Worksheet: Special Quadrilaterals

In this worksheet, we will practice using the properties of special quadrilaterals to solve problems involving algebraic expressions, equations, and coordinates.

Q1:

Given that 𝐢𝑀=16cm, determine the length of 𝐴𝐢.

Q2:

Find the value of 𝑧 in the following parallelogram.

Q3:

In parallelogram 𝐴𝐡𝐢𝐷, 𝐡𝐢=89, 𝑀𝐡=46, and 𝑀𝐢=78. What is the perimeter of △𝐴𝑀𝐷?

Q4:

What can you say about the diagonals of a parallelogram?

  • AThey are equal in length.
  • BThey bisect each other.
  • CThey are perpendicular.

Q5:

Is any quadrilateral whose diagonals bisect each other a parallelogram?

  • Ayes
  • Bno

Q6:

Are the two diagonals of a parallelogram perpendicular?

  • Ano
  • Byes

Q7:

If 𝑄𝑅𝑆𝑇 is a parallelogram, find the value of 𝑧.

Q8:

The given figure shows a parallelogram 𝐴𝐡𝐢𝐷.

Using what you know about alternate angles, determine which angle will have the same measure as ∠𝐴𝐡𝐷.

  • A∠𝐡𝐷𝐢
  • B∠𝐡𝐴𝐷
  • C∠𝐡𝐢𝐷
  • D∠𝐢𝐡𝐴
  • E∠𝐡𝐷𝐴

Which angle will be equal in measure to ∠𝐴𝐷𝐡?

  • A∠𝐡𝐷𝐢
  • B∠𝐡𝐴𝐷
  • C∠𝐢𝐡𝐷
  • D∠𝐢𝐡𝐴
  • E∠𝐴𝐷𝐢

𝐡𝐷 is a common side to triangle 𝐴𝐡𝐷 and triangle 𝐢𝐷𝐡. Using the information from the earlier parts of the question, can we prove that triangles 𝐴𝐡𝐷 and 𝐢𝐷𝐡 are congruent? If yes, by which congruence criteria?

  • AYes, by ASA
  • BNo
  • CYes, by SAS
  • DYes, by SSS
  • EYes, by HL

What will be true of 𝐴𝐡 and 𝐢𝐷 and of 𝐡𝐢 and 𝐴𝐷?

  • A𝐴𝐡=𝐢𝐷, 𝐴𝐷≠𝐡𝐢
  • B𝐴𝐡≠𝐢𝐷, 𝐴𝐷=𝐡𝐢
  • C𝐴𝐡=𝐢𝐷, 𝐴𝐷=𝐡𝐢
  • D𝐴𝐡≠𝐢𝐷, 𝐴𝐷≠𝐡𝐢

What will be true of angles ∠𝐡𝐴𝐷 and ∠𝐡𝐢𝐷 and of angles ∠𝐴𝐡𝐢 and ∠𝐴𝐷𝐢?

  • Aπ‘šβˆ π΅π΄π·=π‘šβˆ π΅πΆπ·, π‘šβˆ π΄π΅πΆ=π‘šβˆ π΄π·πΆ
  • Bπ‘šβˆ π΅π΄π·β‰ π‘šβˆ π΅πΆπ·, π‘šβˆ π΄π΅πΆ=π‘šβˆ π΄π·πΆ
  • Cπ‘šβˆ π΅π΄π·β‰ π‘šβˆ π΅πΆπ·, π‘šβˆ π΄π΅πΆβ‰ π‘šβˆ π΄π·πΆ
  • Dπ‘šβˆ π΅π΄π·=π‘šβˆ π΅πΆπ·, π‘šβˆ π΄π΅πΆβ‰ π‘šβˆ π΄π·πΆ

Q9:

Given that 𝐴𝐡𝐢𝐷 is a parallelogram and 𝐢𝑀=8.6cm, find the perimeter of △𝐴𝐡𝐢.

Q10:

In a parallelogram 𝐴𝐡𝐢𝐷, the coordinates of 𝐴, 𝐡, and 𝐢 are (9,0), (11,0), and (βˆ’4,9), respectively. Find the coordinates of the point at which the two diagonals intersect, and then determine the coordinates of point 𝐷.

