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Understanding Parallelograms: Properties and Applications

Explore the properties of parallelograms, solve for unknown angles, and identify different types of parallelograms such as rectangles, rhombuses, and squares. Learn to apply conditions for determining if a quadrilateral is a parallelogram.

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Understanding Parallelograms: Properties and Applications

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  1. Warm Up: • Solve for x and y in the following parallelogram. What properties of parallelograms did you use when solving? • What is the measure of CD? • What is the measure of Angle C? • What is the sum of the interior angles of a dodecagon? 1. B C (16x – 4)o 5y – 1 2y + 8 (14x + 34)o A D 2. 3. 4.

  2. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its opposite sides are congruent.

  3. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its opposite sides are congruent. • IF both pairs of opposite sides of a quadrilateral are congruent, THEN the quadrilateral is a parallelogram.

  4. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its opposite angles are congruent.

  5. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its opposite angles are congruent. • IF both pairs of opposite angles of a quadrilateral are congruent, THEN the quadrilateral is a parallelogram.

  6. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its consecutive angles are supplementary. B C xo (180 – x)o (180 – x)o xo A D

  7. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its consecutive angles are supplementary. • IF an angle of a quadrilateral is supplementary to both of its consecutive angles, THEN the quadrilateral is a parallelogram. B C xo (180 – x)o (180 – x)o xo A D

  8. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its diagonals bisect each other.

  9. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF a quadrilateral is a parallelogram, THEN its diagonals bisect each other. • IF the diagonals of a quadrilateral bisect each other, THEN the quadrilateral is a parallelogram.

  10. Conditions of Parallelograms (6.3) and Special Parallelograms (6.4) • IF one pair of opposite sides of a quadrilateral are parallel AND congruent, THEN the quadrilateral is a parallelogram.

  11. Show that ABCD is a parallelogram for m = 12 and n = 9.5; which one of the conditions of parallelograms did you use? B C (7m – 29)o (12n + 11)o (2m + 31)o A D

  12. Are each of the given quadrilaterals also parallelograms? Justify your answer. # 1 # 2 # 3 7 7

  13. Find x and y so the quadrilateral is a parallelogram. (x – 12)o B C (3y – 4)o (4x – 8)o A D (1/2 y)o

  14. RECTANGLES

  15. RECTANGLE • Four Right Angles • Congruent Diagonals • Properties of a Parallelogram

  16. RHOMBUSES

  17. RHOMBUS • Four Congruent Sides • Perpendicular Diagonals • Diagonals Bisect Opposite • Angles • Properties of a Parallelogram

  18. SQUARES

  19. SQUARE • Properties of a Rectangle • Properties of a Rhombus

  20. ABCD is a rhombus. Find the measure of Angle B. (y + 2)o B C (2y + 10)o A D

  21. Show the diagonals • ofsquare ABCD • are congruent • perpendicular • bisectors of each • other. • A (-1, 0) • B (-3, 5) • C (2, 7) • D (4, 2)

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