Question

dont forget!
rectangles, rhombuses, and squares are all special forms of parallelograms.
so, they have the properties of a parallelogram.
therefore, adding the conditions below (1·2) to a parallelogram makes it either a rectangle, or a rhombus, or a square.
1 the figure below shows what kind of conditions can be added to a parallelogram that will make it a rectangle or a rhombus. choose the conditions that apply from the box below and write a symbol from the following (a), (b), (c), or (d), in the blanks.
2 the figure below shows what kind of conditions can be added to a rectangle and a rhombus that will make it a square. choose the conditions that apply from the box below and write a symbol from the following (a), (b), (c), or (d), in the blanks.
(a) ∠a = 90° (b) ac ⊥ bd (c) ac = bd (d) ab = bc

Explanation:

Question 1

To determine the conditions to make a parallelogram a rectangle or a rhombus:

• For Rectangle (from Parallelogram):
• A parallelogram with one right angle (\(\angle A = 90^\circ\)) is a rectangle (condition (a)).
• A parallelogram with equal diagonals (\(AC = BD\)) is a rectangle (condition (c)).
• For Rhombus (from Parallelogram):
• A parallelogram with perpendicular diagonals (\(AC \perp BD\)) is a rhombus (condition (b)).
• A parallelogram with equal adjacent sides (\(AB = BC\)) is a rhombus (condition (d)).

So, for the blanks:

• (1) and (2) (leading to rectangle) can be (a) and (c) (order can vary, but common: (1) (a), (2) (c) or vice versa).
• (3) and (4) (leading to rhombus) can be (b) and (d) (order can vary, but common: (3) (b), (4) (d) or vice versa).

Question 2

To determine the conditions to make a rectangle or a rhombus a square:

• For Rectangle to Square:
• A rectangle with equal adjacent sides (\(AB = BC\)) is a square (condition (d)).
• A rectangle with perpendicular diagonals (\(AC \perp BD\)) is a square (condition (b)).
• For Rhombus to Square:
• A rhombus with one right angle (\(\angle A = 90^\circ\)) is a square (condition (a)).
• A rhombus with equal diagonals (\(AC = BD\)) is a square (condition (c)).

So, for the blanks:

• From Rectangle (top path): (1) and (2) can be (d) and (b) (order can vary).
• From Rhombus (bottom path): (3) and (4) can be (a) and (c) (order can vary).

Final Answers (Sample for Question 1)

• (1): (a) \(\angle A = 90^\circ\)
• (2): (c) \(AC = BD\)
• (3): (b) \(AC \perp BD\)
• (4): (d) \(AB = BC\)

(Sample for Question 2)

• (1): (d) \(AB = BC\)
• (2): (b) \(AC \perp BD\)
• (3): (a) \(\angle A = 90^\circ\)
• (4): (c) \(AC = BD\)

(Note: Orders can vary based on diagram interpretation, but the conditions are as above.)

Answer:

Question 1

To determine the conditions to make a parallelogram a rectangle or a rhombus:

• For Rectangle (from Parallelogram):
• A parallelogram with one right angle (\(\angle A = 90^\circ\)) is a rectangle (condition (a)).
• A parallelogram with equal diagonals (\(AC = BD\)) is a rectangle (condition (c)).
• For Rhombus (from Parallelogram):
• A parallelogram with perpendicular diagonals (\(AC \perp BD\)) is a rhombus (condition (b)).
• A parallelogram with equal adjacent sides (\(AB = BC\)) is a rhombus (condition (d)).

So, for the blanks:

• (1) and (2) (leading to rectangle) can be (a) and (c) (order can vary, but common: (1) (a), (2) (c) or vice versa).
• (3) and (4) (leading to rhombus) can be (b) and (d) (order can vary, but common: (3) (b), (4) (d) or vice versa).

Question 2

To determine the conditions to make a rectangle or a rhombus a square:

• For Rectangle to Square:
• A rectangle with equal adjacent sides (\(AB = BC\)) is a square (condition (d)).
• A rectangle with perpendicular diagonals (\(AC \perp BD\)) is a square (condition (b)).
• For Rhombus to Square:
• A rhombus with one right angle (\(\angle A = 90^\circ\)) is a square (condition (a)).
• A rhombus with equal diagonals (\(AC = BD\)) is a square (condition (c)).

So, for the blanks:

• From Rectangle (top path): (1) and (2) can be (d) and (b) (order can vary).
• From Rhombus (bottom path): (3) and (4) can be (a) and (c) (order can vary).

Final Answers (Sample for Question 1)

• (1): (a) \(\angle A = 90^\circ\)
• (2): (c) \(AC = BD\)
• (3): (b) \(AC \perp BD\)
• (4): (d) \(AB = BC\)

(Sample for Question 2)

• (1): (d) \(AB = BC\)
• (2): (b) \(AC \perp BD\)
• (3): (a) \(\angle A = 90^\circ\)
• (4): (c) \(AC = BD\)

(Note: Orders can vary based on diagram interpretation, but the conditions are as above.)