  • A(13,βˆ’9), (6,9)
  • B(5,9), (6,βˆ’9)
  • Cο€Ό132,βˆ’92, (βˆ’6,βˆ’9)
  • Dο€Ό52,92, (βˆ’6,9)

Q11:

𝐴𝐡𝐢𝐷 is a parallelogram where the coordinates of 𝐴 are (7,7) and the coordinates of 𝐢 are (βˆ’1,5). Find the coordinates of the point of intersection of the two diagonals of 𝐴𝐡𝐢𝐷.

  • A(7,7)
  • B(3,6)
  • C(4,6)
  • D(βˆ’1,5)

Q12:

In parallelogram 𝐴𝐡𝐢𝐷, the coordinates of 𝐴 are (1,3) and those of the intersection of the diagonals are (4,βˆ’3). What are the coordinates of point 𝐢?

  • Aο€Ό12,2
  • Bο€Ό52,0
  • C(βˆ’9,7)
  • D(3,βˆ’6)
  • E(7,βˆ’9)

Q13:

Find the value of 𝑧 in the following parallelogram.

Q14:

In parallelogram 𝐴𝐡𝐢𝐷, 𝐡𝐢=47, 𝑀𝐡=43, and 𝑀𝐢=33. What is the perimeter of △𝐴𝑀𝐷?

Q15:

If 𝑄𝑅𝑆𝑇 is a parallelogram, find the value of 𝑧.

Q16:

𝐴𝐡𝐢𝐷 is a parallelogram where the coordinates of 𝐴 are (7,5) and the coordinates of 𝐢 are (βˆ’1,3). Find the coordinates of the point of intersection of the two diagonals of 𝐴𝐡𝐢𝐷.

  • A(7,5)
  • B(3,4)
  • C(4,4)
  • D(βˆ’1,3)

Q17:

Given that 𝐴𝐡𝐢𝐷 is a parallelogram and 𝐢𝑀=5.8cm, find the perimeter of △𝐴𝐡𝐢.

Q18:

In a parallelogram 𝐴𝐡𝐢𝐷, the coordinates of 𝐴, 𝐡, and 𝐢 are (βˆ’8,5), (6,5), and (13,8), respectively. Find the coordinates of the point at which the two diagonals intersect, and then determine the coordinates of point 𝐷.

  • A(βˆ’21,βˆ’3), (1,8)
  • B(5,13), (1,βˆ’8)
  • Cο€Όβˆ’212,βˆ’32, (βˆ’1,βˆ’8)
  • Dο€Ό52,132, (βˆ’1,8)

Q19:

The perimeter of the rectangle is the same as the perimeter of the square.

What is the area of the square?

Q20:

In the figure below, given that 𝐴𝐡𝐢𝐷 is a rectangle, 𝑂𝐡𝐢𝐻 is a parallelogram, and 𝐴𝑂=3.7cm, find the length of 𝐻𝐴.

Q21:

If 𝐹𝐽=6π‘₯+8𝑦, 𝐹𝑀=3π‘₯+5𝑦, 𝐺𝐻=42, and 𝐺𝑀=24, what values of π‘₯ and 𝑦 make parellelogram 𝐹𝐺𝐻𝐽 a rectangle?

  • Aπ‘₯=42, 𝑦=3
  • Bπ‘₯=42, 𝑦=24
  • Cπ‘₯=42, 𝑦=48
  • Dπ‘₯=3, 𝑦=3
  • Eπ‘₯=3, 𝑦=24

Q22:

Given that 𝐴𝐡𝐢𝐷 is a rectangle, where 𝐸𝐢=6π‘₯βˆ’7 and 𝐴𝐸=2π‘₯+5, find 𝐷𝐸.

Q23:

In this rectangle, 𝑋𝑍=7𝑐, π‘π‘Œ=21, and π‘‹π‘Œ=28. Find the value of 𝑐.

Q24:

The lengths of two adjacent sides of a parallelogram are 8𝑑 units and 8𝑐 units. Write an expression for its perimeter.

  • A2(8𝑑+8𝑐) units
  • B(8𝑑+16𝑐) units
  • C(8𝑑+8𝑐) units
  • D(16𝑑+8𝑐) units
  • E2(𝑑+𝑐) units

Q25:

𝐴𝐡𝐢𝐷 is a parallelogram. Given that the perimeter of 𝐴𝐡𝐢𝐷 is 54, find the length of 𝐴𝐡.

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