Geometry 1 Unit 6Quadrilaterals

Geometry 1 Unit 66.1 Polygons

PolygonsPolygonA closed figure in a planeFormed by connecting line segments endpoint to endpointEach segment intersects exactly 2 othersClassified by the number of sides they haveNamed by listing vertices in consecutive orderSides Line segments in a polygonVertex Each endpoint in a polygon

Polygons

polygonsnot polygons

PolygonsPentagon ABCDE or pentagon CDEAB

PolygonsSidesName3Triangle4Quadrilateral5Pentagon6Hexagon7Heptagon8Octagon9Nonagon10Decagon11Undecagon12Dodecagonnn-gon(a 19 sided polygon is a 19-gon)

PolygonsDiagonals Line segments that connect non-consecutive vertices.

PolygonsConvex polygons Polygons with no diagonals on the outside of the polygon

PolygonsConcave polygonsA polygon is concave if at least one diagonal is outside the polygonThese are also called nonconvex.

PolygonsExample 1Identify the polygon and state whether it is convex or concave.

PolygonsEquilateral Polygonall sides the same length

Equiangular Polygon all angles equal measure

Regular Polygon equilateral and equiangular

Polygons

Equilateral Equiangular Regular

PolygonsExample 2Decide whether the polygon is regular.

PolygonsInterior Angles of a Quadrilateral TheoremThe sum of the measures of the interior angles of a quadrilateral is 360.

m1 + m2 + m3 + m4 = 360

PolygonsExample 3Find mF, mG, and mH.

PolygonsExample 4Use the information in the diagram to solve for x

Geometry 1 Unit 66.2 Properties of Parallelograms

Properties of ParallelogramsParallelogramQuadrilateral with two pairs of parallel sides.

Properties of ParallelogramsOpposite Sides of a Parallelogram TheoremIf a quadrilateral is a parallelogram, then its opposite sides are congruent.

Properties of ParallelogramsOpposite Angles in a Parallelogram TheoremIf a quadrilateral is a parallelogram, then its opposite angles are congruent.P R and Q S

Properties of ParallelogramsConsecutive Angles in a Parallelogram TheoremIf a quadrilateral is a parallelogram, then its consecutive angles are supplementaryPQSRmP + mQ = 180mQ + mR = 180mR + mS = 180mS + mP = 180

Properties of ParallelogramsDiagonals in a Parallelogram TheoremIf a quadrilateral is a parallelogram, then its diagonals bisect each other.

Properties of ParallelogramsExample 1GHJK is a parallelogram. Find each unknown lengthJHLH

Properties of ParallelogramsExample 2In ABCD, mC = 105. Find the measure of each angle.mAmD

Properties of ParallelogramsExample 3WXYZ is a parallelogram. Find the value of x.WYZX3x + 184x 9

Properties of ParallelogramsExample 4Given: ABCD is a parallelogram.Prove:2 4

Properties of ParallelogramsExample 5Given: ACDF is a parallelogram. ABDE is a parallelogram.Prove:BCD EFA

Properties of ParallelogramsExample 6A four-sided concrete slab has consecutive angle measures of 85, 94, 85, and 96. Is the slab a parallelogram? Explain.

Geometry 1 Unit 66.3 Proving Quadrilaterals are Parallelograms

Proving Quadrilaterals are ParallelogramsInvestigating Properties of ParallelogramsCut 4 straws to form two congruent pairs.Partly unbend two paperclips, link their smaller ends, and insert the larger ends into two cut straws. Join the rest of the straws to form a quadrilateral with opposite sides congruent.Change the angles of your quadrilateral. Is your quadrilateral a parallelogram?

Proving Quadrilaterals are ParallelogramsConverse of the Opposite Sides of a Parallelogram TheoremIf a opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.ABCD is a parallelogram

Proving Quadrilaterals are ParallelogramsConverse of the Opposite Angles in a Parallelogram TheoremIf both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.ABCD is a parallelogram.

Proving Quadrilaterals are ParallelogramsConverse of the Consecutive Angles in a Parallelogram TheoremIf an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.ABCD is a parallelogram

Proving Quadrilaterals are ParallelogramsConverse of the Diagonals in a Parallelogram TheoremIf the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.ABCD is a parallelogram

Proving Quadrilaterals are ParallelogramsExample 1Given:PQT RSTProve:PQRS is a parallelogram.

Proving Quadrilaterals are ParallelogramsExample 2A gate is braced as shown. How do you know that opposite sides of the gate are congruent?

Proving Quadrilaterals are ParallelogramsCongruent and Parallel Sides TheoremIf one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.ABCD is a parallelogram.

