1. Chapter 6
Quadrilaterals

2. Types of Polygons Triangle – 3 sides Quadrilateral – 4 sides
Pentagon – 5 sides
Hexagon – 6 sides
Heptagon – 7 sides
Octagon – 8 sides
Nonagon – 9 sides
Decagon – 10 sides
Dodecagon – 12 sides
All other polygons = n-gon

3. Lesson 6.1 : Angles of Polygons
Interior Angle Sum Theorem
The sum of the measures of the interior angles of a polygon is found by S=180(n-2)
Ex: Hexagon
Exterior Angle Sum Theorem
The sum of the measures of the exterior angles of a polygon is 360 no matter how many sides.

4. Lesson 6.1 : Angles of Polygons
Find the measure of an interior and an exterior angle for each polygon.
24-gon
3x-gon
Find the measure of an exterior angle given the number of sides of a polygon
260 sides

5. Lesson 6.1: Angles of Polygons
The measure of an interior angle of a polygon is given. Find the number of sides.
175
168.75
A pentagon has angles (4x+5), (5x-5), (6x+10), (4x+10), and 7x. Find x.

6. A. Find the value of x in the diagram.

7. Lesson 6.2: Parallelograms
Opposite angles in a parallelogram are congruent
Opposite sides of a parallelogram are congruent
Consecutive angles in a parallelogram are supplementary
Properties of Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel
If a parallelogram has 1 right angle, it has 4 right angles.
The diagonals of a parallelogram split it into 2 congruent triangles
The diagonals of a parallelogram bisect each other

8. ____ ? ? ?

9. A. ABCD is a parallelogram. Find AB.
B. ABCD is a parallelogram. Find mC.
C. ABCD is a parallelogram. Find mD.

10. A. If WXYZ is a parallelogram, find the value of r, s and t.

11. What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)?

12. Lesson 6.3 : Tests for Parallelograms
If…
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent
The diagonals bisect each other
One pair of opposite sides is congruent and parallel
Then the quadrilateral is a parallelogram

13. Determine whether the quadrilateral is a parallelogram
Determine whether the quadrilateral is a parallelogram. Justify your answer.

14. Which method would prove the quadrilateral is a parallelogram?

15. Determine whether the quadrilateral is a parallelogram.

16. Determine whether the quadrilateral is a parallelogram.

17. Find x and y so that the quadrilateral is a parallelogram.

18. COORDINATE GEOMETRY Graph quadrilateral QRST with vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.

19. Given quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram.
(The graph does not determine for you)

20. Foldable
Fold the construction paper in half both length and width wise
Unfold the paper and hold width wise
Fold the edges in to meet at the center crease
Cut the creases on the tabs to make 4 flaps

21. Lesson 6.4 : Rectangles Characteristics of a rectangle:
Both sets of opp. Sides are congruent and parallel
Both sets opp. angles are congruent
Diagonals bisect each other
Diagonals split it into 2 congruent triangles
Consecutive angles are supplementary
If one angle is a right angle then all 4 are right angles
In a rectangle the diagonals are congruent.
If diagonals of a parallelogram are congruent, then it is a rectangle.

22. Quadrilateral EFGH is a rectangle
Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

23. Quadrilateral RSTU is a rectangle
Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

24. Quadrilateral EFGH is a rectangle
Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.

25. Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.

26. Lesson 6.5 : Rhombi (special type of parallelogram)
A quadrilateral with 4 congruent sides
Characteristics of a rhombus:
Both sets of opp. sides are congruent and parallel
Both sets of opp. angles are congruent
Diagonals bisect each other
Diagonals split it into 2 congruent triangles
Consecutive angles are supplementary
If an angle is a right angle then all 4 angles are right angles
In a rhombus:
Diagonals are perpendicular
Diagonals bisect the pairs of opposite angles

27. 6.5: Squares (special type of parallelogram)
A quadrilateral with 4 congruent sides
Characteristics of a square:
Both sets of opp. sides are congruent and parallel
Both sets of opp. angles are congruent
Diagonals bisect each other
Diagonals split it into 2 congruent triangles
Consecutive angles are supplementary
If an angle is a right angle then all 4 angles are right angles
Diagonals bisect the pairs of opposite angles
A square is a rhombus and a rectangle.

28. A. The diagonals of rhombus WXYZ intersect at V. If mWZX = 39
A. The diagonals of rhombus WXYZ intersect at V. If mWZX = 39.5, find mZYX.
B. The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

29. A. ABCD is a rhombus. Find mCDB if mABC = 126.
B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

30. QRST is a square. Find n if mTQR = 8n + 8.

31. QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11.

33. Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.

34. 6.6: Trapezoids Diagonals of an isosceles trapezoid are congruent
A quadrilateral with exactly 1 pair of opposite parallel sides (bases), 2 pairs of base angles, and 1 pair of non-parallel sides (legs)
Isosceles Trapezoid:
A trapezoid with congruent legs and congruent base angles
Diagonals of an isosceles trapezoid are congruent
Median (of a trapezoid):
The segment that connects the midpoints of the legs
The median is parallel to the bases A B
AC = BD
base D C
leg
leg
Base angle
Base angle
base
Median = ½ (base + base)

35. A. Each side of the basket shown is an isosceles trapezoid
A. Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6 feet, find mMJK.
B. Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN.

36. In the figure, MN is the midsegment of trapezoid FGJK
In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x.

37. WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25.

38. A. If WXYZ is a kite, find mXYZ.

39. A. If WXYZ is a kite, find mXYZ.