1. Special Quadrilaterals
Unit 6
Special Quadrilaterals

2. Polygon Angle Sum
The sum of the measures of the interior angles of an n-gon is (n-2)180, where n represents the number of sides
Example –
What is the sum of the interior angles of a 13 sided figure
(13-2) 180
(11)(180)
1980°

3. You Try Sum of the interior angles of a Heptagon
900°
Sum of the interior angles of a 17-gon?
2700°
Sum of the interior angles of a Quadrilateral?
360°

4. Types of Polygons Equilateral Polygon – All sides are congruent
Equiangular Polygon – All angles are congruent
Regular polygon – All angles and all sides are congruent

5. Finding One Interior Angle of a Regular Polygon
The measure of each interior angle of a regular n-gon is [(n-2)180] ⁄ n, where n represents the number of sides
Example:
Find the measure of one interior angle of a regular hexagon
[(6-2)180]/6
[(4)180]/6
720/6
120°

6. You Try Find the measure of one interior angle of a regular 16 –gon.
157.5°
Find the measure of one interior angle of a regular nonagon.
140°
Find the measure of one interior angle of a regular 11 – gon.

7. Polygon Exterior Angle Theorem
The sum of the measures of the exterior angles of a polygon, with one angle at each vertex, is always 360°.
Example:
What is one exterior angle of a regular octagon?
360/8
45°

12. Parallelograms

13. Parallelogram
A parallelogram is a special quadrilateral with both pair of opposite sides parallel.

14. Properties of a parallelogram
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent
Consecutive angles are supplementary
Diagonals Bisect Each other

19. How to prove a quadrilateral is a parallelogram
There are 6 ways to prove that a quadrilateral is a parallelogram
By the Definition of a parallelogram that states if both pairs of opposite sides are parallel then a quadrilateral is a parallelogram
If both pairs of opposite sides are congruent then quadrilateral is a parallelogram

20. If both pairs of opposite angles are congruent then quadrilateral is a parallelogram
If consecutive angles are supplementary then quadrilateral is a parallelogram
If diagonals bisect each other then quadrilateral is a parallelogram
If one pair of opposite sides are congruent and parallel then quadrilateral is a parallelogram

21. Rectangle, Rhombus, Square

22. Rectangle A rectangle is a parallelogram with four right angles.
If a parallelogram is a rectangle then all parallelogram properties apply.

23. Properties of Rectangles
If a parallelogram is a rectangle then the diagonals are congruent.

24. Rhombus A Rhombus is a parallelogram with four congruent sides
If a parallelogram is a rhombus then all parallelogram properties apply.

25. Properties of a Rhombus
If a parallelogram is a rhombus, then its diagonals are perpendicular
If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles

26. Square
A square is a parallelogram with four congruent sides and four right angles.
A square is a parallelogram, rectangle, and rhombus so all the properties apply!

30. Proving a Quadrilateral is a Rectangle
To prove a quadrilateral is a rectangle you must first prove it is a parallelogram.
Then:
If all angles are right angles then parallelogram is a rectangle.
If the diagonals of the parallelogram are congruent then it is a rectangle.

31. Proving a Quadrilateral is a Rhombus
To prove a quadrilateral is a rhombus you must first prove it is a parallelogram.
Then:
If all sides are congruent then parallelogram is a rhombus.
If the diagonals of the parallelogram are perpendicular then it is a rhombus.
If the diagonals of a parallelogram bisect opposite angles then it is a rhombus.

32. Proving a Quadrilateral is a Square
To prove a quadrilateral is a square you must first prove it is a parallelogram.
Then:
Prove that parallelogram is a rhombus.
Prove that parallelogram is a rectangle.
Then quadrilateral is a Square!

35. Trapezoid / Kite

36. Trapezoid
A trapezoid is a quadrilateral with one pair of parallel sides, know as the bases. The non-parallel sides are know as the legs.

37. Isosceles Trapezoid
A trapezoid with legs that are congruent.

38. Properties of Isosceles Trapezoid
If a quadrilateral is an isosceles trapezoid then each pair of base angles are congruent
If a quadrilateral is an isosceles trapezoid then diagonals are congruent

39. Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to the bases and is half the sum of the lengths of the bases.

40. Kite
A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.
If a quadrilateral is a kite then the diagonals are perpendicular.