Given the following information about rectangle ABCD what triangle criterion will you use to prove ADC BCD.

Transcription

1 A B D Given the following information about rectangle ABCD what triangle criterion will you use to prove ADC BCD. ADC and BCD are right angles because ABCD is a rectangle ADC BCD because all right angles are congruent ABCD is a parallelogram because all rectangles are parallelograms AD BC because opposite sides of a parallelogram are congruent DC DC because of the Reflexive Property C

2 A B S 1 S 1 ADC BCD by SAS D A 1 S 2 A 1 C Given the following information about rectangle ABCD what triangle criterion will you use to prove ADC BCD. ADC and BCD are right angles because ABCD is a rectangle ADC BCD because all right angles are congruent CONGRUENT ANGLES (A 1 in picture) ABCD is a parallelogram because all rectangles are parallelograms AD BC because opposite sides of a parallelogram are congruent CONGRUENT SIDES (S 1 in picture) DC DC because of the Reflexive Property CONGRUENT SIDES (S 2 in picture)

3 Is it a property of a square? Parallelogram with perpendicular diagonals

4 Is it a property of a square? Parallelogram with perpendicular diagonals YES

5 Is it a property of a square? Quadrilateral with diagonals that bisect each other

6 Is it a property of a square? Quadrilateral with diagonals that bisect each other YES

7 Is it a property of a square? Quadrilateral with perpendicular diagonals that bisect opposite angles

8 Is it a property of a square? Quadrilateral with perpendicular diagonals that bisect opposite angles YES

9 Is it a property of a square? Diagonals bisect opposite sides

10 Is it a property of a square? Diagonals bisect opposite sides NO (not sides!!)

11 Is it a property of a rectangle? Parallelogram with congruent diagonals

12 Is it a property of a rectangle? Parallelogram with congruent diagonals YES

13 Is it a property of a rectangle? Parallelogram with perpendicular diagonals

14 Is it a property of a rectangle? Parallelogram with perpendicular diagonals NO- Rhombus property

15 Does this property only describe a rectangle? Quadrilateral with diagonals that bisect each other

16 Does this property only describe a rectangle? Quadrilateral with diagonals that bisect each other No that could be a parallelogram, square, or rhombus too!

17 Does this property only describe a rectangle? Quadrilateral with perpendicular diagonals that bisect opposite angles

18 Does this property only describe a rectangle? Quadrilateral with perpendicular diagonals that bisect opposite angles No- Rhombus Property

19 XWZY is a Rhombus. m WZX = 35, m ZYW = 5x 10y and m YZX = 5x + 15y. Find x and y.

20 XWZY is a Rhombus. m WZX = 35, m ZYW = 5x 10y and m YZX = 5x + 15y. Find x and y. First equation- Three angles in a triangle add to 180 5x 10y + 5x + 15y + 90 = x + 5y + 90 = x + 5y = 90 Second equation- Diagonals bisect opposite angles in a rhombus 5x + 15y = 35 Solve the system 10x + 5y = 90 10x + 5( 0.8) = 90-2 ( 5x + 15y = 35) 10x 4 = 90 10x = 94 10x + 5y = 90 x = x 30y = 70 25y = 20 y = 0.8 x = 9.4 and y = (5x 10y) (5x + 15y)

21 Use the coordinates of the quadrilateral to find the slope and length of each side. Then using that information, what name(s) describe ABCD. The vertices are A(1, 3), B(-3, 1), C(-1, -3), and D(3,-1).

22 Use the coordinates of the quadrilateral to find the slope and length of each side. Then using that information, what name(s) describe ABCD. The vertices are A(1, 3), B(-3, 1), C(-1, -3), and D(3,-1). AB= 2 5 or 4.47 BC= 2 5 or 4.47 CD= 2 5 or 4.47 AD= 2 5 or 4.47 m AB = 2 4 = 1 2 m BC = 4 2 = 2 m CD = 2 4 = 1 2 m AD = 4 2 = 2 Work to find side length (all the same numbers) = c = c 2 20 = c 2 20 = c or 4.47 = c To calculate slope count rise over run Sides are equal in measure. Slopes show opposite sides are parallel (same slope) Slopes show consecutive sides are perpendicular (slopes are opposite reciprocals)-which make right angles 2 B A 2 D 2 Names: Quadrilateral (4 sides), Parallelogram (opposite sides parallel), Rectangle (parallelogram with right angles), Rhombus (parallelogram with equal sides), and Square (both a rhombus and square) 2 C 4

23 Rectangle ABCD has the vertices A(1, 3), B(-3, 1), and C(-1, -3). What is the location of point D? A B C

24 Rectangle ABCD has the vertices A(1, 3), B(-3, 1), and C(-1, -3). What is true about the location of point D? To calculate slope count rise over run 4 A 2 m AB = 2 4 Using the same slope count find the location of D B 2 4 D D is at (2, 1) C

25 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and WY XZ Unit

26 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and WY XZ NO WXZ YZX alternate interior angles congruent give one set of parallel lines WY XZ diagonals congruent Unit

27 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and WZX YXZ Unit

28 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and WZX YXZ Yes- Both pairs of opposite sides are parallel WXZ YZX alternate interior angles congruent give one set of parallel lines WZX YXZ alternate interior angles congruent give the other set of parallel lines Unit

29 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YXZ, XW XY, and ZW ZY Unit

30 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YXZ, XW XY, and ZW ZY NO WXZ YXZ one pair congruent angles XW XY and ZW ZY consecutive sides congruent Unit

31 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and XWY ZYW Unit

32 Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and XWY ZYW NO WXZ YXZ and XWY ZYW are alternate interior for the same pair of parallel lines XW YZ. One pair of opposite sides parallel lines does not make a parallelogram. Unit

33 A B D Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square P Is the additional piece of information enough to complete the proof? C AD BC Unit

34 Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square Is the additional piece of information enough to complete the proof? AD BC A D P B C NO one pair of opposite sides congruent and diagonals congruent is not enough information. You need all sides congruent. Unit

35 A B D Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square P Is the additional piece of information enough to complete the proof? C APB is a right angle Unit

36 Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square A B Is the additional piece of information enough to complete the proof? P APB is a right angle D C Yes, If APB is a right angle then diagonals are perpendicular. Diagonals perpendicular (makes a rhombus) and congruent (makes a rectangle). Unit

37 A B D Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square P Is the additional piece of information enough to complete the proof? C ABC is a right angle Unit

38 Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square A B Is the additional piece of information enough to complete the proof? ABC is a right angle D P C No, If ABC is a right angle is a property of a rectangle. Need information to prove the parallelogram is a rhombus. (AC BD congruent diagonals is a property of a rectangle) Unit

39 A B D Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square Is the additional piece of information enough to complete the proof? C AC and BD bisect each other Unit

40 Given: Parallelogram ABCD and AC BD Prove: Parallelogram ABCD is a square A B Is the additional piece of information enough to complete the proof? P AC and BD bisect each other D C No, If AC and BD bisect each other is a property of a parallelogram. Need information to prove the parallelogram is a rhombus. (AC BD congruent diagonals is a property of a rectangle) Unit