Geometry. Unit 6. Quadrilaterals. Unit 6

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1 Geometry Quadrilaterals

2 Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections are the vertices of the polygon. No three consecutive vertices of a polygon are collinear.

3 Common Polygons Triangle: three sides Quadrilateral: four sides Pentagon: five sides Hexagon: six sides Heptagon: seven sides Octagon: eight sides Nonagon: nine sides Decagon: ten sides Dodecagon: twelve sides n-gon: n sides

4 Convex vs. Concave Convex All points of any segment joining two points in the interior of the polygon must also be in the interior of the polygon. Concave At least one segment joining two points in the interior of the polygon also contain points in the exterior of the polygon. A B A B

5 Polygon Interior/Exterior Angles Polygon Interior Angles Theorem The sum of the interior angles of a convex polygon with n sides is: n Regular Polygon The measure of each interior angle is: n n The measure of the Central Angle is: 360 n Polygon Exterior Angles Theorem The sum of the measures of the exterior angles, one at each vertex, of a convex polygon is 360º

6 Quadrilaterals Parallelogram: a quadrilateral with both pairs of opposite sides parallel. Rectangle: a parallelogram with four right angles. Rhombus: a parallelogram with four congruent sides. Square: a parallelogram with four right angles and four congruent sides. Trapezoid: a quadrilateral with exactly one pair of parallel sides. Kite: a quadrilateral with two distinct pairs of congruent, adjacent sides.

7 Quadrilateral Interior/Exterior Angles Quadrilateral Sum Theorem The sum of the interior angles of a quadrilateral is 360º. Quadrilateral Exterior Angles The sum of the measures of the exterior angles, one at each vertex, of a convex quadrilateral is 360º. Regular Quadrilateral (Square) The measure of each interior angle is 90º.

8 Properties of Parallelograms Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other.

9 Proving a Quadrilateral is a Parallelogram Prove that both pairs of opposite sides are parallel. Prove that both pairs of opposite sides are congruent. Prove that one pair of opposite sides is both congruent and parallel. Prove that both pairs of opposite angles are congruent. Prove that consecutive angles are supplementary. Prove that the diagonals bisect each other. The quadrilateral formed by joining the midpoints of the consecutive sides of another quadrilateral is a parallelogram.

10 Common Parallelograms Parallelogram: a quadrilateral with both pairs of opposite sides parallel. Rectangle: a parallelogram with four right angles. Rhombus: a parallelogram with four congruent sides. Square: a parallelogram with four right angles and four congruent sides.

11 Summary of Properties Rectangles Has all the properties of a parallelogram. All angles are right angles. The diagonals are congruent. Rhombuses Has all the properties of a parallelogram. All sides are congruent. The diagonals are perpendicular. The diagonals bisect opposite angles. Squares Has all the properties of a parallelogram, rectangle, and rhombus.

12 Properties of a Trapezoid One pair of opposite sides are parallel and are called the bases. Each pair of base angles of an isosceles trapezoid is congruent and the legs are congruent. The diagonals of an isosceles trapezoid are congruent. base legs legs base

13 Midsegment Theorem of Trapezoids and Triangles Trapezoids The median (or midsegment) of a trapezoid is parallel to the bases. The length of the midsegment is equal to half the sum of the lengths of the bases (average). legs base Midsegment b 1 b 2 2 legs Triangles The midsegment of a triangle is parallel to the third side. The length of the midsegment of a triangle is one-half the length of the third side. A line contains the midpoint of one side of a triangle and is parallel to another side bisects the third side. base

14 Properties of a Kite a quadrilateral with exactly two pairs of consecutive congruent sides diagonals are perpendicular

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