1. **Find all other angles inside the rectangles.** (i) Rectangle ABCD with diagonal AC and angle at A = 30°. - Step 1: Recall that in a rectangle, all angles are 90°. - Step 2: Diagonals of a rectangle are equal and bisect each other. - Step 3: Triangle ABC is formed by diagonal AC. - Step 4: Since angle at A is 30°, angle at C (opposite vertex) is also 30° because diagonals bisect each other. - Step 5: Angles at B and D adjacent to the diagonal are each 90° - 30° = 60°. (ii) Rectangle PQRS with diagonal PR and angle at intersection X = 110°. - Step 1: Diagonals bisect each other, so angles at intersection are supplementary pairs. - Step 2: Given angle X = 110°, the adjacent angle at intersection is 180° - 110° = 70°. - Step 3: Since diagonals bisect each other, opposite angles at intersection are equal. - Step 4: Therefore, the four angles at intersection are 110°, 70°, 110°, and 70°. 2. **Draw quadrilaterals with diagonals of length 8 cm bisecting each other at angles:** - Step 1: The diagonals bisecting each other means the quadrilateral is a parallelogram. - Step 2: The angle between diagonals is given (30°, 40°, 90°, 140°). - Step 3: Construct diagonals of length 8 cm intersecting at midpoint. - Step 4: Use the given angle between diagonals to complete the parallelogram. 3. **Figure APML formed by two perpendicular diameters PL and AM in a circle with center O.** - Step 1: PL and AM are diameters and perpendicular. - Step 2: Points A, P, M, L lie on the circle. - Step 3: Quadrilateral APML has all vertices on the circle and opposite sides are diameters. - Step 4: Since diameters are perpendicular, APML is a rectangle inscribed in the circle. 4. **Making an exact 90° angle using two sticks of equal length and a thread.** - Step 1: Tie the thread to form a triangle with the two sticks. - Step 2: Adjust the thread length to satisfy the Pythagorean theorem (e.g., 3-4-5 triangle). - Step 3: The angle opposite the longest side (thread) will be exactly 90°. 5. **Is having opposite sides parallel and equal a sufficient definition of a rectangle?** - Step 1: Opposite sides parallel and equal define a parallelogram. - Step 2: A rectangle is a parallelogram with all angles 90°. - Step 3: Therefore, not every parallelogram with opposite sides equal and parallel is a rectangle. 6. **Is it possible to construct a quadrilateral with three 90° angles and the fourth not 90°?** - Step 1: Sum of angles in a quadrilateral is 360°. - Step 2: If three angles are 90°, sum is 270°. - Step 3: Fourth angle must be 360° - 270° = 90°. - Step 4: Hence, the fourth angle must also be 90°, so such a quadrilateral is not possible. Final answers: - Angles in rectangle ABCD: 30°, 60°, 30°, 60° at diagonal intersections. - Angles at diagonal intersection in PQRS: 110° and 70° alternating. - Quadrilateral APML is a rectangle. - 90° angle can be made using sticks and thread by forming a right triangle. - Opposite sides parallel and equal define parallelogram, not necessarily rectangle. - Quadrilateral with three 90° angles must have fourth 90° angle.