In a parallelogram ABCD, AB=10 cm, BC=14 cm, and angle B is 50 degrees. D and E are the midpoints of AB and AC respectively. Calculate DE and angle ADE.

Question 4 4.1 A rhombus is a parallelogram with all four sides equal in length. 4.2.1 Step 1: Identify the relationship between DE and BC. In ABC, D is the midpoint of AB and E is the midpoint of AC. By the Midpoint Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. DE = (1)/(2) BC Step 2: Substitute the given value and calculate DE. DE = (1)/(2) (14 cm) DE = 7 cm 4.2.2 Step 1: Identify the relationship between ADE and B. Since D and E are midpoints, DE BC by the Midpoint Theorem. When two parallel lines are intersected by a transversal (AB), the corresponding angles are equal. Step 2: State the size of ADE and the reason. ADE = B ADE = 50^ Reason: Corresponding angles (DE BC). Question 5 5.1 Step 1: Identify properties of a parallelogram ABCD. In parallelogram ABCD, AB DC and AB = DC (opposite sides of a parallelogram are parallel and equal). Step 2: Use the given perpendicular lines. Given BP AC and DQ AC, this means APB = 90^ and CQD = 90^. Step 3: Prove congruence using AAS. Consider ABP and CDQ: • APB = CQD = 90^ (Given BP AC and DQ AC) • AB = CD (Opposite sides of parallelogram ABCD) • BAP = DCQ (Alternate interior angles, since AB DC and AC is a transversal) Therefore, ABP CDQ (AAS congruence criterion). 5.2 Step 1: Use the result from 5.1. Since ABP CDQ (proven in 5.1), their corresponding sides are equal. Therefore, BP = DQ. Step 2: Determine the relationship between BP and DQ. Given BP AC and DQ AC. If two lines are perpendicular to the same line, then they are parallel to each other. Therefore, BP DQ. Step 3: Conclude that BPCQ is a parallelogram. Since one pair of opposite sides (BP and DQ) in quadrilateral BPCQ is both equal in length and parallel, BPCQ is a parallelogram. Send me the next one 📸