Rhombus properties (equal sides, opposite angles congruent) lead to congruent triangles XNW and ZXNY by SSS congruence, proving XNW ∠≅ ZXNY.
The given statement is "Fill in the missing blanks in the proof: Given: Rhombus WXYZ. Prove: XNW = ZXNY".
Here's how we can prove it step-by-step:
Step 1: Identify the properties of a rhombus.
A rhombus is a special type of parallelogram with the following properties:
All four sides are equal in length.
Opposite angles are congruent.
Diagonals bisect each other perpendicularly, creating four right triangles.
Step 2: Analyze the givens.
We are given that WXYZ is a rhombus. This means we can apply the properties of rhombi mentioned above to solve for the missing elements.
Step 3: Break down the statement.
We need to prove that XNW is congruent to ZXNY. This means we need to show that these two triangles have the same corresponding side lengths and angles.
Step 4: Fill in the blanks.
Now, let's look at the missing steps in the proof:
Blank 2:We are given that XY ≅ WZ and XW ≅ YZ. This is because opposite sides of a rhombus are congruent (property 1).
Blank 3: Since XN is half of XZ and ZN is half of ZY, and we know XZ ≅ YZ (from property 1), we can conclude that XN ≅ ZN and WN ≅ YN. This is because corresponding segments of congruent triangles are proportional when cut by a transversal.
Blank 4: We can now use the Side-Side-Side (SSS) congruence rule. We have XN ≅ ZN (from step 3) and WN ≅ YN (from step 3), and NW ≅ NY (because opposite sides of a rhombus are congruent from property 1). Therefore, triangle XNW ≅ triangle ZXNY by SSS congruence.
Blank 5: Finally, since we have proven that triangle XNW ≅ triangle ZXNY in step 4, we can conclude that XNW ∠≅ ZXNY.
By filling in the missing steps with the justifications given above, we have successfully proven that XNW ≅ ZXNY. This completes the proof for the given statement.
In summary, the key steps to solving this proof involve understanding the properties of rhombi, analyzing the givens, breaking down the statement into smaller parts, and applying appropriate congruence rules.