Chapter 9 Quadrilaterals

Lesson #1 Polygons and Classifying Quadrilaterals Essential Question #1 Explain why a square is a rectangle, but a rectangle is not a square.

Polygons Polygon Not a Polygon Polygon ABCDEF A closed plane figure with at least 3 sides The sides intersect only at their endpoints Polygon Not a Polygon A B C F E D Polygon ABCDEF

Diagonals The diagonals of a polygon are the segments that connects one vertex of a polygon to another non adjacent vertex. A B C F E D

A polygon is concave if a diagonal passes outside of the polygon. Convex vs. Concave A polygon is convex if no diagonals contain points outside the polygon. A polygon is concave if a diagonal passes outside of the polygon.

Names of Polygons Polygon # of Sides 3 4 5 6 7 Polygon # of Sides 8 9 10 12 n

Polygon Angle Sum Theorem Sides Triangles Degrees 3 4 5 6 7 Sides Triangles Degrees 8 9 10 12 n The sum of the measures of the interior angles of a polygon with n sides is 180(n – 2).

Find the sum of the measures of the interior angles of each polygon. Example 1 Find the sum of the measures of the interior angles of each polygon. 15-gon 20-gon

Example 2 Find x:

Each exterior angle is supplementary to its adjacent interior angle. Exterior Angles An angle drawn on the outside of a polygon that is formed by extending one side. Exterior Angle Each exterior angle is supplementary to its adjacent interior angle.

Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°

Regular Polygons A polygon whose sides are the same length and angles are the same measure.

Example 3 Find the measure of an interior angle and an exterior angle of a regular dodecagon.

Homework Practice Worksheet 6-1

Quadrilaterals A quadrilateral is a polygon with 4 sides. Depending upon its characteristics, a quadrilateral may have a special name.

A quadrilateral with exactly one pair of parallel sides QUADRILATERALS No pairs of parallel sides Two pairs of parallel sides One pairs of parallel sides TRAPEZOID KITES PARALLELOGRAM A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent A quadrilateral with exactly one pair of parallel sides A quadrilateral with both pairs of opposite sides parallel RECTANGLE RHOMBUS ISOSCELES TRAPEZOID SQUARE A parallelogram with four congruent sides and four congruent angles A trapezoid whose nonparallel sides are congruent A parallelogram with four congruent angles A parallelogram with four congruent sides

QUADRILATERALS No pairs of parallel sides Two pairs of parallel sides One pairs of parallel sides

Determine the most precise name for quadrilateral LMNP. Example 4 Determine the most precise name for quadrilateral LMNP. L M N P

Example 5 Find the value of the variables of the quadrilateral at the left is a kite. I E K T 2y + 5 x + 6 3x - 5 2x + 4

Homework Page 94 #16, 18, 25-27

Properties of Parallelograms Lesson #2 Properties of Parallelograms Essential Question #2 Explain how the diagonal properties of a rhombus differ from those of a parallelogram.

Properties of a Parallelogram Opposite sides are parallel (definition) Opposites sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other.

Find the value of the missing angles. Example 1 Find the value of the missing angles. 22° 3 2 1 45° 113° 3 2 1 4 47° 72°

Example 2 In parallelogram RSTV, diagonals RT and SV intersect at Q. If RQ = 5x – 1 and QT = 3x + 15, find QT.

QUADRILATERALS No pairs of parallel sides Two pairs of parallel sides One pairs of parallel sides

Properties of a Rectangles HAS ALL OF THE PROPERTIES OF A PARALLELOGRAM. 4 Congruent Angles (by definition) Diagonals are congruent.

Example 3 Find x.

Properties of a Rhombus HAS ALL OF THE PROPERTIES OF A PARALLELOGRAM. Has 4 congruent sides (by definition) Each diagonal bisects two angles The diagonals are perpendicular.

Example 4 Find x.

Properties of a Square ALL OF THE PROPERTIES OF A PARALLELOGRAM. ALL OF THE PROPERTIES OF A RECTANGLE. ALL OF THE PROPERTIES OF A RHOMBUS.

Example 5 Find x.

Find each value of each missing angle. Example 6 Find each value of each missing angle.

Find each value of each missing measure. Example 7 Find each value of each missing measure.

Homework Practice Worksheet 6.2, 6.4, 6.5

Lesson #3 Trapezoids, Isosceles Trapezoids, and Kites Essential Question #3 Can two consecutive angles of a kite be supplementary? Why or why not?

Trapezoids BASE LEG LEG BASE ANGLES BASE

Properties of a Trapezoid Exactly one pair of opposite sides are parallel (by definition) Exactly two pairs of consecutive angles are supplementary

Properties of a Isosceles Trapezoid ALL OF THE PROPERTIES OF A TRAPEZOID The legs are congruent (by definition) The base angles are congruent Diagonals are congruent

Find the missing angles. Example 1 Find the missing angles.

Properties of Kites Two sets of adjacent sides congruent (by definition) The angles formed by the sides of differing lengths are congruent The diagonals are perpendicular The diagonal connecting the non-congruent angles bisects the other diagonal.

Find the missing angles. Example 2 Find the missing angles.

Example 3 Find the side measures.

Homework Page 473 #1-5, 13-16