QUADRILATERALSChapter 8

8.1 – Find Angle Measures in Polygons

• Two vertices that are endpoints of the same side are called consecutive vertices in polygons

• Diagonal• Segment that joins two non-consecutive vertices

• Theorem 8.1 – Polygon Interior Angles Theorem• The sum of the measures of the interior angles of a convex n-gon is

(n – 2)*180o where n is the number of sides

• Corollary to Thrm 8.1 - Interior angles of a quadrilateral:• Sum of measures of interior angles of a quadrilateral is 360 degrees

Examples• Example 1

• Find the sum of the measures of the interior angles of a convex octagon

• Example 2• The sum of the measures of the interior angles of a convex

polygon is 900 degrees. Classify the polygon by the number of sides

• Example 3• Find the value of x (on board)

• GP #1-4

Exterior Angles• Sum of exterior angle measures does not depend on

number of sides of polygon

• Theorem 8.2 – Polygon Exterior Angles Theorem• Sum of measures of exterior angles of a convex polygon, one

angle at each vertex, is 360 degrees

• Example 4• What is the value of x ? (on board)

Example 5• If you have a trampoline in the shape of a regular

dodecagon, find the followinga) Measure of each interior angle

b) Measure of each exterior angle

• GP #5-6

8.2 – Use Properties of Parallelograms

• Parallelogram• Quadrilateral with both pairs of opposite sides parallel

• Theorem 8.3• If a quadrilateral is a parallelogram, then its opposite sides are

congruent

• Theorem 8.4• If a quadrilateral is a parallelogram, then its opposite angles are

congruent

• Example 1: find values of x and y (on board)

Interior Angles• Consecutive interior angle theorem states that if two

parallel lines are cut by a transversal, then consecutive interior angles are supplementary• This holds true for parallelograms as well

• Theorem 8.5• If a quadrilateral is a parallelogram, then its consecutive angles are

supplementary

• Theorem 8.6• If a quadrilateral is a parallelogram, then its diagonals bisect each

other

Example 3• The diagonals of parallelogram LMNO at point P.

• What are the coordinates of P? (on board)

• GP #1-6

8.3 – Show a Quad. is a Parallelogram

• Converses of theorems 8.3 & 8.4 are stated below• Can be used to show a quadrilateral with certain properties is a

parallelogram

• Theorem 8.7• If both pairs of opposite sides of a quadrilateral are congruent, then

the quadrilateral is a parallelogram

• Theorem 8.8• If both pairs of opposite angles of a quadrilateral are congruent,

then the quadrilateral is a parallelogram

More Theorems!• Theorem 8.9

• If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram

• Theorem 8.10• If the diagonals of a quadrilateral bisect each other, then the

quadrilateral is a parallelogram

• Example 3• For what value of x is CDEF a parallelogram? (on board)

• GP #2-5

Ways to Prove a Quad. is a Parallelogram

1. Show both pairs of opposite sides are parallel (DEFINITION)

2. Show both pairs of opposite sides are congruent (THEOREM 8.7)

3. Show both pairs of opposite angles are congruent (THEOREM 8.8)

4. Show one pair of opposite sides are congruent and parallel (THEOREM 8.9)

5. Show the diagonals bisect each other (THEOREM 8.10)

8.4 – Properties of Rhombuses, Rectangles, & Squares• Three special types of quadrilaterals exist:

• Rhombus• Parallelogram with four congruent sides

• Rectangle• Parallelogram with four congruent angles

• Square• Parallelogram with four congruent sides and four congruent angles

Corollaries• Rhombus Corollary

• A quadrilateral is a rhombus if and only if it has four congruent sides

• Rectangle Corollary• A quadrilateral is a rectangle if and only if it has four congruent

angles

• Square Corollary• A quadrilateral is a square if and only if it is a rhombus and a

rectangle

Venn Diagram of Parallelograms

Diagonals of Rhombuses & Rectangles

• Theorem 8.11• A parallelogram is a rhombus if and only if its diagonals are

perpendicular

• Theorem 8. 12• A parallelogram is a rhombus if and only if each diagonal bisects a

pair of opposite angles

• Theorem 8.13• A parallelogram is a rectangle if and only if its diagonals are

congruent

8.5 – Use Properties of Trapezoids & Kites

• Other types of special quadrilaterals exist

• Trapezoid• Quadrilateral with exactly one pair of parallel sides• Parallel sides are called bases, non-parallel sides are called legs• Has two pairs of base angles

• Example 1• Show that ORST is a trapezoid (on board)

Isosceles Trapezoids• Isosceles trapezoid

• A trapezoid is isosceles when the legs are congruent

• Theorem 8.14• If a trapezoid is isosceles, then each pair of base angles is

congruent

• Theorem 8.15• If a trapezoid has a pair of congruent base angles, then it is an

isosceles trapezoid

• Theorem 8.16• A trapezoid is isosceles if and only if its diagonals are congruent

Midsegments• Midsegment of a trapezoid

• Segment that connects the midpoints of its legs

• Theorem 8.17 – Midsegment Theorem for Trapezoids• The midsegment of a trapezoid is parallel to each base and its

length is one half the sum of the lengths of the bases (average of the bases)

• Example 3• In the diagram (on board), MN is the midsegment of trapezoid

PQRS. Find length of MN

Kites• Kite

• Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent

• Theorem 8.18• If a quadrilateral is a kite, then its diagonals are perpendicular

• Theorem 8.19• If a quadrilateral is a kite, then exactly one pair of opposite angles

are congruent

Example 4• Find m <D in the kite (on board)

• GP #5 & 6

8.6 – Identify Special Quadrilaterals