Question

Olivia wrote a paragraph proof to prove that the diagonals of parallelogram MNOP bisect each other. Drag and drop the statements and reasons of her proof to put them in the correct order, starting with the first step at the top. Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.

Answer 1

1. Let the diagonals MO and NP intersect at X. — Reason: Definition of intersection.

2. MN ∥ OP and NP ∥ MO. — Reason: Opposite sides of a parallelogram are parallel.

3. ∠MNX = ∠OPX. — Reason: If two lines are parallel, alternate interior angles formed by a transversal are congruent (MN ∥ OP with transversal NP).

4. ∠MXN = ∠OXP. — Reason: Vertical angles are congruent.

5. ΔMXN ∼ ΔOXP. — Reason: Two pairs of equal angles ⇒ triangles are similar (AA).

6. MN/OP = NX/PX = MX/XO. — Reason: Corresponding sides of similar triangles are proportional.

7. MN = OP. — Reason: Opposite sides of a parallelogram are congruent.

8. From MN/OP = NX/PX and MN = OP, conclude NX = PX. — Reason: If the ratio MN/OP = 1, then NX/PX = 1, so NX = PX.

9. From NX = PX and MN/OP = NX/PX = MX/XO, conclude MX = XO. — Reason: Substitution into the proportion gives MX/XO = 1, so MX = XO.

10. Therefore X is the midpoint of both MO and NP, so the diagonals bisect each other. — Reason: Definition of bisect/midpoint.