Lesson Worksheet: Special Quadrilaterals Mathematics

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

Q1:

Given that 𝐶𝑀=16cm, determine the length of 𝐴𝐶.

Q2:

Find the value of 𝑧 in the following parallelogram.

Q3:

In parallelogram 𝐴𝐵𝐶𝐷, 𝐵𝐶=89, 𝑀𝐵=46, and 𝑀𝐶=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 𝐴𝐵𝐶𝐷.

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

  • A∠𝐵𝐷𝐶
  • B∠𝐵𝐴𝐷
  • C∠𝐵𝐶𝐷
  • D∠𝐶𝐵𝐴
  • E∠𝐵𝐷𝐴

Which angle will be equal in measure to ∠𝐴𝐷𝐵?

  • A∠𝐵𝐷𝐶
  • B∠𝐵𝐴𝐷
  • C∠𝐶𝐵𝐷
  • D∠𝐶𝐵𝐴
  • E∠𝐴𝐷𝐶

𝐵𝐷 is a common side to triangle 𝐴𝐵𝐷 and triangle 𝐶𝐷𝐵. Using the information from the earlier parts of the question, can we prove that triangles 𝐴𝐵𝐷 and 𝐶𝐷𝐵 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 𝐴𝐵 and 𝐶𝐷 and of 𝐵𝐶 and 𝐴𝐷?

  • A𝐴𝐵=𝐶𝐷, 𝐴𝐷≠𝐵𝐶
  • B𝐴𝐵≠𝐶𝐷, 𝐴𝐷=𝐵𝐶
  • C𝐴𝐵=𝐶𝐷, 𝐴𝐷=𝐵𝐶
  • D𝐴𝐵≠𝐶𝐷, 𝐴𝐷≠𝐵𝐶

What will be true of angles ∠𝐵𝐴𝐷 and ∠𝐵𝐶𝐷 and of angles ∠𝐴𝐵𝐶 and ∠𝐴𝐷𝐶?

  • A𝑚∠𝐵𝐴𝐷=𝑚∠𝐵𝐶𝐷, 𝑚∠𝐴𝐵𝐶=𝑚∠𝐴𝐷𝐶
  • B𝑚∠𝐵𝐴𝐷≠𝑚∠𝐵𝐶𝐷, 𝑚∠𝐴𝐵𝐶=𝑚∠𝐴𝐷𝐶
  • C𝑚∠𝐵𝐴𝐷≠𝑚∠𝐵𝐶𝐷, 𝑚∠𝐴𝐵𝐶≠𝑚∠𝐴𝐷𝐶
  • D𝑚∠𝐵𝐴𝐷=𝑚∠𝐵𝐶𝐷, 𝑚∠𝐴𝐵𝐶≠𝑚∠𝐴𝐷𝐶

Q9:

Given that 𝐴𝐵𝐶𝐷 is a parallelogram and 𝐶𝑀=8.6cm, find the perimeter of △𝐴𝐵𝐶.

Q10:

In a parallelogram 𝐴𝐵𝐶𝐷, the coordinates of 𝐴, 𝐵, and 𝐶 are (9,0), (11,0), and (−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,−9), (6,9)
  • B(5,9), (6,−9)
  • C132,−92, (−6,−9)
  • D52,92, (−6,9)

Q11:

𝐴𝐵𝐶𝐷 is a parallelogram where the coordinates of 𝐴 are (7,7) and the coordinates of 𝐶 are (−1,5). Find the coordinates of the point of intersection of the two diagonals of 𝐴𝐵𝐶𝐷.

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

Q12:

In parallelogram 𝐴𝐵𝐶𝐷, the coordinates of 𝐴 are (1,3) and those of the intersection of the diagonals are (4,−3). What are the coordinates of point 𝐶?

  • A12,2
  • B52,0
  • C(−9,7)
  • D(3,−6)
  • E(7,−9)

Q13:

Find the value of 𝑧 in the following parallelogram.

Q14:

In parallelogram 𝐴𝐵𝐶𝐷, 𝐵𝐶=47, 𝑀𝐵=43, and 𝑀𝐶=33. What is the perimeter of △𝐴𝑀𝐷?

Q15:

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

Q16:

𝐴𝐵𝐶𝐷 is a parallelogram where the coordinates of 𝐴 are (7,5) and the coordinates of 𝐶 are (−1,3). Find the coordinates of the point of intersection of the two diagonals of 𝐴𝐵𝐶𝐷.

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

Q17:

Given that 𝐴𝐵𝐶𝐷 is a parallelogram and 𝐶𝑀=5.8cm, find the perimeter of △𝐴𝐵𝐶.

Q18:

In a parallelogram 𝐴𝐵𝐶𝐷, the coordinates of 𝐴, 𝐵, and 𝐶 are (−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(−21,−3), (1,8)
  • B(5,13), (1,−8)
  • C−212,−32, (−1,−8)
  • D52,132, (−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 𝐴𝐵𝐶𝐷 is a rectangle, 𝑂𝐵𝐶𝐻 is a parallelogram, and 𝐴𝑂=3.7cm, find the length of 𝐻𝐴.

Q21:

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

  • A𝑥=42, 𝑦=3
  • B𝑥=42, 𝑦=24
  • C𝑥=42, 𝑦=48
  • D𝑥=3, 𝑦=3
  • E𝑥=3, 𝑦=24

Q22:

Given that 𝐴𝐵𝐶𝐷 is a rectangle, where 𝐸𝐶=6𝑥−7 and 𝐴𝐸=2𝑥+5, find 𝐷𝐸.

Q23:

In this rectangle, 𝑋𝑍=7𝑐, 𝑍𝑌=21, and 𝑋𝑌=28. Find the value of 𝑐.

Q24:

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

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

Q25:

𝐴𝐵𝐶𝐷 is a parallelogram. Given that the perimeter of 𝐴𝐵𝐶𝐷 is 54, find the length of 𝐴𝐵.

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