To fill in the missing parts and complete the proof, we need to make sure each statement logically leads to the next one, ensuring that we have a clear argument supporting each reason given. Here’s a completed version of your proof outline:
Proof Outline:
| Statement | Reason | |-------------------------------------|------------------------------------------------------------------------| | 1) LMNO is a rectangle | 1) Given | | 2) LM ≅ ON and LN ≅ MO | 2) Opposite sides of a rectangle are congruent | | 3) ∠MLO ≅ ∠ONM | 3) All angles of a rectangle are congruent, right angles | | 4) ΔLMO ≅ ΔONM | 4) SAS Congruence Theorem (LM ≅ ON, LN ≅ MO, ∠MLO ≅ ∠ONM) | | 5) Corresponding parts of congruent triangles are congruent | 5) CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
Explanation of Each Step:
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Given - It is stated that LMNO is a rectangle, which is the basis for all subsequent claims.
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Opposite sides of a rectangle are congruent - According to the properties of rectangles, we know that opposite sides are equal in length.
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All angles of a rectangle are congruent, right angles - This states that all angles in a rectangle measure 90 degrees, hence they are congruent to each other.
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SAS Congruence Theorem - Since we have two pairs of sides that are equal in length (LM ≅ ON and LN ≅ MO) and the included angle (∠MLO ≅ ∠ONM) is also congruent, we can apply the SAS (Side-Angle-Side) criterion for triangle congruence.
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CPCTC - Since the triangles ΔLMO and ΔONM are congruent, all corresponding parts of these triangles are congruent (hence the use of ‘CPCTC’).
This completed proof outlines the logical reasoning that links each statement with its corresponding reason, fully demonstrating the congruence of triangles formed from the rectangle.