1. Sum of Interior Angles of a Polygon

2. Th. 6.1 – Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex n-gon is 180 o (n – 2) Let  = sum   = 180 o (n – 2) Sum of Interior Angles of a Polygon Corollary to Th. 6.1 The measure of each interior angle of a regular n – gon is 1/n(180 o )(n – 2) m  = 1/n(180 o )(n – 2)

3. Th. 6.2 Polygon Exterior Angles Theorem – The sum of the measures of the exterior angles, one from each vertex, of a convex polygon is 360 o.  = 360 Corollary to Th. 6.2 – The measure of each exterior angle of a regular n – gon is 1/n(360 o ). m  = 1/n(360 o ).

4. Ex. 1. The measure of each angle of a regular n – gon is 160 o. How many sides does the polygon have? (what is n?) Ex. 2. The measure of each exterior angle of a regular polygon is 72 o. How many sides does the polygon have? Ex. 3 Find the measure of each interior angle of the quadrilateral shown below. A B C D x + 30 x + 90 x + 60 x

5. 276/ 11 – 33 odd, 34 You are shown part of a convex n-gon. The pattern of congruent angles continues around the polygon. Find n. (hint, consider exterior angles) 142 o 158 o

6. Parallelograms

7. Proving Quadrilaterals are Parallelograms W12 o N – 562.58’

8. A B CD Definition – If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. AB || DC, AD || BC

9. A B CD Th. 6.7 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. AB  DC, AD  BC

10. A B CD Th. 6.8 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.  A   C,  B   D

11. A B CD Th. 6.9 – If an angle of a quadrilateral is supplementary to both consecutive angles, then the quadrilateral is a parallelogram.  A is supplementary to  B and  D

12. A B CD Th. 6.10 – If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. AC bisects DB, DB bisects AC, AE  EC, BE  ED E

13. A B CD Th. 6.11 – If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. AB  DC, AB || DC

14. PQ U RS G:, PU  UR P: PQRS is a parallelogram 1., PT  UR 1. Given If diagonals bisect each other the quadrilateral is a parallelogram 290/1 – 18, 20, 22 – 26

15. Sid Gilman Bill Walsh Sam Wyche Fassel Mike Holmgren Dennis Green George Seifert Wayne Coslet Mike Shanahan Brad Musgrave Steve Mariucci Ray Rhodes Mike Sherman

16. Quadrilaterals Parallelograms Trapezoids Perpendicular Diagonals Others No Properties Rectangles Rhombus Squares Kites Others T

17. Special Parallelograms Rhombus – a parallelogram that has all 4 sides congruent. Rectangle – a parallelogram that has 4 right angles. Square – a parallelogram that is both a rhombus and a rectangle.

18. Proving Special Parallelograms 6.12 – A parallelogram is a rhombus iff its diagonal are perpendicular. 6.13 – A parallelogram is a rhombus iff, each diagonal bisects a pair of opposite angles. 6.15 – A quadrilateral is a rhombus iff it has 4 congruent sides.

19. 6.14 – A parallelogram is a rectangle iff, its diagonals are congruent. CD BA AC  BD 6.16 – A quadrilateral is a rectangle iff it has 4 right angles. Proving Special Parallelograms

20. 296/1 – 12, 21 – 24, 34

21. Trapezoids A quadrilateral with exactly one pair of parallel sides. Base Leg Isosceles Trapezoid – Legs are congruent Trapezoid has two pairs of base angles

22. Trapezoids 6.17 – If a trapezoid is isosceles, then each pair of base angles is congruent. 6.19 – If a trapezoid has one pair of congruent base angles, then it is isosceles. 6.18 - If a trapezoid is isosceles, then its diagonals are congruent. 6.20 – If a trapezoid has congruent diagonals, then it is isosceles.

23. Trapezoids YX DC BA Midsegment of a Trapezoid – connects the midpoints of the legs of the trapezoid 6.21 – The midsegment of a trapezoid is parallel to each base, and its length is half the sum of the lengths of the bases. b1b1 b2b2 18 12 ? ? 20 27 2x + 2 3x – 1 3x + 3 302/1 – 10, 11 – 31 odd, 37 – 41odd

24. A B CD AB  DC,  A   C