 § 8.1 Quadrilaterals  § 8.4 Rectangles, Rhombi, and Squares  § 8.3 Tests for Parallelograms  § 8.2 Parallelograms  § 8.5 Trapezoids

You will learn to identify parts of quadrilaterals and find the sum of the measures of the interior angles of a quadrilateral. 1) Quadrilateral 2) Consecutive 3) Nonconsecutive 4) Diagonal

A quadrilateral is a closed geometric figure with ____ sides and ____ vertices. four The segments of a quadrilateral intersect only at their endpoints. QuadrilateralsNot Quadrilaterals Special types of quadrilaterals include squares and rectangles.

A B C D Quadrilaterals are named by listing their vertices in order. There are several names for the quadrilateral below. Some examples: quadrilateral ABCD quadrilateral BCDA quadrilateral CDAB or quadrilateral DABC

S R Q P Any two _______ of a quadrilateral are either __________ or _____________. consecutive nonconsecutive sidesvertices angles

S R Q P Segments that join nonconsecutive vertices of a quadrilateral are called ________. diagonals S and Q are nonconsecutive vertices. R and P are nonconsecutive vertices.

Q T S R Name all pairs of consecutive sides: Name all pairs of nonconsecutive angles: Name the diagonals:

D C B A Considering the quadrilateral to the right. What shapes are formed if a diagonal is drawn? ___________ two triangles Use the Angle Sum Theorem (Section 5-2) to find m  1 + m  2 + m  Use the Angle Sum Theorem (Section 5-2) to find m  4 + m  5 + m  Find m  1 + m  2 + m  3 + m  4 + m  5 + m  This leads to the following theorem.

Theorem 8-1 The sum of the measures of the angles of a quadrilateral is ____. 360 a°a° d°d° c°c° b°b° a + b + c + d = 360

A D C B m  A + m  B + m  C + m  D = 360 x + 2x + x – = 360 Find the measure of  B in quadrilateral ABCD if  A = x,  B = 2x,  C = x – 10, and  D = 50. 4x + 40 = 360 4x = 320 x = 80  B = 2x  B = 2(80)  B = 160

You will learn to identify and use the properties of parallelograms. 1) Parallelogram

A parallelogram is a quadrilateral with two pairs of ____________. parallel sides A B D C In parallelogram ABCD below, and Also, the parallel sides are _________. congruent Knowledge gained about “parallels” (chapter 4) will now be used in the following theorems.

Theorem 8-2 Theorem 8-3 Theorem 8-4 Opposite angles of a parallelogram are ________. Opposite sides of a parallelogram are ________. The consecutive angles of a parallelogram are ____________. A B D C A B D C A B D C  A   C and  B   D m  A + m  B = 180 m  D + m  C = 180 congruent supplementary

In RSTU, RS = 45, ST = 70, and  U = 68. R U S T ° Find: RU = ____ UT = _____ m  S = _____ m  T = _____ 70 Theorem Theorem ° Theorem ° Theorem 8-4

Theorem 8-5 The diagonals of a parallelogram ______ each other. A D B C E bisect In RSTU, if RT = 56, find RE. R U S T E RE = 28

A D B C In the figure below, ABCD is a parallelogram. DB  BD Since AD || BC and diagonal DB is a transversal, then  ADB   CBD. (Alternate Interior angles) Since AB || DC and diagonal DB is a transversal, then  BDC   DBA. (Alternate Interior angles) ASA Theorem

Theorem 8-6 A diagonal of a parallelogram separates it into two _________________. A D B C congruent triangles

The Escher design below is based on a _____________. You can use a parallelogram to make a simple Escher-like drawing. Change one side of the parallelogram and then translate (slide) the change to the opposite side. The resulting figure is used to make a design with different colors and textures. parallelogram

You will learn to identify and use tests to show that a quadrilateral is a parallelogram. Nothing New!

