Geo 9 h 5 5. Quadrilaterals Parallelograms/Real World Visual Illusions ef: If a quadrilateral is a parallelogram, then both pair of the opposite sides are parallel. Theorem 5-: If you have a parallelogram, then the opposite sides are congruent. Proof: Given : Prove: GH HG, G H H G Theorem 5.: If you have a parallelogram, then the opposite angles are congruent. H G Theorem 5.: If you have a parallelogram, then the diagonals bisect each other. H G Summary If....
Geo 9 h 5 xamples: Given parallelograms.. y x 50 x 5 y 70 5 5 z -. x + 5y x + y S K R y + x + y + 0 Prove: SJ P QK J Q. PQRS PJ RK POWRPOINT XTR XMPLS
Geo 9 h 5 5- Ways to Prove Quadrilaterals are Parallelograms Theorem 5- : Given: TS QR; TQ SR Prove: QRST is a parallelogram. T S Q R Theorem 5-5 If one pair of opposite sides of a quadrilateral are both congruent and parallel then the quad is a parallelogram. raw a picture with markings. Theorem 5-6 If both pairs of opposite sides of a quadrilateral are congruent, then the quad is a parallelogram. raw a picture with markings. Theorem 5-7 If the diagonals of a quadrilateral bisect each other, then the quad is a parallelogram. raw a picture with markings.
Geo 9 h 5 Therefore, the 5 ways to prove that a quad is a parallelogram are:. Show that both pairs of. Show that both pairs of opposite.. Show that one pair of opposite sides are both and.. Show that both pairs of. 5. Show that diagonals. PROOS ) 5 6 Prove: is a parallelogram. 5. Given ) Prove: is a parallelogram. is midpoint of. Given is midpoint of is a parallelogram
Geo 9 h 5 5 ) Prove: is a parallelogram O. O O. given ) 5 N M 6 Prove: MN is a parallelogram. is a parallelogram. given N bi sects M bi sects 5 6 ould I still do it if I didn t give you 5 6?
Geo 9 h 5 6 5) Prove: is a parallelogram. is a parallelogram. given =
Geo 9 h 5 7 5- Theorems involving Parallel Lines Theorem 5-8: If two lines are parallel, then the points one line are equidistant from the other line. Theorem 5-9: If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on all transversals. Theorem 5-0: If a line that contains the midpoint of one side of a triangle and is parallel to another, Then the line passes through the midpoint of the third side. M N Theorem 5-: The segment that joins the midpoints of sides of a triangle is... M N
Geo 9 h 5 8 Practice Problems. Given R, S and T are midpoints of the sides of. omplete the table SR TR ST a) 8 b) 5 0 c) 0 9 7.5 R S. Given R S T, RS ST R T S T omplete: a) If RS = then ST = b) if = 8 then = c) If = 0 then = d) If = 0x then =. Given points X, Y and Z are the midpoints of, and. X Y Z a) If = then ZY = b) if = k then YX = c) If XZ = k+ then = d) If = 9, = 8, = 6 then the perimeter of XYZ is. e) If the perimeter of XYZ =, then the perimeter of =.
Geo 9 h 5 9 5- Special Parallelograms RTNGL is a quadrilateral with RHOMUS is a quadrilateral with SQUR is a quadrilateral with ef: If rectangle, then angles. with right ef: If rhombus, then adjacent sides congruent. with ef: If square, then rectangle that is a rhombus............. 5. 5. 5. 6. 6. 6. 7. 7. 8. The midpoint of the hypotenuse of a right triangle is Property Parallelogram Rectangle Rhombus Square Opp sides Opp sides Opp s iag forms s iag bisect each other iag are iag are perp diag bisects s ll s are rt s ll sides
Geo 9 h 5 0 Quad is a rhombus. ind the measure of each angle. (m = 6). m =. m =. m =. m = Quad MNOP is a rectangle. omplete, if m NML = 9. NL = 6 M N L P O. m PON = 5. m PMO = 6. PL = 7. MO = is a right triangle. M is the midpoint of M 9. If M = 7 then M =, and M =, 0. If = x, then M = M = and M =., What about the angles? Given the right triangle, with W the mp of YZ. Y W. if m = m find m.. If YW = x, and WZ = x + 8 find YZ. X or - I Z. If m = 0, find m, m, m.
