A photo shows a rectangle with the top right one labeled M, bottom right, N, top left, L, bottom left, O. A line is drawn from L to N. A line is drawn from M to O. Where the two lines intersect is labeled P.

Question

Prove that the diagonals of rectangle LMNO are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements.

Answer 1

Two-Column Proof

| Statement | Reason | |------------------------------------------------------|----------------------------------------------------------------------------------------| | 1. Define LMNO as a rectangle with vertices L, M, N, O. | By definition of a rectangle, opposite sides are equal and all angles are 90 degrees. | | 2. Draw diagonals LN and MO. | By definition, diagonals are segments connecting opposite vertices of a polygon. | | 3. Triangle LMO and triangle NMP are formed by the diagonals. | The diagonals intersect at point P. | | 4. LM = ON and LO = MN (opposite sides of rectangle). | Properties of rectangles (opposite sides are equal). | | 5. Angle LMO = angle NMP (both are vertical angles). | Vertical angles are always congruent. | | 6. Triangle LMO ≅ triangle NMP by the Side-Angle-Side (SAS) postulate. | Two sides and the angle between them are congruent in both triangles. | | 7. By the congruence of triangles LMO and NMP, it follows that the diagonals LN and MO are congruent. | Corresponding parts of congruent triangles are congruent (CPCTC). |

Conclusion

Therefore, the diagonals of rectangle LMNO, LN and MO, are congruent.

Answer 2

Can you do it in paragraph proof, please? Thank you!! :D

Answer 3

Certainly! Here's a paragraph proof demonstrating that the diagonals of rectangle LMNO are congruent:

To prove that the diagonals of rectangle LMNO are congruent, we start by noting the properties of rectangles, which state that opposite sides are equal and that all angles are right angles. We draw the diagonals, LN and MO, which intersect at point P. This creates two triangles: triangle LMO and triangle NMP. Because LM = ON and LO = MN, we can say that these sides are equal as they are opposite sides of the rectangle. Additionally, the angles LMO and NMP are vertical angles formed by the intersection of the diagonals and are therefore congruent. From these observations, we can apply the Side-Angle-Side (SAS) postulate, which tells us that if two sides and the angle between them in one triangle are congruent to two sides and the angle between them in another triangle, the triangles are congruent. Hence, triangles LMO and NMP are congruent. Finally, by the corresponding parts of congruent triangles are congruent (CPCTC), we conclude that the diagonals LN and MO must be congruent. Thus, we have proven that the diagonals of rectangle LMNO are indeed congruent.