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Name ___________ Geometry 1 Unit 6: Quadrilaterals 1 2 Geometry 1 6.1 Polygons Polygon Unit 6: Quadrilaterals Sides Vertex polygons not polygons How to name a Polygon: A E B D C 3 tri- 3 hepta- 7 quadr- 4 oct- 8 penta- 5 nona- 9 hexa- 6 dec- 10 Use the prefix chart above and the given definition to write a definition of the boldfaced word. 1. triplet: One of three children born at one birth. Quadruplet 2. quadrennial: Happening once every four years. Octennial 3. decathalon: An athletic contest that consists of ten events for each participant. Pentathalon 4. hexapod: Having six legs or feet. Tripod 5. octogenarian: A person between eighty and ninety years of age. Nonagenarian 6. pentagram: A five-pointed star. Hexagram 7. octahedron: A solid geometric figure with eight plane faces. Decahedron 8. nonagon: A polygon with nine sides and nine angles. Heptagon 9. heptameter: A line of verse consisting of seven metrical feet. Pentameter 4 Polygons Number of Sides 3 4 5 6 7 8 9 Name 10 11 12 n Diagonals Convex polygons Concave polygons Example 1 Identify the polygon and state whether it is convex or concave. 5 Can a polygon be equiangular and not equilateral? Draw an example. 6 Equilateral Polygon Equiangular Polygon Regular Polygon Example 2 Decide whether the polygon is regular. Interior Angles of a Quadrilateral Theorem 2 1 __________________________________________ 4 3 7 8 H Example 3 Find mF, mG, and mH. G x E 55 ° x F Example 4 Use the information in the diagram to solve for x 100° 120° 2x + 30 3x – 5 9 10 Geometry 1 6.2 Properties of Parallelograms Parallelogram Unit 6: Quadrilaterals Opposite Sides of a Parallelogram Theorem Q P Opposite Angles in a Parallelogram Theorem R S Q P Consecutive Angles in a Parallelogram Theorem R S Add to equal 180° Diagonals in a Parallelogram Theorem Q R M P Example 1 GHJK is a parallelogram. Find each unknown length JH K LH 6 8 G S J L H 11 12 Example 2 In parallelogram ABCD, mC = 105°. Find the measure of each angle. mA mD Example 3 WXYZ is a parallelogram. Find the value of x. 3x + 18° 4x – 9° Example 4 Given:: ABCD is a parallelogram. Prove: 2 4 Statement Reasons ABCD is a parallelogram AD || BC A B 2 1 4 D 3 2 1 AB || CD Alternate interior angles theorem C 2 4 13 14 Example 5 Given: ACDF is a parallelogram. ABDE is a parallelogram. Prove: ∆BCD ∆EFA A B Statement Reason ACDF is a parallelogram. ABDE is a parallelogram. C Opposite sides of a parallelogram are congruent AC = DF AB = DE AC = AB + BC F E D DF = DE + EF AC = DE + DF AB + BC = AB + EF BC = EF Def of Congruent ∆BCD ∆EFA Example 6 A four-sided concrete slab has consecutive angle measures of 85°, 94°, 85°, and 96°. Is the slab a parallelogram? Explain. 15 Investigating Properties of Parallelograms Cut 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? 16 Geometry 1 6.3 Proving Quadrilaterals are Parallelograms Unit 6: Quadrilaterals Converse of the Opposite Sides of a Parallelogram Theorem A B D Converse of the Opposite Angles in a Parallelogram Theorem Converse of the Consecutive Angles in a Parallelogram Theorem C A B D C A x° (180 – x)° x° B C D Converse of the Diagonals in a Parallelogram Theorem A B M D C 17 18 Example 1 Given: ∆PQT ∆RST Prove: PQRS is a parallelogram. P Q Statements Reasons ∆PQT ∆RST CPCTC PT = RT ST = QT Def. of bisect T S PQRS is a parallelogram R Example 2 A gate is braced as shown. How do you know that opposite sides of the gate are congruent? Congruent and Parallel Sides Theorem B A C D To determine if a quadrilateral is a parallelogram, you need to know one of the following: 19 20 Example 3 Show that A(-1,2), B(3,2), C(1,-2), and D(-3,-2) are the vertices of a parallelogram. 21 Answer Sometimes, Always or Never 1. A square is a rectangle. 2. A rectangle is a square. 3. A rhombus is a rectangle. 4. A square is a rhombus. 5. A rhombus is a rectangle. 22 Geometry 1 6.4 Rhombuses, Rectangles, and Squares Unit 6: Quadrilaterals Rectangle Rhombus Square 23 24 Example 1 Decide 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 Example 2 Given FROG is a rectangle, what else do you know about FROG? F R G Example 3 O EFGH 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? Rhombus Corollary Rectangle Corollary Square Corollary Perpendicular Diagonals of a Rhombus Theorem B C A D 25 26 Diagonals Bisecting Opposite Angles Theorem. B C A Diagonals in a Rectangle Theorem D A B D C Example 4 You 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? 27 28 29 30 31 32 Geometry 1 6.5 Trapezoids and Kites Unit 6: Quadrilaterals Trapezoid Bases Pairs of Base Angles Legs Isosceles Trapezoid Base Angles of an Isosceles Trapezoid Theorem Congruent Base Angles in a Trapezoid Theorem. Diagonals in an Isosceles Trapezoid Theorem Midsegment of a trapezoid A B D A C B D A D C B C 33 34 Midsegment Theorem for Trapezoids B M A Kite Diagonals of a Kite Theorem Opposite Angles in a Kite Theorem Example 4 GHJK is a kite. Find HP. √29 5 G H J P K Example 5 RSTU is a kite. Find mR, mS, and mT. S R x + 30° x° T 125° U 35 C N D 36 37 38 39 40 Geometry 1 6.6 Special Quadrilaterals Property Unit 6: Quadrilaterals Parallelogram Rectangle Rhombus Square Trapezoid Kite 1.Both pairs of opposite sides are congruent 2. Diagonals are congruent 3. Diagonals are perpendicular 4. Diagonals bisect each other 5. Consecutive angles are supplementary 6. Both pairs of opposite angles are congruent 41 42 43 44 Geometry 1 6.7 Areas of Triangles and Quadrilaterals Unit 6: Quadrilaterals Example 1 Example 2 What is the base of a triangle that has an area of 48 and a height of 3? Example 3 A rectangle has an area of 100 square meters and a height of 25 meters. Are all the rectangles with these dimensions congruent? Example 4 45 Example 5 What is the height of a parallelogram that has an area of 96 square feet and a base length of 8 feet? Example 6 Find the area of trapezoid EFGH. E(-2, 3), F(2, 4), G(2, -2), H(-2, -1) Example 7 46 Example 8 Example 9 Example 10 Find the area of rhombus EFGH if EG = 10 and FH = 15. 47 48