Theorem 8-7 If both pairs of opposite sides of a quadrilateral are _________, then the quadrilateral is a parallelogram. A D C B congruent

You can use the properties of congruent triangles and Theorem 8-7 to find other ways to show that a quadrilateral is a parallelogram. In quadrilateral PQRS, PR and QS bisect each other at T. Show that PQRS is a parallelogram by providing a reason for each step. Definition of segment bisector Vertical angles are congruent SAS Corresp. parts of Congruent Triangles are Congruent Theorem 8-7 T P S R Q

Theorem 8-8 If one pair of opposite sides of a quadrilateral is _______ and _________, then the quadrilateral is a parallelogram. A D C B congruent parallel

Theorem 8-9 If the diagonals of a quadrilateral ________________, then the quadrilateral is a parallelogram. bisect each other A DC B E

Determine whether each quadrilateral is a parallelogram. If the figure is a parallelogram, give a reason for your answer. A D C B Given Alt. Int. Angles Therefore, quadrilateral ABCD is a parallelogram. Theorem 8-8

You will learn to identify and use the properties of rectangles, rhombi, and squares. 1) Rectangle 2) Rhombus 3) Square

A closed figure, 4 sides & 4 vertices Quadrilateral Opposite sides parallel opposite sides congruent Parallelogram Parallelogram with 4 right angles Rectangle Parallelogram with 4 congruent sides Rhombus Parallelogram with 4 congruent sides and 4 right angles Square

Identify the parallelogram below. D C B A Parallelogram ABCD has 4 right angles, but the four sides are not congruent. Therefore, it is a _________ rectangle Identify the parallelogram below. rhombus

Theorem 8-10 The diagonals of a rectangle are _________. congruent A D B C

Theorem 8-11 The diagonals of a rhombus are ____________. perpendicular A B D C

Theorem 8-12 Each diagonal of a rhombus _______ a pair of opposite angles. bisects D C B A

Use square XYZW to answer the following questions: W Z Y X O 1) If YW = 14, XZ = ____ 2) m  YOX = ____ A square has all the properties of a rectangle, and the diagonals of a rectangle are congruent. 14 A square has all the properties of a rhombus, and the diagonals of a rhombus are perpendicular. 90 3) Name all segments that are congruent to WO. Explain your reasoning. The diagonals are congruent and they bisect each other. OY, XO, and OZ

Quadrilaterals Parallelograms RhombiRectangles Squares Use the Venn diagram to answer the following questions: T or F 1) Every square is a rhombus: ___ 2) Every rhombus is a square: ___ 3) Every rectangle is a square: ___ 4) Every square is a rectangle: ___ 5) All rhombi are parallelograms: ___ 6) Every parallelogram is a rectangle: ___ T T T F F F

You will learn to identify and use the properties of trapezoids and isosceles trapezoids. 1) Trapezoid

A trapezoid is a ____________ with exactly one pair of ____________. quadrilateral parallel sides T P A R The parallel sides are called ______. bases base The non parallel sides are called _____. legs leg Each trapezoid has two pair of base angles. base angles  T and  R are one pair of base angles.  P and  A are the other pair of base angles.

Theorem 8-13 The median of a trapezoid is parallel to the _____, bases and the length of the median equals _______________ of the lengths of the bases. one-half the sum C N B D M A

C N D M B A Find the length of median MN in trapezoid ABCD if AB = 16 and DC =

If the legs of a trapezoid are congruent, the trapezoid is an _________________. isosceles trapezoid In lesson 6 – 4, you learned that the base angles of an isosceles triangle are congruent. There is a similar property of isosceles trapezoids.

Theorem 8-14 Each pair of __________ in an isosceles trapezoid is congruent. base angles Z Y X W

T P A R 60° Find the missing angle measures in isosceles trapezoid TRAP.  P   A m  P = m  A 60 = m  A 60° Theorem 8 – 14  T   R  P +  A + 2(  T) = 360 Theorem 8 – (  T) = (  T) = 360 2(  T) = 240  T = °  R = °