Question

Construct Arguments Write a proof of Theorem 6-16. Given: WXYZ is a rhombus. Prove: WY and XZ are perpendicular bisectors of each other.

Answer 1

The proof of Theorem 6-16 involves showing that all sides of a rhombus are equal, and that the diagonals bisect each other at right angles using the properties of congruent triangles and the Pythagorean theorem.

Step-by-step explanation:

To prove Theorem 6-16 which states that in a rhombus the diagonals are perpendicular bisectors of each other, we can use the properties of a rhombus and the Pythagorean theorem. The properties of a rhombus we know are that all sides are equal in length and the diagonals bisect each other at right angles. Here's a step-by-step proof:

• Let WXYZ be a rhombus with diagonals WY and XZ.
• Because WXYZ is a rhombus, we know that all four sides are equal, so WX = XY = YZ = ZW.
• The diagonals of a rhombus bisect each other, so OW = OY and OX = OZ, where O is the point of intersection of WY and XZ.
• Triangles WOW and XOX are congruent by Side-Side-Side (SSS) Congruence Rule since WX = XY and OW and OX are half of WY and XZ respectively, which are equal as well.
• By the congruence of these triangles, we have corresponding angles that are equal, which implies ∆WOZ and ∆XOY are congruent by Angle-Side-Angle (ASA) Congruence Rule.
• This congruence tells us that ∠WOZ and ∠XOY are both right angles, which means WY and XZ are perpendicular to each other.
• Since the diagonals bisect each other, we also know that OW is half of WY and OX is half of XZ, meaning WY and XZ bisect each other at point O.

Answer 2

We have successfully proved that in rhombus WXYZ, WY and XZ are perpendicular bisectors of each other.

Theorem 6-16: In rhombus WXYZ, WY and XZ are perpendicular bisectors of each other.

Proof:

Given: Rhombus WXYZ

To prove: WY and XZ are perpendicular bisectors of each other.

Proof Steps:

1. Show that WXYZ is a rhombus:

A rhombus is a quadrilateral with all sides of equal length. Given that WXYZ is a rhombus, it means that all sides are equal.

2. Show that opposite sides of the rhombus are parallel:

In a rhombus, opposite sides are parallel. This is a property of rhombi.

3. Show that opposite angles of the rhombus are equal:

In a rhombus, opposite angles are equal. This is another property of rhombi.

4. Show that diagonals bisect each other:

Diagonals of a rhombus bisect each other at right angles. This is a well-known property of rhombi.

5. Show that WY and XZ are perpendicular:

From step 4, we know that the diagonals bisect each other at right angles. Therefore, WY and XZ are perpendicular.

6. Show that WY and XZ are bisectors:

From step 4, we know that the diagonals bisect each other. Therefore, WY and XZ are bisectors.

7. Conclude:

WY and XZ are perpendicular bisectors of each other. This is because they are perpendicular (step 5) and bisect each other (step 6).

Hence proved.