GEOMETRY CHAPTER 6 STUDY GUIDE A. Vocabulary Term Definition & symbols Picture and/or example Parallelogram (pg 306) Rhombus (pg 306) Rectangle (pg 306) Square (pg 306) Kite (pg 306) Trapezoid (pg 306) Geometry Ch. 6 Note taking guide pg 1 of 12
Isosceles Trapezoid (pg 306) Consecutive angles (pg 312) Base angles of a trapezoid (pg 336) Midsegment of a trapezoid (pg 348) B. REVIEW FORMULAS FROM CH. 1 SLOPE OF A LINE: DISTANCE BETWEEN TWO POINTS: MIDPOINT: Geometry Ch. 6 Note taking guide pg 2 of 12
C. Coordinate Proofs: 1) Determine the most precise name for quadrilateral ABCD with vertices A(-3, 3), B(2, 4), C(3, -1), and D(-2, -2). Prove using appropriate formulas. 4 2-2 2) Prove that ABCD form a parallelogram. A(-1, 1), B(-2, -1), C(2, -3), and D(3, -1). 4 2-2 Geometry Ch. 6 Note taking guide pg 3 of 12
D. Relationships Among Special Quadrilaterals (See chart on pg 307) Write the name of the quadrilateral and mark the relevant features (congruent sides, parallel sides, right angles, congruent angles) No pairs of // sides 2 pairs of // sides 1 pair of // sides E. Draw a Venn diagram to show the relationships among the special quadrilaterals. Geometry Ch. 6 Note taking guide pg 4 of 12
F. PROPERTIES OF SPECIAL QUADRILATERALS: Check each box if the property is ALWAYS TRUE for the given special quadrilaterals. Property Only one pair of opposite sides are //. Both pairs of opposite sides are //. Only one pair of opposite sides are ". Both pairs of opposite sides are ". All sides are ". Only 1 pair of opposite "s are ". Both pairs of opposite "s are " All "s are ". Parallelogram Rectangle Rhombus Square Kite Trapezoid Isosceles Trapezoid All 4 pairs of consecutive "s are supplementary. Only 2 pairs of consecutive "s are supplementary. Diagonals bisect each other. Diagonals are congruent. Diagonals are perpendicular. Each diagonal bisect a pair of opposite angles. Base angles are congruent. Geometry Ch. 6 Note taking guide pg 5 of 12
G. Ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are. (Definition of a parallelogram) 2. Show that both pairs of opposite sides are. (Theorem 6-5) 3. Show that both pairs of opposite angles are. (Theorem 6-6) 4. Show that diagonals. (Theorem 6-7) 5. Show that one pair of opposite sides is both and. (Theorem 6-8) Examples: 1. Given: "ADB " "DBC, "ABD " "BDC Prove: ABCD is a parallelogram. A B D C Geometry Ch. 6 Note taking guide pg 6 of 12
2. Use only a compass and a straightedge, construct a parallelogram. Explain why your construction is valid. 3. Given three vertices of a parallelogram A(1, -2), B(2, 2), C(5, 0), determine all possible sets of coordinates of a fourth point that would form a parallelogram with the given vertices. 8 6 4 2-5 5 10-2 -4-6 Geometry Ch. 6 Note taking guide pg 7 of 12
H. Ways to prove that a parallelogram is a rhombus. 1. Show that one of the parallelogram bisects two. (Theorem 6-12) 2. Show that the diagonals are. (Theorem 6-13) 3. Show that all four sides are. (Def. of rhombus) Ex: 1) Proof of Theorem 6-13. Given: Parallelogram ABCD; AC " BD at E. Prove: ABCD is a rhombus. D A E C B 2) Use only a compass and a straightedge, construct a rhombus. Explain why your construction is valid. Geometry Ch. 6 Note taking guide pg 8 of 12
I. Way to prove that a parallelogram is a rectangle. 1. Show that the diagonals of the parallelogram are (Theorem 6-14) 2. Show that all four angles are. (Def. of a rectangle) Examples: 1. Builders use properties of diagonals to square up rectangular shapes like building frames and playing-field boundaries. Suppose you are on a building team and helping to lay out a rectangular patio. Explain how to use properties of diagonals to locate the four corners of a rectangle. 2. Use only a compass and a straightedge, construct a rectangle. Explain why your construction is valid. Geometry Ch. 6 Note taking guide pg 9 of 12
I. Trapezoid Midsegment Theorem (Theorem 6-18, pg 348): 1. The midsegment of a trapezoid is to the. M R A N 2. The length of the midsegment of a trapezoid is the sum of the lengths of the two bases. T MN //, MN //, and MN = P Examples: 1) Let RA = 24, TP = 30, MN =? 2) Let RA = 5, MN = 9, TP =? 3) Let MN = 12, RA = x + 4, TP = 2x. Find the values of x, RA, and TP. Geometry Ch. 6 Note taking guide pg 10 of 12
J. Coordinate Geometry: When working with a figure in the coordinate plane, it generally is good practice to place a vertex at the origin and one side on an axis. You can also use multiples of 2 to avoid fractions when finding midpoints. 1. Find the most convenient place to draw a rectangle in the coordinate plane with side lengths 2 and 5. 6 4 2-5 5-2 -4-6 2. Give coordinates for unlabeled points without using any new variables. Rectangle Square B(0, b) D(a, 0) D(a, 0) Parallelogram Isosceles Trapezoid B (b, h) B (b, h) D(a, 0) D(a, 0) Geometry Ch. 6 Note taking guide pg 11 of 12
3. a. Draw a square whose diagonals of length 2b lie on the x- and y-axes. b. Give the coordinates of the vertices of the square. c. Compute the length of a side of the square. d. Find the slopes of two adjacent sides of the square. e. Do the slopes show that the sides are perpendicular? Explain. 4. Given an ordinary quadrilateral shown to the right, find the coordinates of the midpoint of each side. Prove that the segments joining the midpoints form a parallelogram. B (b, h) C (c, e) A(0, 0) D(d, 0) Geometry Ch. 6 Note taking guide pg 12 of 12