass width, place the compass at P and draw an arc that intersects the first arc at point Q. • Draw a ray from O through Q. This ray, OQ, forms a 60^ angle with OA (i.e., QOA = 60^). Step 2: Construct a 90^ angle. • With the same compass width, place the compass at Q and draw an arc that intersects the initial large arc at point R. (This creates a 120^ angle, ROA). • Place the compass at Q and draw an arc above the initial large arc. • Place the compass at R and draw another arc that intersects the arc drawn from Q. Let this intersection point be S. • Draw a ray from O through S. This ray, OS, forms a 90^ angle with OA (i.e., SOA = 90^). Step 3: Bisect the angle between 60^ and 90^. The angle between OQ (60^) and OS (90^) is SOQ = 90^ - 60^ = 30^. • Place the compass at Q (where the initial arc intersects OQ) and draw an arc inside SOQ. • Let T be the point where the initial large arc intersects OS. Place the compass at T and draw another arc that intersects the arc drawn from Q. Let this intersection point be U. • Draw a ray from O through U. This ray, OU, bisects SOQ. Therefore, UOQ = (1)/(2) × 30^ = 15^. Step 4: Identify the 75^ angle. The angle UOA is the sum of UOQ and QOA. UOA = UOQ + QOA UOA = 15^ + 60^ UOA = 75^ The final answer is The angle UOA = 75^. Last free one today — make it count tomorrow, or type /upgrade for unlimited.