1. Chapter 8: Quadrilaterals
Section: 8.1 – Find Angle Measures in Polygons Aim: To find angle measures in polygons. Date: 2/17/12

2. What is a polygon?
It is a closed plane figure that is formed by three or more sides.
Convex Polygons:
If you continue a line from each side of the polygon and it does not contain a point inside the polygon .
Concave Polygons:
If you continue a line from each side of the polygon and it does contain a point inside the polygon .

3. Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a convex n-gon is (n - 2) • 180°. 1 2 6 3 5 4

4. An octagon has 8 sides. Use the Polygon Interior Angles Theorem.
Find the sum of the measures of the interior angles of a convex octagon.
SOLUTION
An octagon has 8 sides. Use the Polygon Interior Angles Theorem.
(n – 2) 180° =
(8 – 2) 180°
Substitute 8 for n.
= °
Subtract.
= 1080°
Multiply.

5. EXAMPLE 2
Example 2
The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides.
SOLUTION
(n –2) 180° =
900°
Polygon Interior Angles Theorem
n –2 = 5
Divide each side by 180°.
n = 7
Add 2 to each side.

6. What is a Quadrilateral?
A four sided figure
With four angles that all add up to 360°
There are six most well known types of quadrilaterals:
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Kites

7. Find the value of x in the diagram shown.
EXAMPLE 3
Find the value of x in the diagram shown.
The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve.
x° + 108° + 121° + 59° =
360°
Corollary to Theorem 8.1
x =
360
Combine like terms.
x = 72
Subtract 288 from each side.

8. Use the diagram at the right. Find m S and m T.
103°, 103°
The measures of three of the interior angles of a quadrilateral are 89°, 110°, and 46°. Find the measure of the fourth interior angle.
115°

9. Polygon Exterior Angles Theorem
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°

10. Use the Polygon Exterior Angles Theorem to write and
SOLUTION
Use the Polygon Exterior Angles Theorem to write and
solve an equation.
x° + 2x° + 89° + 67° =
360°
Polygon Exterior Angles Theorem
3x =
360
Combine like terms.
x = 68
Solve for x.

11. Chapter 8: Quadrilaterals
Section: 8.2 – Use Properties of Parallelograms Aim: To find angle and side measures in a parallelogram. Date: 2/27

12. What is a Parallelogram?
A quadrilateral
Opposite sides are both parallel and equal
Theorem 8.3: if a quadrilateral is a parallelogram, then its opposite sides are congruent.
Opposite angles are equal
Theorem 8.4: if a quadrilateral is a parallelogram, then its opposite angles are congruent.

13. Find the values of x and y.
72° x°
ABCD is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x. 36
AB = CD
Opposite sides of a are .
y - 8 = 36
Substitute y - 8 for AB and 36 for CD.
x = 44
Add 8 from each side.
By Theorem 8.3, A C,
or m A = m C.
So, x ° = 72°.
In ABCD, x = 72 and y = 44.

14. GUIDED PRACTICE
Solve on your own:
Find FG and m G. 1.
8, 60°
Find the values of x and y. 2.
25, 15

15. Theorem 8.5
If quadrilateral is a parallelogram, then its consecutive angles are supplementary. y° x° x° y°

16. GUIDED PRACTICE
Find the indicated measure in JKLM. NM 3.
m JML 5.
ANSWER 2
ANSWER
70° KM 4.
m KML 6.
ANSWER 4
ANSWER
40°

17. Chapter 8: Quadrilaterals
Section: 8.3 – Show that a Quadrilateral is a Parallelogram Aim: To use properties to identify a parallelogram. Date: 3/8/11

18. Theorems
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

19. Theorems
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

20. Examples
For what value of x is quadrilateral CDEF a parallelogram?

21. Quadrilateral CDEF is a parallelogram when x = 4.
FN = DN
Set the segment lengths equal.
5x – 8 = 3x
Substitute 5x –8 for FN and 3x for DN.
2x – 8 =
Subtract 3x from each side.
2x = 8
Add 8 to each side.
x = 4
Divide each side by 2.
When x = 4,
FN = 5(4) –8 =
12 and DN =
3(4) =
12.
Quadrilateral CDEF is a parallelogram when x = 4.

22. GUIDED PRACTICE
For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning.
2; The diagonals of a parallelogram bisect
each other so solve 2x = 10 – 3x for x.

