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7.3 Notes: Proving Quadrilaterals are Parallelograms

Learn 5 ways to prove quadrilaterals are parallelograms, using properties like opposite sides, angles, and diagonals. Master the art of proving shapes are parallelograms!

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7.3 Notes: Proving Quadrilaterals are Parallelograms

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  1. 7.3 Notes:Proving Quadrilaterals are Parallelograms Objective: To use the properties of parallelograms to prove when a quadrilateral is a parallelogram

  2. A B E There are 5 ways we can prove a quadrilateral is a parallelogram. D C 1. If the diagonals of a quadrilateral bisecteach other, then it is a parallelogram.

  3. There are 5 ways we can prove a quadrilateral is a parallelogram. B A 2. If one pair of opposite sides of a quadrilateral is both paralleland congruent, then it is a parallelogram. C D

  4. A B There are 5 ways we can prove a quadrilateral is a parallelogram. D C 3. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

  5. A B There are 5 ways we can prove a quadrilateral is a parallelogram. D C 4. If both pairs of opposite anglesof a quadrilateral are congruent, then it is a parallelogram.

  6. A B There are 5 ways we can prove a quadrilateral is a parallelogram. D C 5. If both pairs of opposites sides of a quadrilateral are parallel, then it is a parallelogram.

  7. These 5 will be the “if-then statements” you will use to justify your answers: 1. If diagonalsbisecteach other, then it is a parallelogram. 2. If one pair of opposite sides is both parallel and congruent, then it is a parallelogram. 3. If both pairs of opposite sides are congruent, then it is a parallelogram. 4. If both pairs of opposite angles are congruent, then it is a parallelogram. 5. If both pairs of opposites sides are parallel, then it is a parallelogram.

  8. a. b. c. d. Ex. 1 Based on the information given, can you determine that the quadrilateral must be a parallelogram? Justify your answer with an “if-then statement”. If no, explain why not. a. b. Yes! If both pairs of opposite sides are parallel, then it is a parallelogram. (use alt int’s) No! Opposite sides are NOT congruent. d. c. Yes! If one pair of opposite sides are both parallel and congruent, then it is a parallelogram. No! Diagonals do NOT bisect each other.

  9. X Y Ex. 2 Find the values of x and y for which XYZW must be a parallelogram. y + 9 8x + 12 N 2y – 80 Diagonals must bisect each other 10x – 24 W Z

  10. Q R a° Ex. 3 Find the values of a and c for which PQRS must be a parallelogram. (a + 40)° 3c - 3 c + 1 One pair of opposite sides must be both parallel and congruent P S

  11. Proving Other Parallelograms **You must first prove a figure is a parallelogram BEFORE proving they are a rhombus or rectangle** Rhombus: If the diagonals bisect the angles they go to, then it is a rhombus. If the diagonals are perpendicular, then it is a rhombus. You only need to have one diagonal to prove.

  12. Proving Other Parallelograms **You must first prove a figure is a parallelogram BEFORE proving they are a rhombus or rectangle** Rectangle: If the diagonals are congruent, then it is a rectangle

  13. Proving Other Parallelograms **You must first prove a figure is a parallelogram BEFORE proving they are a rhombus or rectangle** Square: Any combination of rhombus and rectangle will give you a square

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