1. **Problem 11: Rectangle JKLM with diagonals intersecting at N** Given: Rectangle JKLM, diagonals intersect at N. We know in a rectangle, diagonals are equal and bisect each other. - Given $JN = x + 3$ - Given $JL = 3x + 1$ Since $N$ is the midpoint of diagonal $JL$, $JN = NL$. Also, diagonal $JL$ is the full diagonal, so $JL = JN + NL = 2 \times JN$. Using this, we get: $$JL = 2 \times JN$$ Substitute the expressions: $$3x + 1 = 2(x + 3)$$ Simplify: $$3x + 1 = 2x + 6$$ $$3x - 2x = 6 - 1$$ $$x = 5$$ Now find values: - $JN = x + 3 = 5 + 3 = 8$ - $JL = 3x + 1 = 3(5) + 1 = 16$ - Since $N$ is midpoint, $NL = JN = 8$ Check options: A. $NL = 4$? No, $NL=8$ B. $JN = 5$? No, $JN=8$ C. $NM = 8$? $NM$ is not defined in the problem, but if $NM$ is half of the other diagonal, it equals $JN=8$ (assuming symmetry). So yes. D. $KM = 10$? $KM$ is a side of the rectangle, no info given, cannot confirm. E. $JL = 16$? Yes. **Correct answers for 11: C and E** 2. **Problem 12: Parallelogram ABCD with diagonals intersecting at E** Given: Parallelogram ABCD, diagonals intersect at E. Triangles: $\triangle AED$ and $\triangle CEB$. Properties: - Diagonals of a parallelogram bisect each other, so $AE = EC$ and $DE = EB$. - Opposite sides are parallel and equal. To prove $\triangle AED \cong \triangle CEB$, consider congruence theorems: - **ASA (Angle-Side-Angle):** Two angles and the included side are equal. - **SAS (Side-Angle-Side):** Two sides and the included angle are equal. - **SSS (Side-Side-Side):** All three sides are equal. - **AA (Angle-Angle):** Only proves similarity, not congruence. Since diagonals bisect each other, two pairs of sides are equal. Also, angles at $E$ are vertical angles and equal. Therefore, possible congruence theorems: - ASA (angle at $A$ equals angle at $C$, side $AE=EC$, angle at $E$ vertical angles) - SAS (side $AE=EC$, angle $AED=CEB$, side $DE=EB$) - SSS (all three sides equal by parallelogram properties and diagonal bisection) AA is not sufficient for congruence. **Correct answers for 12: B (ASA), C (SAS), D (SSS)**