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Comprehensive Guide to Parallelograms and Related Theorems

This project explores the properties and theorems related to parallelograms, including key definitions and examples. It outlines the essential characteristics of parallelograms, such as the congruency of opposite sides and angles, and the properties of special parallelograms including rectangles, rhombuses, and squares. The project also covers theorems involving parallel lines and trapezoids, offering a solid understanding of these geometric concepts. Practice problems are included to enhance learning and application of the theorems presented.

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Comprehensive Guide to Parallelograms and Related Theorems

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  1. Math Project AriellaLindenfeld & Nikki Colona

  2. 5.1 - Parallelograms EFGH EF = HG;FG=EH 1. H • A parallelogram is a quadrilateral with both pairs of opposite parallel sides. • Theorem 5-1: Opposite sides of the parallelogram are congruent. • Theorem 5-2: Opposite angles of a parallelogram are congruent. • Theorem 5-3: Diagonals of a parallelogram bisect each other. G 3 1 2. EFGH <E = <G 4 2 3. EFGH <3 = <4 E F

  3. Example: 38 X y 7 120 X equals 120 degrees because of OAC which says that Opposite Angles are congruent Y equals 22 because it supplementary to 120 So 60 minus 38 is 22

  4. 5.2 – Proving Quadrilaterals are Parallelograms • Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent , then the quadrilateral is a parallelogram. • Theorem 5-5: If one pair of opposite sides of a quadrilateral are both parallel, then the quadrilateral is a parallelogram. • Theorem 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • Theorem 5-7: If the diagonals of a quadrilateral bisects each other, then the quadrilateral is a parallelogram. T < S 3 2 << M << 1 4 < Q R

  5. 5.3 – Theorems Involving Parallel Lines • Theorem 5-8: If two lines are parallel, then all points on the line are equidistant from the other line. A B > L > M C D AB=CD

  6. 5.3 continued… • Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A X B 1 2 3 Y C 4 5 Z AX II BY II CZ and AB = BC XY = YZ

  7. Section 3 – theorems Involving Parallel Lines • Theorem 5-10 • A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side • Theorem 5-11 • The segment that joins the midpoints of two sides of a triangle • Is parallel to the third side • Is half as long as the third side

  8. Example: 2(4x+3) = 13x+1 8x+6=13x+1 5=5x X=1 5y+7 4x+3 5y+7 = 6y +3 4=y 6y+3 13x+1 X=1, Y=4

  9. 5-4 Special parallelograms • Rectangle  is a quadrilateral with four right angles. • Therefore, every rectangle is a parallelogram. • Rhombus  A quadrilateral with four congruent sides. • Therefore, every rhombus is a parallelogram.

  10. 5-4 Special parallelograms • Square  a quadrilateral with four right angles and four congruent sides. • Therefore, every square is a rectangle, a rhombus, and a parallelogram *** since rectangles, rhombuses, and squares are parallelograms, they have all the properties of parallelograms.

  11. 5-4 Special parallelograms • Theorem 5-12 • The diagonals of a rectangle are congruent • Theorem 5-13 • The diagonals of a rhombus are perpendicular • Theorem 5-14 • Each diagonal of a rhombus are perpendicular

  12. 5-4 Special parallelograms • Theorem 5-15 • The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices • Theorem 5-16 • If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle • Theorem 5-17 • If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus

  13. 5-5 Trapezoids • Trapezoid a quadrilateral with exactly one pair of parallel sides • Bases  the parallel sides • Legs  the other sides • Isosceles Trapezoid  a trapezoid with congruent legs • Base angles are congruent

  14. 5-5 Trapezoids • Theorem 5-18 • Base angles of an isosceles trapezoid are congruent *** Median (of a trapezoid )the segment that joins the midpoints of the legs • Theorem 5-19 • The Median of a Trapezoid • Is parallel to the bases • Has a length equal to the average of the base lengths

  15. Practice Questions: 1. 4x-y 7x-4y 5 9 2. z X y 75

  16. Practice Question 42 3. 26 3x-2y 4x+y

  17. Proof: T S J Q R K

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