1. Quadrilaterals
Chapter 8

2. 8.1 – Find Angle Measures in Polygons
Two vertices that are endpoints of the same side are called consecutive vertices in polygons
Diagonal
Segment that joins two non-consecutive vertices
Theorem 8.1 – Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a convex n-gon is (n – 2)*180o where n is the number of sides
Corollary to Thrm Interior angles of a quadrilateral:
Sum of measures of interior angles of a quadrilateral is 360 degrees

3. Examples Example 1 Example 2 Example 3 GP #1-4
Find the sum of the measures of the interior angles of a convex octagon
Example 2
The sum of the measures of the interior angles of a convex polygon is 900 degrees. Classify the polygon by the number of sides
Example 3
Find the value of x (on board)
GP #1-4

4. Exterior Angles
Sum of exterior angle measures does not depend on number of sides of polygon
Theorem 8.2 – Polygon Exterior Angles Theorem
Sum of measures of exterior angles of a convex polygon, one angle at each vertex, is 360 degrees
Example 4
What is the value of x ? (on board)

5. Example 5
If you have a trampoline in the shape of a regular dodecagon, find the following
Measure of each interior angle
Measure of each exterior angle
GP #5-6

6. 8.2 – Use Properties of Parallelograms
Quadrilateral with both pairs of opposite sides parallel
Theorem 8.3
If a quadrilateral is a parallelogram, then its opposite sides are congruent
Theorem 8.4
If a quadrilateral is a parallelogram, then its opposite angles are congruent
Example 1: find values of x and y (on board)

7. Interior Angles
Consecutive interior angle theorem states that if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary
This holds true for parallelograms as well
Theorem 8.5
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary
Theorem 8.6
If a quadrilateral is a parallelogram, then its diagonals bisect each other

8. Example 3 The diagonals of parallelogram LMNO at point P. GP #1-6
What are the coordinates of P? (on board)
GP #1-6

9. 8.3 – Show a Quad. is a Parallelogram
Converses of theorems 8.3 & 8.4 are stated below
Can be used to show a quadrilateral with certain properties is a parallelogram
Theorem 8.7
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Theorem 8.8
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

10. More Theorems! Theorem 8.9 Theorem 8.10 Example 3 GP #2-5
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram
Theorem 8.10
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
Example 3
For what value of x is CDEF a parallelogram? (on board)
GP #2-5

11. Ways to Prove a Quad. is a Parallelogram
Show both pairs of opposite sides are parallel (DEFINITION)
Show both pairs of opposite sides are congruent (THEOREM 8.7)
Show both pairs of opposite angles are congruent (THEOREM 8.8)
Show one pair of opposite sides are congruent and parallel (THEOREM 8.9)
Show the diagonals bisect each other (THEOREM 8.10)

12. 8.4 – Properties of Rhombuses, Rectangles, & Squares
Three special types of quadrilaterals exist:
Rhombus
Parallelogram with four congruent sides
Rectangle
Parallelogram with four congruent angles
Square
Parallelogram with four congruent sides and four congruent angles

13. Corollaries Rhombus Corollary Rectangle Corollary Square Corollary
A quadrilateral is a rhombus if and only if it has four congruent sides
Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four congruent angles
Square Corollary
A quadrilateral is a square if and only if it is a rhombus and a rectangle

14. Venn Diagram of Parallelograms

15. Diagonals of Rhombuses & Rectangles
Theorem 8.11
A parallelogram is a rhombus if and only if its diagonals are perpendicular
Theorem 8. 12
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles
Theorem 8.13
A parallelogram is a rectangle if and only if its diagonals are congruent

16. 8.5 – Use Properties of Trapezoids & Kites
Other types of special quadrilaterals exist
Trapezoid
Quadrilateral with exactly one pair of parallel sides
Parallel sides are called bases, non-parallel sides are called legs
Has two pairs of base angles
Example 1
Show that ORST is a trapezoid (on board)

17. Isosceles Trapezoids Isosceles trapezoid Theorem 8.14 Theorem 8.15
A trapezoid is isosceles when the legs are congruent
Theorem 8.14
If a trapezoid is isosceles, then each pair of base angles is congruent
Theorem 8.15
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid
Theorem 8.16
A trapezoid is isosceles if and only if its diagonals are congruent

18. Midsegments Midsegment of a trapezoid
Segment that connects the midpoints of its legs
Theorem 8.17 – Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases (average of the bases)
Example 3
In the diagram (on board), MN is the midsegment of trapezoid PQRS. Find length of MN

19. Kites Kite Theorem 8.18 Theorem 8.19
Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent
Theorem 8.18
If a quadrilateral is a kite, then its diagonals are perpendicular
Theorem 8.19
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent

20. Example 4
Find m <D in the kite (on board)
GP #5 & 6

21. 8.6 – Identify Special Quadrilaterals