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4.1 Quadrilaterals

4.1 Quadrilaterals. Quadrilateral. Parallelogram. Trapezoid. Isosceles Trapezoid. Rhombus. Rectangle. Square. 4.1 Properties of a Parallelogram. Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A. B. D. C.

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4.1 Quadrilaterals

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  1. 4.1 Quadrilaterals Quadrilateral Parallelogram Trapezoid IsoscelesTrapezoid Rhombus Rectangle Square

  2. 4.1 Properties of a Parallelogram • Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A B D C

  3. 4.1 Properties of a Parallelogram • Properties of a parallelogram: • Opposite angles are congruent • Opposite sides are congruent • Diagonals bisect each other • Consecutive angles are supplementary

  4. 4.1 Properties of a Parallelogram • In the following parallelogram:AB = 7, BC = 4, • What is CD? • What is AD? • What is mABC? • What is mDCB? A B D C

  5. 4.2 Proofs • Proving a quadrilateral is a parallelogram: • Show both pairs of opposite sides are parallel (definition) • Show one pair of opposite sides are congruent and parallel • Show both pairs of opposite sides are congruent • Show the diagonals bisect each other

  6. 4.2 Kites • Kite - a quadrilateral with two distinct pairs of congruent adjacent sides. • Theorem: In a kite, one pair of opposite angles is congruent.

  7. 4.2 Midpoint Segments • The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to ½ the length of the third side. A M N B C

  8. 4.3 Rectangle, Square, and Rhombus • Rectangle - a parallelogram that has 4 right angles. • The diagonals of a rectangle are congruent. • A square is a rectangle that has all sides congruent (regular quadrilateral).

  9. 4.3 Rectangle, Square, and Rhombus • A rhombus is a parallelogram with all sides congruent. • The diagonals of a rhombus are perpendicular.

  10. 4.3 Rectangles: Pythagorean Theorem • Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2Note: You can use this to get the length of the diagonal of a rectangle. a c b

  11. 4.4 The Trapezoid • Definition: A trapezoid is a quadrilateral with exactly 2 parallel sides. Base Leg Leg Base Base angles

  12. 4.4 The Trapezoid • Isosceles trapezoid: • 2 legs are congruent • Base angles are congruent • Diagonals are congruent

  13. 4.4 The Trapezoid A B • Median of a trapezoid:connecting midpointsof both legs M N D C

  14. 4.4 Miscellaneous Theorems • If 3 or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any transversal.

  15. 5.1 Ratios, Rates, and Proportions • Ratio - sometimes written as a:b Note: a and b should have the same units of measure. • Rate - like ratio except the units are different(example: 50 miles per hour) • Extended Ratio: Compares more than 2 quantitiesexample: sides of a triangle are in the ratio 2:3:4

  16. 5.1 Ratios, Rates, and Proportions • two rates or ratios are equal (read “a is to b as c is to d”) • Means-extremes property:product of the means = product of the extremeswhere a,d are the extremes and b,c are the means(a.k.a. “cross-multiplying”)

  17. 5.1 Ratios, Rates, and Proportions • b is the geometric mean of a & c • …..used with similar triangles

  18. 5.1 Ratios, Rates, and Proportions • Ratios – property 2:(means and extremes may be switched) • Ratios – property 3:Note: cross-multiplying will always work, these may lead to a solution faster sometimes

  19. 5.2 Similar Polygons • Definition: Two Polygons are similar  two conditions are satisfied: • All corresponding pairs of angles are congruent. • All corresponding pairs of sides are proportional.Note: “~” is read “is similar to”

  20. 5.2 Similar Polygons • Given ABC ~ DEF with the following measures, find the lengths DF and EF: E 10 5 B D A 6 4 C F

  21. 5.3 Proving Triangles Similar • Postulate 15: If 3 angles of a triangle are congruent to 3 angles of another triangle, then the triangles are similar (AAA) • Corollary: If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangle, then the triangles are similar. (AA)

  22. 5.3 Proving Triangles Similar • AA - If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangle, then the triangles are similar. • SAS~ - If a an angle of one triangle is congruent to an angle of a second triangle and the pairs of sides including the two angles are proportional, then the triangles are similar

  23. 5.3 Proving Triangles Similar • SSS~ - If the 3 sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar • CSSTP – Corresponding Sides of Similar Triangles are Proportional (analogous to CPCTC in triangle congruence proofs) • CASTC – Corresponding angles of similar triangles are congruent.

  24. 5.3 Proving Triangles Similar • (example proof using CSSTP)

  25. 5.4 Pythagorean Theorem • Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2 • Converse of Pythagorean Theorem: If for a triangle, c2 = a2 + b2 then the  opposite side c is a right angle and the triangle is a right triangle. c a b

  26. 5.4 Pythagorean Theorem • Pythagorean Triples: 3 integers that satisfy the Pythagorean theorem • 3, 4, 5 (6, 8, 10; 9, 12, 15; etc.) • 5, 12, 13 • 8, 15, 17 • 7, 24, 25

  27. 5.4 Classifying a Triangle by Angle • If a, b, and c are lengths of sides of a triangle with c being the longest, • c2 > a2 + b2the triangle is obtuse • c2 < a2 + b2the triangle is acute • c2 = a2 + b2the triangle is right c a b

  28. 5.5 Special Right Triangles • 45-45-90 triangle: • Leg opposite the 45 angle = a • Leg opposite the 90 angle = 45 a 90 45 a

  29. 5.5 Special Right Triangles • 30-60-90 triangle: • Leg opposite 30 angle = a • Leg opposite 60 angle = • Leg opposite 90 angle = 2a 60 2a a 30 90

  30. 5.6 Segments Divided Proportionally • If a line is parallel to one side of a triangle and intersects the other two sides, then it divides these sides proportionally A D E B C

  31. 5.6 Segments Divided Proportionally • When 3 or more parallel lines are cut by a pair of transversals, the transversals are divided proportionally by the parallel lines A D B E C F

  32. 5.6 Segments Divided Proportionally • Angle Bisector Theorem: If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the length of the 2 sides which form that angle. C A B D

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