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
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)
Is it a property of a square? Parallelogram with perpendicular diagonals
Is it a property of a square? Parallelogram with perpendicular diagonals YES
Is it a property of a square? Quadrilateral with diagonals that bisect each other
Is it a property of a square? Quadrilateral with diagonals that bisect each other YES
Is it a property of a square? Quadrilateral with perpendicular diagonals that bisect opposite angles
Is it a property of a square? Quadrilateral with perpendicular diagonals that bisect opposite angles YES
Is it a property of a square? Diagonals bisect opposite sides
Is it a property of a square? Diagonals bisect opposite sides NO (not sides!!)
Is it a property of a rectangle? Parallelogram with congruent diagonals
Is it a property of a rectangle? Parallelogram with congruent diagonals YES
Is it a property of a rectangle? Parallelogram with perpendicular diagonals
Is it a property of a rectangle? Parallelogram with perpendicular diagonals NO- Rhombus property
Does this property only describe a rectangle? Quadrilateral with diagonals that bisect each other
Does this property only describe a rectangle? Quadrilateral with diagonals that bisect each other No that could be a parallelogram, square, or rhombus too!
Does this property only describe a rectangle? Quadrilateral with perpendicular diagonals that bisect opposite angles
Does this property only describe a rectangle? Quadrilateral with perpendicular diagonals that bisect opposite angles No- Rhombus Property
XWZY is a Rhombus. m WZX = 35, m ZYW = 5x 10y and m YZX = 5x + 15y. Find x and y.
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 = 180 10x + 5y + 90 = 180 10x + 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 = 9.4 10x 30y = 70 25y = 20 y = 0.8 x = 9.4 and y = 0.8 35 (5x 10y) (5x + 15y)
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).
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) 2 2 + 4 2 = c 2 4 + 16 = c 2 20 = c 2 20 = c 2 2 5 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 -4 4-4 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
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
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
Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and WY XZ Unit 2 16.1
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 2 16.1
Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and WZX YXZ Unit 2 16.1
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 2 16.1
Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YXZ, XW XY, and ZW ZY Unit 2 16.1
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 2 16.1
Given the quadrilateral XWZY. Does the given information prove it is a parallelogram? WXZ YZX and XWY ZYW Unit 2 16.1
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 2 16.1
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 2 16.4
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 2 16.4
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 2 16.4
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 2 16.4
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 2 16.4
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 2 16.4
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 2 16.4
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 2 16.4