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.
give me paragraphs and statements for those
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.
give me paragraphs and statements for those
Answers
GPT-4o mini
**Paragraph Proof:**
In a rectangle, by definition, opposite sides are equal in length and all angles are right angles (90 degrees). Let's consider rectangle LMNO. We want to show that the diagonals LN and MO are congruent.
1. **Definition of a Rectangle:** Since LMNO is a rectangle, we know that opposite sides are equal: \( LM = NO \) and \( LN = OM \). (Definition of a rectangle)
2. **Right Angles:** Furthermore, the angles at the corners of the rectangle, \( \angle L = \angle M = \angle N = \angle O\), are all right angles. (Definition of rectangle)
3. **Triangle Congruence:** By drawing the diagonals LN and MO, we create two triangles: triangle LMO and triangle NMO. In triangle LMO, we have \( LM = NO \), \( LO = MN \) (which are also equal by the properties of rectangles), and \( MO\) is a common side. Therefore, by the Side-Side-Side (SSS) postulate, triangle LMO is congruent to triangle NMO.
4. **Corresponding Parts of Congruent Triangles:** Since triangle LMO is congruent to triangle NMO, we can conclude that the corresponding parts of congruent triangles are equal. Therefore, \( LN \cong MO \). (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)
This proves that the diagonals \( LN \) and \( MO \) of rectangle LMNO are congruent.
**Statements and Reasons:**
| Statements | Reasons |
|---------------------------------------------------|------------------------------------------------|
| Rectangle LMNO has opposite sides that are equal: \( LM = NO \) and \( LN = OM \). | Definition of a rectangle. |
| All angles in rectangle LMNO are right angles. | Definition of a rectangle. |
| Diagonals \( LN \) and \( MO \) create triangles \( LMO \) and \( NMO \). | Diagonals of a quadrilateral form triangles. |
| Side \( LM = NO \) and side \( LO = MN \) are equal, and diagonal \( MO \) is common. | Properties of a rectangle; \( LM = NO \) and \( LO = MN \). |
| Triangles \( LMO \) and \( NMO \) are congruent by SSS. | SSS Postulate (Side-Side-Side congruence). |
| \( LN \) and \( MO \) are congruent. | CPCTC (Corresponding Parts of Congruent Triangles are Congruent). |
This proof confirms that the diagonals of rectangle LMNO are congruent.
In a rectangle, by definition, opposite sides are equal in length and all angles are right angles (90 degrees). Let's consider rectangle LMNO. We want to show that the diagonals LN and MO are congruent.
1. **Definition of a Rectangle:** Since LMNO is a rectangle, we know that opposite sides are equal: \( LM = NO \) and \( LN = OM \). (Definition of a rectangle)
2. **Right Angles:** Furthermore, the angles at the corners of the rectangle, \( \angle L = \angle M = \angle N = \angle O\), are all right angles. (Definition of rectangle)
3. **Triangle Congruence:** By drawing the diagonals LN and MO, we create two triangles: triangle LMO and triangle NMO. In triangle LMO, we have \( LM = NO \), \( LO = MN \) (which are also equal by the properties of rectangles), and \( MO\) is a common side. Therefore, by the Side-Side-Side (SSS) postulate, triangle LMO is congruent to triangle NMO.
4. **Corresponding Parts of Congruent Triangles:** Since triangle LMO is congruent to triangle NMO, we can conclude that the corresponding parts of congruent triangles are equal. Therefore, \( LN \cong MO \). (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)
This proves that the diagonals \( LN \) and \( MO \) of rectangle LMNO are congruent.
**Statements and Reasons:**
| Statements | Reasons |
|---------------------------------------------------|------------------------------------------------|
| Rectangle LMNO has opposite sides that are equal: \( LM = NO \) and \( LN = OM \). | Definition of a rectangle. |
| All angles in rectangle LMNO are right angles. | Definition of a rectangle. |
| Diagonals \( LN \) and \( MO \) create triangles \( LMO \) and \( NMO \). | Diagonals of a quadrilateral form triangles. |
| Side \( LM = NO \) and side \( LO = MN \) are equal, and diagonal \( MO \) is common. | Properties of a rectangle; \( LM = NO \) and \( LO = MN \). |
| Triangles \( LMO \) and \( NMO \) are congruent by SSS. | SSS Postulate (Side-Side-Side congruence). |
| \( LN \) and \( MO \) are congruent. | CPCTC (Corresponding Parts of Congruent Triangles are Congruent). |
This proof confirms that the diagonals of rectangle LMNO are congruent.