Quadrilaterals

More About Quadrilaterals

A 4-gon Conclusion

Lesson 16-1: Proving a Quadrilateral Is a Parallelogram
Learning Targets:
  • Develop criteria for showing that a quadrilateral is a parallelogram.
  • Prove that a quadrilateral is a parallelogram.
Suggested Learning Strategies:
  • Think-Pair-Share
  • Group Presentation
  • Discussion Groups
  • Visualization
Quadrilateral CHIA:
  1. Steps: Verify if CHIA is a Parallelogram
    • a. Find the slope of each side.
    • b. Use the slopes to explain how you know quadrilateral CHIA is a parallelogram.
Quadrilateral SKIP Properties:
  1. Properties Given: SK = IP and KI = SP
    • a. ∠PSI ≅ _. Explain.
    • b. ∠SIP ≅ ___ and ∠PSI ≅ ___. Explain.
    • c. SK || IP and KI || SP because .
    • d. Theorem Completion: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a ___.
Quadrilateral WALK:
  1. Coordinates Given: W(8, 7), A(11, 3), L(4, 1), K(1, 5)
    • Use the theorem from Item 2 to show that quadrilateral WALK is a parallelogram.
Slope Formula:
  • Given A(x1, y1) and B(x2, y2)
  • Slope of AB: m = (y2 - y1) / (x2 - x1)
Proof Justification:
  • Math Tip: Once a theorem has been proven, it can be used to justify other steps or statements in proofs.
Quadrilateral WXYZ:
  1. Properties Given: WX || ZY and WX ≅ ZY
    • a. ∠WZX ≅ ____. Explain.
    • b. Construct viable arguments. Explain why WZ || XY.
    • c. Theorem Completion: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a .
Quadrilateral GOLD:
  1. Coordinates Given: G(−1, 0), O(5, 4), L(9, 2), D(3, −2)
    • Use the theorem from Item 4 to show that quadrilateral GOLD is a parallelogram.
Theorem on Angles:
  • A Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
    • Given: Quadrilateral POLY with ∠P ≅ ∠L and ∠O ≅ ∠Y
    • Prove: Quadrilateral POLY is a parallelogram.
    • Statements and Reasons:
    1. Quadrilateral POLY with ∠P ≅ ∠L and ∠O ≅ ∠Y (Given)
    2. m∠P = m∠L and m∠O = m∠Y (Definition of congruent angles)
    3. m∠P + m∠O + m∠L + m∠Y = 360° (Sum of interior angles)
    4. m∠P + m∠O + m∠P + m∠O = 360° (Substitution property)
    5. 2m∠P + 2m∠O = 360° (Simplify)
    6. m∠P + m∠O = 180° (Division property of equality)
    7. m∠P + m∠Y + m∠P + m∠Y = 360° (Substitution property)
    8. 2m∠P + 2m∠Y = 360° (Simplify)
Quadrilateral PLAN:
  1. Properties Given: Diagonals PA and LN bisect each other.
    • a. ∠LEP ≅ and ∠LEA ≅ . Explain.
    • b. ∠ALE ≅ and ∠ELP ≅ . Explain.
    • c. Explain how information in part b can be used to prove that quadrilateral PLAN is a parallelogram.
    • d. Theorem Completion: If the diagonals of a quadrilateral , then the quadrilateral is a .
Quadrilateral THIN:
  1. Coordinates Given: T(3, 3), H(5, 9), I(6, 5), N(4, −1)
    • a. Find the coordinates for the midpoint of each diagonal.
    • b. Do the diagonals bisect each other? Explain.
    • c. The best name for this quadrilateral is:
      • A. quadrilateral
      • B. kite
      • C. trapezoid
      • D. parallelogram
Proof on Quadrilateral TRUC:
  • Try These: Write a proof using the theorem from Example 1 as the last reason. Given: RT ≅ RK, ∠RKT ≅ ∠U, ∠1 ≅ ∠2
  • Prove: Quadrilateral TRUC.
    • Statements and Reasons:
    1. m∠P + m∠Y = 180° (Division property of equality)
    2. PY || OL and PO || YL (If two lines are intersected by a transversal and a pair of consecutive interior angles are supplementary, then the lines are parallel)
    3. Quadrilateral POLY is a parallelogram (Definition of a parallelogram).
Lesson 16-1 Practice:
Quadrilateral RSTU:

