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Good Morning Unit 6 Quadrilaterals Part 1 Parallelograms Definition • A parallelogram is a quadrilateral whose opposite sides are parallel. B C D A • Its symbol is a small figure: AB CD and BC AD Naming a Parallelogram • A parallelogram is named using all four vertices. • You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction. • For example, this can be either ABCD or ADCB. B A C D Basic Properties • There are four basic properties of all parallelograms. • These properties have to do with the angles, the sides and the diagonals. Opposite Sides Theorem Opposite sides of a parallelogram are congruent. B C A D • That means that AB CD and BC . AD • So, if AB = 7, then _____ = 7? Opposite Angles • One pair of opposite angles is A and C. The other pair is B and D. B A C D Opposite Angles Theorem Opposite angles of a parallelogram are congruent. • Complete: If m A = 75 and m B = 105, then m C = ______ and m D = ______ . B A C D Consecutive Angles • Each angle is consecutive to two other angles. A is consecutive with B and D. B A C D Consecutive Angles in Parallelograms Theorem Consecutive angles in a parallelogram are supplementary. • Therefore, m A + m B = 180 and m A + m D = 180. • If m<C = 46, then m B = _____? B A C D Consecutive INTERIOR Angles are Supplementary! Diagonals • Diagonals are segments that join nonconsecutive vertices. • For example, in this diagram, the only two diagonals are . AC and BD B A C D Diagonal Property When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal. • So, P is the midpoint of AC and BD. • Therefore, they bisect each other; so AP PC and BP PD . • But, the diagonals are not congruent! AC BD B C P A D Diagonal Property Theorem The diagonals of a parallelogram bisect each other. B C P A D Parallelogram Summary • By its definition, opposite sides are parallel. Other properties (theorems): • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • The diagonals bisect each other. Examples • 1. Draw HKLP. • 2. Complete: HK = _______ and HP = ________ . • 3. m<K = m<______ . • 4. m<L + m<______ = 180. • 5. If m<P = 65, then m<H = ____, m<K = ______ and m<L =______ . Examples (cont’d) • • • • • 6. Draw in the diagonals. They intersect at M. 7. Complete: If HM = 5, then ML = ____ . 8. If KM = 7, then KP = ____ . 9. If HL = 15, then ML = ____ . 10. If m<HPK = 36, then m<PKL = _____ . Part 2 Tests for Parallelograms Review: Properties of Parallelograms • • • • • Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other. How can you tell if a quadrilateral is a parallelogram? • Defn: A quadrilateral is a parallelogram iff opposite sides are parallel. • Property If a quadrilateral is a parallelogram, then opposite sides are parallel. • Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram. Proving Quadrilaterals as Parallelograms Theorem 1: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram . If EF GH; FG EH, then Quad. EFGH is a parallelogram. H G Theorem 2: E F If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram . If EF GH and EF || HG, then Quad. EFGH is a parallelogram. Theorem: Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. G H If H F and E G, then Quad. EFGH is a parallelogram. M Theorem 4: E F If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram . If M is the midpo int of EG and FH then Quad. EFGH is a parallelogram. EM = GM and HM = FM 5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are || . [definition] 2. Show that both pairs of opposite sides are . 3. Show that one pair of opposite sides are both || and . 4. Show that both pairs of opposite angles are . 5. Show that the diagonals bisect each other . Examples …… Example 1: Find the values of x and y that ensures the quadrilateral y+2 is a parallelogram. 6x = 4x + 8 2y = y + 2 6x 4x+8 2x = 8 y=2 2y x=4 Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram. 2x + 8 = 120 5y + 120 = 180 (2x + 8)° 5y° 120° 2x = 112 5y = 60 x = 56 y = 12 Part 3 Rectangles Lesson 6-3: Rectangles 24 Rectangles Definition: A rectangle is a quadrilateral with four right angles. Is a rectangle is a parallelogram? Yes, since opposite angles are congruent. Thus a rectangle has all the properties of a parallelogram. • • • • • Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Lesson 6-3: Rectangles 25 Properties of Rectangles Theorem: If a parallelogram is a rectangle, then its diagonals are congruent. Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles. A B E D C Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle. Lesson 6-3: Rectangles 26 Properties of Rectangles Parallelogram Properties: Opposite sides are parallel. Opposite sides are congruent. A Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Plus: D All angles are right angles. Diagonals are congruent. B E C Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles Lesson 6-3: Rectangles 27 Examples……. 1. If AE = 3x +2 and BE = 29, find the value of x. x = 9 units 10.5 units 2. If AC = 21, then BE = _______. 3. If m<1 = 4x and m<4 = 2x, find the value of x. x = 18 units 4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6. A B 2 m<1=50, m<3=40, m<4=80, m<5=100, m<6=40 3 1 4 E 5 6 D Lesson 6-3: Rectangles C 28 Part 4 Rhombi and Squares Lesson 6-4: Rhombus & Square 29 Rhombus Definition: A rhombus is a quadrilateral with four congruent sides. Is a rhombus a parallelogram? Yes, since opposite sides are congruent. Since a rhombus is a parallelogram the following are true: • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Lesson 6-4: Rhombus & Square 30 Rhombus Note: The four small triangles are congruent, by SSS. This means the diagonals form four angles that are congruent, and must measure 90 degrees each. So the diagonals are perpendicular. This also means the diagonals bisect each of the four angles of the rhombus So the diagonals bisect opposite angles. Lesson 6-4: Rhombus & Square 31 Properties of a Rhombus Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects a pair of opposite angles. Note: The small triangles are RIGHT and CONGRUENT! Lesson 6-4: Rhombus & Square 32 Properties of a Rhombus Since a rhombus . is a parallelogram the following are true: • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • All four sides are congruent. • Diagonals are perpendicular. • Diagonals bisect opposite angles. • Also remember: the small triangles are RIGHT and CONGRUENT! Lesson 6-4: Rhombus & Square 33 Rhombus Examples ..... Given: ABCD is a rhombus. Complete the following. A 1. 2. 3. 9 units If AB = 9, then AD = ______. B 1 2 3 65° If m<1 = 65, the m<2 = _____. 4 m<3 = ______. 90° 5 D 4. 100° If m<ADC = 80, the m<DAB = ______. 5. 10 If m<1 = 3x -7 and m<2 = 2x +3, then x = _____. Lesson 6-4: Rhombus & Square E C 34 Square Definition:A square is a quadrilateral with four congruent angles and four congruent sides. Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals. • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • Four right angles. • Four congruent sides. • Diagonals are congruent. • Diagonals are perpendicular. • Diagonals bisect opposite angles. 35 Squares – Examples…... Given: ABCD is a square. Complete the following. 1. 10 unitsand DC = _____. 10 units If AB = 10, then AD = _____ A 2. 5 units If CE = 5, then DE = _____. B 1 2 3 4 E 3. 90° m<ABC = _____. 8 7 4. 45° m<ACD = _____. 5. 90° m<AED = _____. D Lesson 6-4: Rhombus & Square 6 5 C 36 Part 5 Trapezoids and Kites Lesson 6-5: Trapezoid & Kites 37 Trapezoid Definition: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases and the non-parallel sides are called legs. Base Leg Trapezoid Leg Base Lesson 6-5: Trapezoid & Kites 38 Median of a Trapezoid The median of a trapezoid is the segment that joins the midpoints of the legs. (It is sometimes called a midsegment.) • Theorem - The median of a trapezoid is parallel to the bases. • Theorem - The length of the median is one-half the sum of the lengths of the bases. b 1 1 median (b1 b2 ) 2 Median b2 Lesson 6-5: Trapezoid & Kites 39 Isosceles Trapezoid Definition: A trapezoid with congruent legs. Isosceles trapezoid Lesson 6-5: Trapezoid & Kites 40 Properties of Isosceles Trapezoid 1. Both pairs of base angles of an isosceles trapezoid are congruent. A B and D C 2. The diagonals of an isosceles trapezoid are congruent. AC DB B A D Lesson 6-5: Trapezoid & Kites C 41 Kite Definition: A quadrilateral with two distinct pairs of congruent adjacent sides. Theorem: Diagonals of a kite are perpendicular. Lesson 6-5: Trapezoid & Kites 42 Flow Chart Quadrilaterals Kite Parallelogram Trapezoid Rhombus Rectangle Isosceles Trapezoid Square Lesson 6-5: Trapezoid & Kites 43