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Quadrilaterals . Chapter 6. Parallelogram. A quadrilateral with both pairs of opposite sides parallel. Rhombus. A parallelogram with four congruent sides. Rectangle. A parallelogram with four right angles. Square . A parallelogram with four congruent side and four right angles.
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Quadrilaterals Chapter 6
Parallelogram • A quadrilateral with both pairs of opposite sides parallel
Rhombus • A parallelogram with four congruent sides.
Rectangle • A parallelogram with four right angles
Square • A parallelogram with four congruent side and four right angles.
Kite • A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent
Trapezoid • A quadrilateral with exactly one pair of parallel sides
Isosceles Trapezoid • A trapezoid whose nonparallel opposite sides are congruent.
Seven important types of quadrilaterals … • Parallelogram-has both pairs of opposite sides parallel • Rhombus-has four congruent sides • Rectangle-has four right angles • Square-has four congruent sides and four right angles • Kite-has two pairs of adjacent sides congruent and no opposite opposite sides congruent.
Continued…. • Trapezoid-has exactly one pair of parallel sides. (you have same side interior angles) • Isosceles trapezoid-is a trapezoid whose non-parallel opposite sides are congruent
Quadrilaterals Parallelograms Kites Rhombuses Rectangles Squares Trapezoids
Classifying by Coordinate Method • Do you remember the slope formula? • Do you remember the distance formula that finds the distance between two points?
Do you remember how to tell if two lines are parallel? • Do you remember how to tell if two lines are perpendicular?
Graph quadrilateral QBHA. First, 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 BH is parallel to QA because their slopes are equal. QB is not parallel to HA because their slopes are not equal. Classifying Quadrilaterals Determine the most precise name for the quadrilateral with vertices Q(–4, 4), B(–2, 9), H(8, 9), and A(10, 4).
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 Classifying Quadrilaterals (continued) One pair of opposite sides are parallel, so QBHA is a trapezoid. Because QB = HA, QBHA is an isosceles trapezoid.
In parallelogram RSTU, mR = 2x – 10 and m S = 3x + 50. Find x. If lines are parallel, then interior angles on the same side of a transversal are supplementary. m R + m S = 180 Draw quadrilateral RSTU. Label R and S. RSTU is a parallelogram. Given ST || RU Definition of parallelogram Classifying Quadrilaterals
(continued) (2x – 10) + (3x + 50) = 180 Substitute 2x – 10 for m R and 3x + 50 for m S. 5x + 40 = 180 Simplify. 5x = 140 Subtract 40 from each side. x = 28 Divide each side by 5. Classifying Quadrilaterals
Theorem 6-1: • Opposite sides of a parallelogram are congruent.
Theorem 6.2 • Opposite angles of a parallelogram are congruent.
Find the measure of the numbered angles for each parallelogram.
Theorem 6.3 • The diagonals of a parallelogram bisect each other.
Theorem 6.4 • If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Then If
Warm Up: • Given: • Prove:
Theorem 6-5: • If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Given: • Prove:
Theorem 6.6: • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Given: • Prove:
Theorem 6.7: • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Given: • Prove:
Theorem 6.8: • If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.
Given: • Prove:
Warm Up: • Given: • Prove:
Warm Up: Show algebraically that ABCD is a parallelogram A (-1, 4) B (3, 2) C (2, -2) D (-2, 0) Hint: Think of the choices you have that prove a quadrilateral is a parallelogram…
Theorem 6.9: • Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem 6.10: • Diagonals of a rhombus are perpendicular.
Theorem 6.11: • Diagonals of a rectangle are congruent.
Example 3 Use Diagonals of a Rhombus ABCDis a rhombus. Find the value ofeach variable.
Finding Angle Measure • MNPQ is a rhombus and m<N =120°. Find the measures of the numbered angles. N P 3 120° 4 1 2 M Q
Finding Angles Measures • Find the measures of the numbered angles in the rhombus. m<1 : m<2 and m<3: 50° 1 m<4: 3 4 2
Checkpoint 90 ANSWER 12 ANSWER 45 ANSWER Use Diagonals Find the value of x. 4. rhombusABCD 5. rectangleEFGH 6. squareJKLM
Given Rectangle EFGH, find the value of x, y, FG, GH, and EH.
Theorem 6.12: • If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.
Theorem 6.13: • If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.