Chapter 7 Quadrilaterals

1. Chapter 7 Quadrilaterals Drew Lukoff

2. All Key Terms Convex and concave polygons Diagonal Isosceles trapezoid Midsegment Parallelogram Point symmetry Rectangle Rhombus Square Trapezoid

3. Lesson 1-Quadrilaterals • Diagonal-a line segment that connects any two nonconsecutive vertices • Theorem 24: The sum of the angles of a quadrilateral is 360 degrees • Corollary to Thm 24: A quadrilateral is equiangular iff it is a rectangle • Rectangle-All 4 angles are right angles

4. More Lesson 1 • Concave and convex polygons • Concave means that from two points inside the polygon, the line segment connecting them does NOT lie entirely inside the polygon • Convex means that from two points inside the polygon, the line segment connecting them does lie entirely inside the polygon

5. Lesson 2-Parallelograms and Point Symmetry • Parallelogram: A quadrilateral whose opposite sides are parallel • All parallelograms have point symmetry • Point symmetry is when the shape is turned upside down and it looks identical • Two points are symmetric with respect to a point iff it is the midpoint of the line segment joining them A B Point symmetry

6. More on parallelograms Theorem 25: The opposite sides and angles are equal Theorem 26: The diagonals of a parallelogram bisect each other

7. Lesson 3-Even more on parallelograms! Theorem 27: A quadrilateral is a parallelogram if its opposite sides are equal Theorem 28: A quadrilateral is a parallelogram if its opposite angles are equal Theorem 29: A quadrilateral is a parallelogram if two opposite sides are both parallel and equal |> |> Opposite sides are equal Opposite angles are equal Two opposite sides are Both parallel and equal

8. Theorem 30 A quadrilateral is a parallelogram if its diagonals bisect each other

9. Lesson 4-Rectangles, Rhombuses, and Squares Square-Quadrilateral whose sides and angles are equal Rhombus-Quadrilateral whose sides are equal All rectangles are parallelograms All rhombuses are parallelograms Rhombus and parallelogram Square, rhombus, parallelogram, and rectangle

10. Diagonals The diagonals of a rectangle are equal The diagonals of a rhombus are perpendicular Diagonal AC is perpendicular to diagonalBD Note: The diagonals of a square are BOTH perpendicular AND equal

11. Lesson 5-Trapezoids Trapezoid-Quadrilateral that has exactly one pair of parallel sides Isosceles trapezoid-Trapezoid whose legs are equal One pair of parallel sides Trapezoid

12. Trapezoid theorems The base angles of an isosceles trapezoid are equal The diagonals of a trapezoid are equal <P=<S and <Q=<R

13. Lesson 6-Midsegment Theorem A Midsegmentof a triangle is a line segment that connects the midpoints of two of its sides. The Midsegment Theorem: A midsegment of a triangle is parallel to the third side and half as long > > AB is ½ of XY and parallel to XY

14. Review ALWAYS, SOMETIMES, NEVER 1. Rhombuses are _______ parallelograms. 2. Rectangles are _______ squares. 3. There are _____ two pairs of parallel sides in a trapezoid. 4. Diagonals of trapezoids, squares, and rectangles are _____ equal.

15. Quick review problems 1. Does a square have point symmetry? 2. If BC equals 10 what does DE equal?

16. Answers 1. Always 1. Yes 2. Sometimes 2. 5 3. Never 4. Always

17. Summary The characteristics of quadrilaterals are all somehow connected. For example, squares have parallel sides just like parallelograms, rhombuses, and rectangles. Trapezoids just have one pair of parallel sides. Then the other quadrilaterals are split up by more specific characteristics like if it has all right angles (squares and rectangles) or if it has equal sides (squares and rhombuses).