Geo 9 h 5 5-5 Trapezoids Warm Up. Start your brains stretching!! ill in with always, sometimes, or never.. square is a rhombus. The diagonals of a parallelogram bisect the angles of the parallelogram.. quadrilateral with one pair of sides congruent and one pair parallel is a parallelogram.. The diagonals of a rhombus are congruent. 5. rectangle has consecutive sides congruent. 6. rectangle has perpendicular diagonals. 7. The diagonals of a rhombus bisect each other. 8. The diagonals of a parallelogram are perpendicular bisectors of each other. TRPZOI is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called. The other sides are. raw a picture. n ISOSLS TRPZOI has. T R raw auxillary line TX so that TX is parallel to R P X If you have an isosceles trapezoid, then
Geo 9 h 5 The median of a trapezoid is the segment that joins the midpoints of the legs. raw a median of a triangle. The median of a trapezoid.. In trapezoid, is a median. (which means.) omplete:. If = 5, =, then =.. If =, = 8, then = and =.. If = 9, and = then =.. If = 7y + 6, = 5y, and = y 5, then y =. 5. In KLM, HJ, JI and IH are the segments joining the midpoints of the sides. Name the three trapezoids in the figure, name an isosceles trapezoid, and find its perimeter. M H J K I 6 L
Geo 9 h 5 6. Quad TUN is an isosceles trapezoid with TU and N as bases. If m u = 6, find the measures of the other four angles of the trapezoid. 7) exceeds by. ind the bases. 5 8) = x 5 G = x + 5 = x + 9 ind x G 9) Prove: is a parallelogram. Isosceles trapezoid
Geo 9 h 5 GOM 9 RVIW SHT HPTR 5 QURILTRLS. Given is an altitude of and are midpoints of and = 0 = 8 = 5 = IN: = = = Perim of = Perim of =. Give the most descriptive name for quad MNOP given the following: a) MN PO; MN PO b) MN PO; NO MP; MO NP c) <M <N <O <P d) MNOP is a rectangle with MN = NO. MN is the median of trapezoid ZOI Z O a) The bases of ZOI are and b) If ZO = 8 and MN = then I = c) If ZO = 8 then TN = M T I N. Given ; and H are midpoints of and a) =, = 0, H =, =, G = GH = b) G =, GH = 5, =, =, = c) G = x-, G = x +, x = d) = x +, = 5x-, H = 6x-0, H =, G = G H
Geo 9 h 5 5 (5) Given: is a parallelogram. (a) = 5x 0, = x + 6 ind x 5 x 5 m = ind m (b) = x 6, = x + ind x (c) m = (x ), m = (5x) (d) = x y + 0, = y x, = x + y, = x + y. ind x, y (6) (7) Given:,, Given:, = and are midpoints, = 6 5 9 0 7 8 ind:,, Prove: is a parallelogram (8) (9) 6 5 9 0 Given: is a parallelogram Given: is a parallelogram bisects 9 0 bisects 7 8 0 5 6 8 7 9 Prove: = Prove: is a parallelogram
Geo 9 h 5 6 5 6 (0) () 7 8 Given: =,, = Given:, = Prove: is a rhombus Prove: is isosceles () () 6 7 5 7 8 0 9 5 6 8 Given: is a rhombus, 5 Given: is a parallelogram =, = Prove: is a rhombus Prove: is a rectangle () (5) 6 5 0 6 5 9 8 7 8 7 9 Given: is a square, = Given:, =, = Prove: is a parallelogram Prove: 5
Geo 9 h 5 7 XTR PROLMS () Given the figure to the right, is a parallelogram. (a) m = (9x + ), m = (5x ) m 6 = (x + ) ind: m 5, m 8 7 6 5 (b) = 5x, = 8x + (c) = x 8, = x + y, = 5x y, = y + 5 ind: x ind: x, y () Given the figure to the right, is a rhombus, m = 5 ind: the measures of the numbered angles 9 0 8 7 6 5 () Given the figure to the right,,, and H are midpoints, =, = 9, = 8, H = 6, = 5 9 G 6 H 8 ind:, G, GH, H,
Geo 9 h 5 8 () Given the figure to the right, is a rectangle,,, G, H are midpoints, = G ind: Perimeter of GH H (5) (6) 9 0 G 5 8 6 7 8 7 6 5 H Given: is a rectangle Given: is a parallelogram,, G, H are midpoints, Prove: GH is a rhombus Prove: is a parallelogram
Geo 9 h 5 9 QURILTRLS....
