Classifying Quadrilaterals Lesson 6-1 Notes A quadrilateral is a polygon with four sides. A parallelogram () is a quadrilateral with both pairs of opposite sides parallel. A rhombus is a  with four congruent sides. A rectangle is a  with four right angles. A square is a  with four congruent sides and four right angles. 6-1

Classifying Quadrilaterals Lesson 6-1 Notes A kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a trapezoid whose nonparallel sides are congruent. 6-1

Classifying Quadrilaterals Lesson 6-1 Notes 6-1

Classifying Quadrilaterals Lesson 6-1 Additional Examples Classifying a Quadrilateral Judging by appearance, classify ABCD in as many ways as possible. ABCD is a quadrilateral because it has four sides. It is a trapezoid because AB and DC appear parallel and AD and BC appear nonparallel. Quick Check 6-1

Classifying Quadrilaterals Lesson 6-1 Additional Examples Classifying by Coordinate Methods 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. First, find the slope of each side. slope of QB = slope of BH = slope of HA = slope of QA = 9 – 4 –2 – (–4) 5 2 = 9 – 9 8 – (–2) 4 – 9 10 – 8 = – 4 – 4 –4 – 10 BH is parallel to QA because their slopes are equal. QB is not parallel to HA because their slopes are not equal. 6-1

Classifying Quadrilaterals Lesson 6-1 Additional Examples (continued) One pair of opposite sides are parallel, so QBHA is a trapezoid. 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 Because QB = HA, QBHA is an isosceles trapezoid. Quick Check 6-1

Classifying Quadrilaterals Lesson 6-1 Additional Examples Using the Properties of Special Quadrilaterals In parallelogram RSTU, m R = 2x – 10 and m S = 3x + 50. Find x. Draw quadrilateral RSTU. Label R and S. RSTU is a parallelogram. Given Definition of parallelogram ST || RU m R + m S = 180 If lines are parallel, then interior angles on the same side of a transversal are supplementary. 6-1

Classifying Quadrilaterals Lesson 6-1 Additional Examples (continued) (2x – 10) + (3x + 50) = 180 Substitute 2x – 10 for m R and 3x + 50 for m S. 5x + 40 = 180 Simplify. Subtract 40 from each side. 5x = 140 x = 28 Divide each side by 5. Quick Check 6-1

Classifying Quadrilaterals Lesson 6-1 Lesson Quiz Judging by appearance, classify the quadrilaterals in Exercises 1 and 2 in as many ways as possible. 1. 2. quadrilateral, kite quadrilateral, parallelogram, rectangle, rhombus, square 4. What is the most precise name for the figure in Exercise 2? 3. What is the most precise name for the figure in Exercise 1? square kite 5. Find the values of the variables in the rhombus to the right. a = 60, x = 6, y = 2 6-1