1. Problem: Construct triangle ABC with AB = 5 cm, angle ABC = 75°, and circumradius = 3.5 cm using ruler and compass. Step 1: Draw segment AB = 5 cm. Step 2: At point B, construct angle ABC = 75° using compass and ruler. Step 3: The circumradius is 3.5 cm, so draw a circle with radius 3.5 cm centered at the circumcenter O (to be found). Step 4: The circumcenter lies on the perpendicular bisector of AB. Construct the perpendicular bisector of AB. Step 5: The point C lies on the ray from B forming 75° and on the circle of radius 3.5 cm centered at O. Step 6: Find O on the perpendicular bisector such that distance OA = OB = OC = 3.5 cm. Step 7: Mark point C on the 75° ray at distance OC = 3.5 cm from O. 2. Problem: Construct triangle ABC with BC = 6.4 cm, CA = 5.8 cm, angle ABC = 60°, draw incircle, measure radius. Step 1: Draw segment BC = 6.4 cm. Step 2: At B, construct angle ABC = 60°. Step 3: From C, draw arc with radius 5.8 cm to intersect the 60° ray from B; mark intersection as A. Step 4: Construct angle bisectors of triangle ABC. Step 5: The incenter I is the intersection of angle bisectors. Step 6: Draw incircle centered at I tangent to all sides. Step 7: Measure radius of incircle (distance from I to any side). 3. Problem: (i) Construct triangle ABC with AB = 4 cm, BC = 6 cm, angle ABC = 90°. Step 1: Draw segment BC = 6 cm. Step 2: At B, construct a 90° angle. Step 3: From B, draw arc radius 4 cm intersecting 90° ray; mark as A. (ii) Construct circle through A, B, C and mark center O. Step 4: Construct perpendicular bisectors of AB and BC. Step 5: Their intersection is circumcenter O. Step 6: Draw circle centered at O passing through A, B, C. 4. Problem: Construct triangle ABC with BC = 6 cm, AB = 5.5 cm, angle ABC = 120°. Step 1: Draw BC = 6 cm. Step 2: At B, construct 120° angle. Step 3: From B, draw arc radius 5.5 cm intersecting 120° ray; mark as A. (i) Construct circumscribing circle. Step 4: Construct perpendicular bisectors of AB and BC. Step 5: Intersection is circumcenter O. Step 6: Draw circle centered at O passing through A, B, C. (ii) Draw cyclic quadrilateral ABCD with D equidistant from B and C. Step 7: Draw circle through A, B, C. Step 8: Find point D on circle such that DB = DC (D lies on perpendicular bisector of BC). 5. Problem: Construct triangle ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm, construct incircle, measure radius. Step 1: Draw BC = 6.5 cm. Step 2: From B, draw arc radius 5.5 cm. Step 3: From C, draw arc radius 5 cm; intersection is A. Step 4: Construct angle bisectors; their intersection is incenter I. Step 5: Draw incircle centered at I tangent to sides. Step 6: Measure radius of incircle. Final answers: All constructions follow ruler and compass methods as described.