  1. Given: R(0, 0), S(−2, 2), T(6, 6), and U(8, 4).

    • a. Show that quadrilateral RSTU is a parallelogram by finding the slope of each side.
    • b. Show that quadrilateral RSTU is a parallelogram by finding the length of each side.
    • c. Show that quadrilateral RSTU is a parallelogram by showing that the diagonals bisect each other.

  2. Write a proof using the theorem in Item 2 of Lesson 16-1 as the last reason:

    • Given: Quadrilateral ABC ≅ Quadrilateral FED, CD ≅ CG, CG ≅ AF
    • Prove: Quadrilateral ACDF.

  3. Write a proof using the theorem in Item 4 of Lesson 16-1 as the last reason:

    • Given: Quadrilateral JKLM, X is the midpoint of JK, Y is the midpoint of ML.
    • Prove: Quadrilateral JXLY.

  4. Identify the insufficient condition to prove a quadrilateral is a parallelogram:

    • A. The diagonals bisect each other.
    • B. One pair of opposite sides are parallel.
    • C. Both pairs of opposite sides are congruent.
    • D. Both pairs of opposite angles are congruent.
Further Practice and Connections:
  • Analyze vertices given for quadrilaterals in practice problems to identify if conditions fulfill that of a parallelogram, rectangle, rhombus, or square.
Lesson 16-2: Proving a Quadrilateral Is a Rectangle
Learning Targets:
  • Develop criteria for showing that a quadrilateral is a rectangle.
  • Prove that a quadrilateral is a rectangle.
Definition of a Rectangle:
  1. A rectangle is a parallelogram with .
Theorem Regarding Angles in a Parallelogram:
  1. a. Complete the theorem: If a parallelogram has one right angle, then it has right angles, and it is a .
    b. Explain why the theorem is true based on the properties of a parallelogram.
Given Quadrilateral WXYZ Properties:
  1. a. If quadrilateral WXYZ is equiangular, find the measure of each angle.
    b. Complete the theorem: If a quadrilateral is equiangular, then it is a .
Hypothesis and Conclusion in Theorem:
  1. Identify the hypothesis and conclusion of the theorem.
Congruent Diagonals in Quadrilateral OKAY:
  1. Given Quadrilateral OKAY with congruent diagonals, OA and KY:
    • a. List the three triangles congruent to triangle OYA, providing reasons.
    • b. List corresponding angles congruent to ∠OYA and their measures.
    • c. Theorem Completion: If the diagonals of a parallelogram are , then the parallelogram is a _.
Further Applications in Lessons:
  • Use coordinates and proofs to ascertain properties of quadrilaterals including squares, rectangles, and rhombuses through various conditions depicted in examples and proofs throughout the activity.
Lesson 16-4: Proving a Quadrilateral Is a Square
Learning Targets:
  • Develop criteria for showing that a quadrilateral is a square.
  • Prove that a quadrilateral is a square.
Additional Requirements for a Square:
  1. Given quadrilateral JKLM:
    • a. What information is needed to prove that JKLM is a square?
    • b. Additional information required to confirm JKLM as a square.
    • c. Information to prove that rectangle JKLM is a square.
    • d. Information to prove that rhombus JKLM is a square.
Checking Understanding:
  • Discuss counterexamples of quadrilaterals that might mislead students in identifying squares or rectangles based solely on measures of angles or lengths of sides without confirming additional required properties.
Conclusion:
  • The document highlights definitions, theorems, and strategies for classifying and proving the characteristics of various quadrilaterals effectively.