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Chapter 3

Chapter 3. Parallel and Perpendicular lines. Quadrilaterals. Chapter 6-1 Obj : To define and classify special types of quadrilaterals. QUADRILATERAL MAPPING. Helpful Facts. Sum of the interior angles: Triangle– 180 Quadrilateral – 360

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Chapter 3

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  1. Chapter 3 Parallel and Perpendicular lines

  2. Quadrilaterals Chapter 6-1 Obj: To define and classify special types of quadrilaterals

  3. QUADRILATERAL MAPPING

  4. Helpful Facts Sum of the interior angles: Triangle– 180 Quadrilateral – 360 Isosceles triangles- two congruent sides and base angles are congruent.

  5. Theorems for Parallelogram • Quadrilateral with both pairs of opposite sides are parallel • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other.

  6. Key Concepts Kites • Two pairs of adjacent sides are congruent • Diagonals are perpendicular

  7. Key Concepts Isosceles Trapezoids • One pair of parallel lines • Base angles are congruent • Non-parallel sides are congruent • Diagonals are congruent

  8. Key concepts Rhombus Properties • Both Pairs of opposite sides are parallel • Four congruent sides • Both pairs of opposite angles are congruent • Consecutive angles are supplementary. • Diagonals bisect each other • Diagonals are perpendicular • Diagonal bisects each angle 1 2 3 4

  9. Key Concepts Rectangles Properties • Both pairs of opposite sides are parallel • Both pairs of opposite sides are congruent • All right angles • Diagonals bisect each other • Diagonals are congruent

  10. Square • All sides are congruent • Both pairs of opposite sides are parallel • All right angles (all angles are congruent) • Diagonals bisect each other • Diagonals are congruent • Diagonals are perpendicular bisectors

  11. Key Concept Distance between the two are congruent DF

  12. Find the distance between the points to the nearest tenth. 1.M (2, –5), N (–7, 1) 2.X (0, 6), Y (4, 9) Find the slope of the line through each pair of points. 1.d = √ (x2 – x1)2 + (y2 – y1)2 = √ ( –7 – 2)2 + (1 – (–5))2 = √( –9)2 + 62 = √ 81 + 36 = √ 117 = 10.8 m= 9-6 = 3 4-0 4

  13. Next, use the distance formula to see whether any pairs of sides are congruent. QB = ( –2 – ( –4))2 + (9 – 4)2 = 4 + 25 = 29 HA = (10 – 8)2 + (4 – 9)2 = 4 + 25 =29 BH = (8 – (–2))2 + (9 – 9)2 = 100 + 0 =10 QA = (– 4 – 10)2 + (4 – 4)2 = 196 + 0 = 14 Find the slope of each side. slope of QB = slope of BH = slope of HA = slope of QA = 4 – 4 –4 – 10 9 – 9 8 – (–2) 4 – 9 10 – 8 9 – 4 –2 – (–4) 5 2 5 2 = = = = – 0 0 Determine the most precise name for the quadrilateral with vertices Q(–4, 4), B(–2, 9), H(8, 9), and A(10, 4). Graph quadrilateral QBHA. QBHA is an isosceles trapezoid.

  14. In parallelogram RSTU, m<R = 2x – 10 and m< S = 3x + 50. Find x. 2X-10 +3X+50 = 180 5X +40 = 180 -40 -40 5X = 140 5 5 X = 28 R 2X-10 U 3X +5 0 S T

  15. Find the measure of the missing angle of the isosceles trapezoid. Y= 156- base angles are congruent. w = 180-156 = 24 - adjacent angles are supplementary (same-side interior angles) Z =24 – base angles are congruent

  16. Find the missing measures of the kite. 1 = Congruent to 72- isosceles triangle have congruent base angles 2= 90 diagonals are perpendicular 3=180-(90+72) =18 sum of the interior angles of a triangle is 180 degrees.

  17. Use KMOQto find m O. 180-35 = 145 mO = 145

  18. Find the value of x in ABCD. Then find m A. Oppostite angles are congruent 135-x = x + 15 + x +x = 2x + 15 -15 -15 120 = 2x 2 60 = x D = 135-60 = 75 A and  D are supplementary 180 – 75 = 105 A = 105

  19. Find the values of x and y in KLMN. Diagonals bisect each other. 2x + 5 = 5y 7y-16 = x 2(7y-16) + 5 = 5y 14y – 32 + 5 = 5y 14y – 27 = 5y -27 = -9y 3 = y X = 7( 3) – 16 = 21 -16 = 5

  20. If AC = 3 and BD = 6, find BF. 6(2) = 12 BF = 12

  21. Try: Find y in .Then find mE, mF, mG, m H. 6y + 4 = 3y + 37 3y = 33 y = 11 mE = mG = 70  mF= m H= 110

  22. Try: Find the Values of a and b. a = b +2 b+10 = 2a -8 Using substitution: b +10 = 2(b+2) -8 b+10 = 2b +4 – 8 10 = b – 4 14 = b a = 14 +2 = 16

  23. Try: n 2.5(3) = 7.5

  24. Find the measures of all missing angles in the rhombus. 1 = 78- diagonal bisects the angles. 2 = 90- diagonals are Perpendicular. 3 = 12- sum of the interior angles of a triangle is 180. 4= 78- opposite angles are congruent.

  25. Try: Find all the measured angles in the rhombus 1 = 90- diagonals are perpendicular 2 = 50- diagonals the angles 3 = 50 opposite angles are congruent 4 = 40 sum of the interior angles of a triangle is 180

  26. Find the length of the diagonal. 8x + 2 = 5x + 11 3x + 2 = 11 3x = 9 X = 3 8(3) +2 = 26 5(3) +11 = 26 both diagonals are 26 units

  27. 5y – 9 = y +5 4y – 9 = 5 4y = 14 y = 7/2 or 3.5 3.5 + 5 = 8.5 units 5(3.5) – 9 = 8.5 units diagonals are 8.5 units

  28. 6-6 Placing figures in the Coordinate Plane To name coordinates of special figures by using their properties.

  29. Find coordinate B for the parallelogram below. Since it is a parallelogram, It must have the same y-coordinate q. The x- coordinate is –x-p Therefore, B(-x-p,q)

  30. Find the coordinate Q for the parallelogram below. Q(b+s, c)

  31. Review Unit test Summative What did you learn today? What is still confusing?

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