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Math Project

Math Project. Ariella Lindenfeld & Nikki Colona. 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.

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Math Project

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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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