23. Chapter 8: Quadrilaterals
Section: 8.4 – Properties of Rhombuses, Rectangles, and Squares. Aim: To use properties of rhombuses, rectangles, and squares. Date: 3/9/11

24. What is a Rhombus? Has four congruent sides Sometimes called a diamond
Diagonals are perpendicular
Diagonals bisect a pair of opposite angles.

25. What is a Rectangle? Rectangles are special parallelograms
Two sets of parallel sides
Opposite sides are parallel and equal
All angles are equal = 90°.
Diagonals are congruent.

26. What is a Square? A square is a special rectangle All angles are equal
Opposite sides are parallel
All angles are equal
Each angle is 90°
All sides are equal

27. EXAMPLE 1
For any rhombus QRST, decide whether the statement is always or sometimes true. a. Q S
By definition, a rhombus is a parallelogram with four congruent sides. By Theorem 8.4, opposite angles of a parallelogram are congruent. So, The statement is always true. Q S

28. EXAMPLE 1
For any rhombus QRST, decide whether the statement is always or sometimes true. Q R b.
If rhombus QRST is a square, then all four angles are congruent right angles. So, if QRST is a square. Because not all rhombuses are also squares, the statement is sometimes true. Q R

29. EXAMPLE 2
Classify the special quadrilateral.
The quadrilateral has four congruent sides. One of the angles is not a right angle, so the rhombus is not also a square.
The quadrilateral is a rhombus.

30. 1. For any rectangle EFGH, is it always or sometimes true that Explain your reasoning.
Sometimes; this is only true if EFGH is a square.

31. A quadrilateral has four congruent sides and four congruent angles.
Sketch the quadrilateral and classify it.
Square

32. Chapter 8: Quadrilaterals
Section: 8.5 – Properties of Trapezoids and Kites. Aim: To use properties of trapezoids. Date: 3/11/11

33. Special Trapezoid: Isosceles Trapezoid
What is a Trapezoid?
One set of parallel sides (which are the bases).
Special Trapezoid: Isosceles Trapezoid
Each pair of base angles are congruent.
Congruent Diagonals.
Can also tell that it’s an Iso. Trap. If there is only one pair of congruent base angles.

34. Midsegments Segment that connects the midpoints of its legs.
Theorem: parallel to each base and its length is one half the sum of the lengths of the bases.

35. In the diagram, MN is the midsegment of trapezoid PQRS. Find MN.
Use Theorem to find MN.
MN (PQ + SR) 1 2 =
Apply Theorem
= ( ) 1 2
Substitute 12 for PQ and 28 for XU.
= 20
Simplify.
The length MN is 20 inches.

36. In 1 and 2, use the diagram of trapezoid EFGH.
1. If EG = FH, is trapezoid EFGH isosceles? Explain.
Yes; if the diagonals are congruent then the trapezoid is isosceles.

37. 2. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles
2. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles? Explain.
Yes;
m EFG = 70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoid by having one pair of base angles congruent.

38. 15 cm; Solve for x to find ML.
GUIDED PRACTICE
3. In trapezoid JKLM, J and M are right angles, and JK = 9 cm. The length of the midsegment NP of trapezoid JKLM is 12 cm. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning. J L K M
9 cm
12 cm N P
( 9 + x ) = 12 1 2
15 cm; Solve for x to find ML.

39. Chapter 8: Quadrilaterals
Section: 8.5 – Properties of Trapezoids and Kites. Aim: To use properties of kites. Date: 3/14/11 PIE DAY!

40. What is a Kite?
Has two pairs of consecutive congruent sides, but opposite sides are not congruent.
Exactly one pair of opposite angles are congruent. The congruent opposite angles are always between the non-congruent sides.
Diagonals are perpendicular.

41. EXAMPLE 4
Find m D in the kite shown at the right.
By Theorem, DEFG has exactly one pair of congruent opposite angles. Because E G, D and F must be congruent. So, m D = m F.Write and solve an equation to find m D.

42. EXAMPLE 4 m D + m F +124o + 80o = 360o m D + m D +124o + 80o = 360o
Corollary to Theorem 8.1
m D + m D +124o + 80o = 360o
Substitute m D for m F.
2(m D) +204o = 360o
Combine like terms.
m D = 78o
Solve for m D.

43. GUIDED PRACTICE
1. In a kite, the measures of the angles are 3xo, 75o, 90o, and 120o. Find the value of x. What are the measures of the angles that are congruent?
25; 75o