MODULE 2

• Discipline


  Good taste<|>Excellence
• Hello Grade 9! In this module, you will learn how to prove theorems on the different kinds of parallelogram (rectangle, rhombus and square)
• You can say that you have understood the lesson in this module if you can already: Prove theorems on the different kinds of parallelogram (rectangle, rhombus and square)
• Prove theorem on different kinds of parallelogram (rectangle, rhombus, square)

  1. Refer to the given figure at the right and answer the following.

  2. Given: MATH is a parallelogram

  3. 𝑀𝐴

  ̅̅̅̅̅ ≅ ___________

  4. ∆𝑀𝐴𝐻 ≅ _________

  5. 𝑀𝑆

  ̅̅̅̅ ≅ __________

  6. ∆𝐴𝑇𝐻 ≅ ________

  7. ∠𝐴𝑇𝐻 ≅ _________

  8. If m∠𝑀𝐻𝑇 = 100, 𝑡ℎ𝑒𝑛 𝑚∠𝑀𝐴𝑇 = _________

  9. If m∠𝐴𝑀𝐻 = 100, 𝑡ℎ𝑒𝑛 𝑚∠𝑀𝐻𝑇 = ___________

  10. If 𝐻𝑀

  ̅̅̅̅̅ =7,then 𝐴𝑇

  ̅̅̅̅ =__________

  11. If 𝐴𝑆

  ̅̅̅̅ =3,then 𝐴𝐻

  ̅̅̅̅= _____________

  12. 10. If 𝑀𝑇

  ̅̅̅̅̅ = 9,then 𝑆𝑀

  ̅̅̅̅ = ____________
• Are you familiar with the different places in our city? Have you visited some of them? What have you noticed about the shapes of the windows, doors, ceilings, and tiles of the floor? Can you imagine if we do not have parallelograms around us? This lesson will discuss the special kinds of parallelograms. It will also show you the proof of the different properties of the different kinds of special parallelograms.
• Prove theorems on rectangle

  1. Given: WINS is a parallelogram with ∠𝑊 is a right angle.

  2. Prove: ∠𝐼, ∠𝑁𝑎𝑛𝑑 ∠𝑆 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠
• Rectangle is a parallelogram with four right angles.
• Prove: The diagonals of a rectangle are congruent.


  1. Given: WINS is a rectangle with diagonals 𝑊𝑁

  ̅̅̅̅̅ and 𝑆𝐼̅

  2. Prove: 𝑊𝑁

  ̅̅̅̅̅ ≅ 𝑆𝐼̅
• I'm totally confused
• I'm ready to move forward
• Proceed with caution (I could use some clarification on _______)
• Proceed with caution (I could use some clarification on)
• What's More

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• What's More

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• What I Have Learned
• Refer to the given figure to answer the following problem.
• Given: Rectangle DESK

  1. KO = 5x + 2 and OE = 3x + 20, Find DS

  2. DS = 4 (x+1) and KE = 3(x+4), Find KO

  3. If m∠DEK = 2x - 7 and m∠KES = 2x + 5, find m∠DEK
• Direction: Read each item carefully and choose the letter of the correct answer from the given choices

  • Given rectangle RSTU, RT ≅ ?
  • The diagonals of a rectangle have lengths (2x + 25) and (5x -11), Find the lengths of the diagonals
  • In rectangle KAYE, YO =18 cm, Find the length of diagonal AE
  • What condition will make parallelogram WXYZ a rectangle?
  • LMNP is a rectangle, LN = x and MP = 2x-4, Find the value of x
• What I Can Do
• Assessment


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• Assessment


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• Additional Activities
• Do as directed


  Parallelogram FARM is a rectangle with diagonals FR = x/3 + 1 and AM = (x-2)/2 + (2x+3)/6, Find FR
• Tell whether the following statements are true or false

  • A rectangle is a parallelogram
  • A rhombus is a parallelogram
  • A rhombus has four congruent sides
  • The diagonals of a rhombus bisect each other
  • The diagonals of a rhombus intersect and meet at their midpoint
• Rhombus can be found in a variety of things around us, such as a kite, windows of a car, rhombus-shaped earring, the structure of a building, mirrors, and even a section of the baseball field.
• The term "rhombus" has been derived from the Greek word rhombos which means something that spins, and this term was eventually derived from the Greek verb rhembo which means to turn round and round.
• Have you seen a shape of a rhombus in the picture?
• Theorem 3. The diagonals of a rhombus are perpendicular.
• Given: Rhombus ROSE


  Prove: RS ⊥ OE
• The diagonals of a rhombus are perpendicular.
• Theorem 4. Each diagonal of a rhombus bisects opposite angles.
• Given: Rhombus VWXY


  Prove: ∠1 ≅ ∠2, ∠3 ≅ ∠4
• MARK is a rhombus, m∠AKR = 360, Find the following:

  1. m∠MKA

  2. m∠MAR

  3. m∠KMA
• Refer to the given figure to solve the following problem.
• 2. Given that RSTV is a rhombus and m∠3 = 55. Find the measure of the following angles:

  1. ∠1

  2. ∠2

  3. ∠TV R

  4. ∠SPT

  5. ∠SRP
• Solve the following:

  If PQRS is a rhombus with m∠PQS = 3x +10 and m∠SQR = x
• Parallelogram

  Supplementary angles
• 720 + m∠KMA = 1800

  1. Substitution

  2. (-720) + m∠KMA = 1800 + (-720)

  3. m∠KMA = 1080