Proving Quadrilaterals are ParallelogramsTo determine if a quadrilateral is a parallelogram, you need to know one of the following:Opposite sides are parallelOpposite sides are congruentOpposite angles are congruentAn angle is supplementary with both of its consecutive anglesDiagonals bisect each otherOne pair of sides is both parallel and congruent

Proving Quadrilaterals are ParallelogramsExample 3Show that A(-1,2), B(3,2), C(1,-2), and D(-3,-2) are the vertices of a parallelogram.

Geometry 1 Unit 66.4 Rhombuses, Rectangles, and Squares

Rhombuses, Rectangles, and SquaresRectangleParallelogram with four congruent anglesRhombusParallelogram with four congruent sidesSquareParallelogram with four congruent angles and four congruent sides

Rhombuses, Rectangles, and SquaresExample 1Decide if each statement is always, sometimes or never true.A rhombus is a rectangle

A parallelogram is a rectangle

A rectangle is a square

A square is a rhombus

Rhombuses, Rectangles, and SquaresExample 2Given FROG is a rectangle, what else do you know about FROG?

Rhombuses, Rectangles, and SquaresExample 3EFGH is a rectangle. K is the midpoint of FH. EG = 8z 16, What is the measure of segment EK?What is the measure of segment GK?

Rhombuses, Rectangles, and SquaresRhombus CorollaryA quadrilateral is a rhombus if and only if it has four congruent sides.

Rhombuses, Rectangles, and SquaresRectangle CorollaryA quadrilateral is a rectangle if and only if it has four right angles.

Rhombuses, Rectangles, and SquaresSquare CorollaryA quadrilateral is a square if and only if it is a rhombus and a rectangle.

Rhombuses, Rectangles, and SquaresPerpendicular Diagonals of a Rhombus TheoremA parallelogram is a rhombus if and only if its diagonals are perpendicular.

Rhombuses, Rectangles, and SquaresDiagonals Bisecting Opposite Angles TheoremA parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

Rhombuses, Rectangles, and SquaresDiagonals in a Rectangle TheoremA parallelogram is a rectangle if and only if its diagonals are congruent.

Rhombuses, Rectangles, and SquaresExample 4You cut out a parallelogram shaped quilt piece and measure the diagonals to be congruent. What is the shape?

An angle formed by the diagonals of the quilt piece measures 90. Is the shape a square?

Geometry 1 Unit 66.5 Trapezoids and Kites

Trapezoids and KitesTrapezoidA quadrilateral with exactly one pair of parallel sides.BasesThe parallel sides of a trapezoid.Pairs of Base AnglesAngles in a trapezoid that share a base.LegsThe nonparallel sides of a trapezoid. Isosceles TrapezoidTrapezoid with congruent legs.

Trapezoids and KitesBase Angles of an Isosceles Trapezoid TheoremIf a trapezoid is isosceles, then each pair of base angles is congruent.A B, C D

Trapezoids and KitesCongruent Base Angles in a Trapezoid TheoremIf a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.ABCD is an isosceles trapezoid.

Trapezoids and KitesDiagonals in an Isosceles Trapezoid TheoremA trapezoid is isosceles if and only if its diagonals are congruent.

Trapezoids and KitesExample 1CDEF is an isosceles trapezoid with CE = 10 and mE = 95. Find DF, mC, mD and mF.

Trapezoids and KitesExample 2The vertices of WXYZ are W(-1, 2), X(3, 0), Y(4, -3) and Z(-4, 1).Show that WXYZ is an isosceles trapezoid.

Trapezoids and KitesMidsegment of a TrapezoidThe segment that connects the midpoints of a trapezoids legs.midsegment

Trapezoids and KitesMidsegment Theorem for TrapezoidsThe midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.BCDNMA

Trapezoids and KitesExample 3A potter crafts a trapezoidal relish dish, placing a divider, shown by AB, in the middle of the dish. How long must the divider be to ensure that it divides the legs it half?

Trapezoids and KitesKite A quadrilateral with two distinct pairs of consecutive congruent sides. Opposite sides are not congruent.

Trapezoids and KitesDiagonals of a Kite TheoremIf a quadrilateral is a kite, then its diagonals are perpendicular.BDAC

Trapezoids and KitesOpposite Angles in a Kite TheoremIf a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Trapezoids and KitesExample 4GHJK is a kite. Find HP.

Trapezoids and KitesExample 5RSTU is a kite. Find mR, mS, and mT.

Geometry 1 Unit 66.6 Special Quadrilaterals

Special QuadrilateralsExample 1ABCD has at least two congruent consecutive sides. What quadrilaterals meet this condition?

Special QuadrilateralsExample 2Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?

Special QuadrilateralsExample 3When you join the midpoints of the sides of an isosceles trapezoid, what special quadrilateral is formed? Why?

