Topic: Characteristics of Quadrilaterals

(Textbook: Chapter 6 Math Nation: Section 9 Topic 1)

Essential Question: How are the various quadrilaterals related to each other?

In each empty square, write the most appropriate description for the quadrilateral.

Identify which quadrilateral(s) meet the following criteria. (Math Nation: Section 9 Topic 3)

DO YOU KNOW YOUR QUADRILATERAL PROPERTIES? (without any help)

  1. State which shape(s) each statement describes, there could be more than one answer.

Choose from parallelogram, rhombus, rectangle, or square.

a.Opposite sides are congruent.

b.All the sides are congruent.

c.Diagonals bisect each other.

d.All angles are 90°.

e.Opposite angles are congruent.

f.Diagonals are perpendicular bisectors of each other.

  1. Always, sometimes, never?

a. A rhombus is a square.b. The diagonals of a rhombus are always congruent

c. A parallelogram is a rectangled. The diagonals of a square are never perpendicular.

e. A square is a rhombus.

Topic:Proving Quadrilaterals

(Textbook:6.2, 6.3, Math Nation: Section 9 Topics Vary) (G-CO.3.11)

Essential Question:How does triangle congruence relate to proving properties about quadrilaterals?

  1. Prove that the opposite sides of parallelogram WXYZare congruent.

2. Prove that the diagonals of parallelogram WXYZ bisect each other.

3. Prove that rectangle WXYZ is a parallelogram.

4. Draw the diagonals of rectangle WXYZ and prove that they are congruent.

5. Prove that opposite angles of parallelogram WXYZ are congruent.

Fill in the blank proofs of quadrilaterals.

Proof Summary:

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Topic: Parallel and Perpendicular Slopes with Quadrilaterals

(Textbook: 3.3, 3.4 Math Nation: Section 1 Topic 7, 8) (G-GPE.2.5)

Essential Question: How does slope prove specific quadrilaterals?

  1. Line a is parallel to line b. Prove that the slope of line a equals the slope of line b.

Note: You may draw axes placing the lines in the coordinate plane if you prefer.

  1. In right trapezoid ABCD, and is contained in the line whose equation is
  1. What is the slope of the line containing? Briefly explain how you got your answer.
  2. Write an equation in slope-intercept form of the line that contains if B is located at (. Show your work to justify your answer.
  1. In rectangle EFGH, and crosses the y-axis at (0, 2). If the equation of the line containing is , write the equation of the line containing in slope-intercept form. Show your work to justify your answer.
  1. Line a is perpendicular to line b. Prove that the slopes of line a and line b are both opposite and reciprocal (or that the product of their slopes is ‒1).

Note: You may draw axes placing the lines in the coordinate plane if you prefer.

  1. In right trapezoid ABCD, and is contained in the line
  1. What is the slope of the line containing? Briefly explain how you got your answer.
  2. Write an equation in slope-intercept form of the line that contains if B is located at (. Show your work to justify your answer.
  1. In rectangle EFGH, and contains the point (0, -4). If the equation of the line containing is , write the equation of the line containing in slope-intercept form. Show your work to justify your answer.

Slopes with Quadrilaterals Summary:

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Topic: Quadrilaterals in Coordinate Geometry

(Textbook: 3.6 Math Nation: Section 9 Topics 1,2,3, Section 10 Topic 6,7) (G-GPE.2.4)

Essential Question: How can I use slope, distance, and midpoint to determine the type of quadrilateral?

  1. A quadrilateral has vertices at A(-3, 2),B(-2, 6),C(2, 7) andD(1, 3). Which, if any, of the following describe quadrilateral ABCD: parallelogram, rhombus, rectangle, square, or trapezoid? Justify your reasoning algebraically.
  1. Three of the vertices of a rectangle have coordinates D(0, 0), A(a, 0), and B(0, b).

a.Find the coordinates of point C, the fourth vertex.

b.Prove that the diagonals of the rectangle are congruent.

  1. Show that the quadrilateral formed by connecting the midpoints of the sides of quadrilateral ABCD (points E, F, G, and H) is a parallelogram.

  1. Quadrilateral ABCD has the following coordinates: . What kind of quadrilateral is ABCD? Prove your answer.
  1. Which of the following conclusions can’t always be drawn using coordinate geometry?
  1. Quinn graphs parallelogram GRIT with the coordinates .

The diagonals meet at point

  1. Consider quadrilateral MATH below. We can prove that MATH is a rectangle by calculating the length of each diagonal. Write the algebraic expression for the length of each diagonal.
  1. Prove that quadrilateral LEAP with vertices is a parallelogram. Which of the following statements help to prove that LEAP is a parallelogram? Select all that apply.
  1. Prove the points represent the vertices of a rhombus. A(-3, 0), B(0, 4), C(4, 1), D(1, -3)
  1. Use coordinate geometry to determine if the quadrilateral is a parallelogram, rectangle, square, rhombus, trapezoid, isosceles trapezoid, or none of these. G(1, 1), E(5, 1), A(4, 8), R(2, 8)

Quadrilaterals in Coordinate Geometry Summary:

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Topic: Quadrilaterals and Algebra

Essential Question: How do the quadrilateral properties assist in solving algebraic equation?

  1. ZACH is an isosceles trapezoid with midsegment . Determine the length of .
  1. MICE is an isosceles trapezoid with midsegment . Determine the length of and .
  1. The length of diagonals of a rectangle are represented by yards and yards. Find the length of each diagonal.
  1. Rectangle JKLMhas diagonals intersecting at P. If , find .
  1. In square ABCD, the diagonals intersect at G. If and . Determine the length of the side of the square.
  1. In square ABCD, the diagonals intersect at G. If and , find values of a and b.
  1. The size of the acute angle of a rhombus is half the size of its obtuse angle. The side length of the rhombus is equal to 20 feet. Find the lengths of the diagonals of the rhombus.
  1. Consider kite WXYZ below.
  2. Determine the lengths of each side of kite WXYZ.
  3. Which diagonal bisects a pair of opposite angles?
  1. Consider kite WXYZ.If and , find If .If and , find
  1. UVWX is a parallelogram.
  1. If XU = 15 and UW = 28, find WZ.
  1. If , find.

W

  1. If and , find x.
  1. EFGH is a rectangle. Find the value of y.

12. Find the following if ZQSC13.Find the value of x. Then find the

is a parallelogram. Find the values measure of all angles.

of .

A B

C

D

14. ABCD is a square. Find values of15. ZXCV is an isosceles trapezoid.

w, t, and Find the values of and

16. QWOP is a kite. Find the value of t. 17. WXYZ is a rectangle. Find x and . Then

Then find the length of . find the length of

18. KITE is a kite.19. ASDF is a square. Find if

Find the length of . and

20.RACH is a parallelogram.21. UVWX is a parallelogram with the

Find the values of b, c, given measurements. Find the following

and .

22. TUWY is a rhombus. Find the value23. Solve for x and y.

of x and .

Algebra and Quadrilaterals Summary:

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