Geo 9 h 5 0 SUPPLMNTRY PROLMS H 5 QURILTRLS. Given that P = (-,-), Q = (,), = (,) and = (6,k), find the value of k that makes the line. (a) parallel to line PQ ; (b) perpendicular to line PQ.. Let = (-6,-), = (,-), = (0,-), and = (-7,-7). a) Show that the opposite sides of the quadrilateral are parallel. Such a quadrilateral is called a parallelogram. b) ind the lengths of all the sides. What is your conclusion? c) ind the point of intersection of and (called the diagonals of the parallelogram ) and call it M. ind M and M. What can you conclude?. How can one tell whether a given quadrilateral is a parallelogram? o the converses of the parallelogram theorems work? re there any other ways? You might want to draw in an auxiliary line to help you.. Given the points = (0,0), = (7,), and = (,), find coordinates for the point that makes quadrilateral a parallelogram. What if the question requested instead? 5. The point on a segment that is equidistant from and is called the midpoint of. or each of the following, find the coordinates for the midpoint of. (a) = (-,5) and = ( 5, -7) (b) = (m,n) and = ( k,l) 6. The midsegment of a triangle is a segment that connects the midpoints of sides of the triangle. Given a triangle with coordinates (, 7), (5,) and (-, ) find the segment that connects that midpoints of sides and, label the midpoints M and N, respectively. (a) ind the length of the midsegment MN and compare it to the length of. (b) What can be said about the lines containing segments and MN? 7. ind the point of intersection of these two lines x + y = 6 and x y =. 8. n equilateral parallelogram is called a rhombus. square is a simple example of a rhombus. Show that the lines x y = -8, x = 0, x y =, and x = form the sides of a rhombus. Support your answer.
Geo 9 h 5 9. In a right triangle, (a) if the legs are and, what is the hypotenuse? (b) What if the sides are 6 and 8? (c) 9 and? (d) 0 and 0? (e) 6 and 8? We call patterns of the Pythagorean theorem that are used often triples. Some are --5 5-- 7--5 8-5-7. (f) ould the hypotenuse be when the legs are and 5? 0. Prove that in a rhombus the diagonals create four congruent triangles. because O Since and are equidistant from and, point and Since is the perpendicular bisector of, segment segment. Similarly, since is the perpendicular bisector of, segment segment. So by we can say that Therefore, four congruent triangles are formed.. The diagram at right shows the graph of x + y =. The shaded figure is a square, three of whose vertices are on the coordinate axes. The fourth vertex is on the line. ind (a) the x- and y- intercepts of the line. (b) the length of the side of the square.
Geo 9 h 5. quadrilateral with only one pair of opposite sides parallel is called a trapezoid. The parallel lines are called bases and the non-parallel sides are called legs. n isosceles trapezoid has congruent legs. raw an isosceles trapezoid, TRP, with the following coordinates; T = ( 6, 8) R = (,8) P = (x,y) and = (-8, ). (a) irst determine (x,y). (b) ind the midpoints of the legs, label them M and N (c) ind TR, MN and P. Is there a relationship? What is it? (d) raw PR. Label the point of intersection with MN as X. ind MX. Notice anything? (e) raw T. Label the point of intersection with MN as Y. ind NY. Notice anything?