Special QuadrilateralsWays to Prove Quadrilaterals are RhombusesUse the definition to show that the quadrilateral is a parallelogram that has four congruent sides. Use the Rhombus Corollary to show that all four sides of the quadrilateral are congruent.Show that the quadrilateral is a parallelogram and that the diagonals are perpendicular.Show that the quadrilateral is a parallelogram and that each diagonal bisects a pair of opposite angles.

Special QuadrilateralsExample 4The coordinates of ABCD are A(-2,5), B(1,8), C(4,5), D(1,2). Show that ABCD is a rhombus.

Special QuadrilateralsExample 5In quadrilateral RSTV, mR = 88, mS = 113, and mV = 113. What type of quadrilateral could RSTV be? Must it be this type? Explain.

Special QuadrilateralsPut an X in the box if the shape always has the given property.

Geometry 1 Unit 66.7 Areas of Triangles and Quadrilaterals

Areas of Triangles and QuadrilateralsArea of a Square PostulateThe area of a square is the square of the length of its side, or A = s2.

Areas of Triangles and QuadrilateralsArea Congruence PostulateIf two polygons are congruent, then they have the same area.

Areas of Triangles and QuadrilateralsArea Addition PostulateThe area of a region is the sum of the areas of its nonoverlapping parts.

Areas of Triangles and QuadrilateralsArea of a RectangleThe area of a rectangle is the product of its base and height.

A = bh

Areas of Triangles and QuadrilateralsArea of a ParallelogramThe area of a parallelogram is the product of a base and its corresponding height.

A = bh

Areas of Triangles and Quadrilaterals

Areas of Triangles and QuadrilateralsArea of a TriangleThe area of a triangle is one half the product of a base and its corresponding height.A = bh

Areas of Triangles and Quadrilaterals

Areas of Triangles and QuadrilateralsExample 1Find the area of RST.

Areas of Triangles and QuadrilateralsExample 2Find the base of a triangle that has an area of 48 and a height of 3.

Areas of Triangles and QuadrilateralsExample 3A triangle has an area of 12 square feet and a base of 4 feet. Are all triangles with these dimensions congruent? Explain.

Areas of Triangles and QuadrilateralsExample 4A rectangle has an area of 100 square meters and a height of 25 meters. Are all the rectangles with these dimensions congruent? Explain.

Areas of Triangles and QuadrilateralsArea of a TrapezoidThe area of a trapezoid is one half the product of the height and the sum of the bases.A = h(b1 + b2)hhhb2b2b2b1b1b1

Areas of Triangles and QuadrilateralsA = triangle 1 + triangle 2A = h(b1) + h(b2)A = h(b1 + b2)

Areas of Triangles and QuadrilateralsArea of a KiteThe area of a kite is one half the product of the lengths of its diagonals.

A = (d1)(d2)

Areas of Triangles and Quadrilaterals d1 and d2A = (d1)(a) + (d1)(b)

A = (d1)(a+b)a + b = d2A = (d1)(d2)abd1d1

Areas of Triangles and QuadrilateralsArea of a RhombusThe area of a rhombus is equal to one half the product of the lengths of its diagonals.A = d1d2

Areas of Triangles and QuadrilateralsExample 5Find the area of trapezoid EFGH with coordinates E(-2, 3), F(2, 4), G(2, -2), and H(-2, -1)

Areas of Triangles and QuadrilateralsExample 6Use the information given in the diagram to find the area of kite ABCD.

Areas of Triangles and QuadrilateralsExample 7The tray below is designed to save space on cafeteria tables. How much table area does the tray use?

JH = 8LH = 6RectangleYesDF = 10, mc = 85, mD = 95, mF = 85The slopes of WX and YZ are -1/2 so they are parallel.The slope of WZ and XY are different so they are not parallel.Therefore WXYZ is a trapezoid.WY = XZ = 50 so the diagonals are congruent and the trapezoid is isosceles.9 in.270, 125, 40Rhombus, square, kiteParallelogram, rhombus, rectangle, square, isosceles trapezoidRhombusABCD is isosceles, AD is congruent to BC so AE is congruent to BG. Also angle A and angle B are congruent.By SAS, triangles AEF and BGF are congruent.EF is congruent to FG.Because opposite sides of a parallelogram are congruent, and using the transitive property of congruence, all four sides of EFGH are congruent so it is a rhombus. AB = BC = CD = AD = 18A kite, no.Angle T = 46 so it is not a parallelogram.No pair of consecutive sides are supplementary so it is not a trapezoid.RSTV will be a kite only if two pairs of consecutive sides are also congruent.See page 367 #2-6Have students cut out triangles from rectangles to see this property in action.616noYes.