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G.9 Quadrilaterals Part 1 Parallelograms Modified by Lisa Palen

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Definition A parallelogram is a quadrilateral whose opposite sides are parallel. Its symbol is a small figure:

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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.

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Basic Properties There are four basic properties of all parallelograms. These properties have to do with the angles, the sides and the diagonals.

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Opposite Sides Theorem Opposite sides of a parallelogram are congruent. That means that. So, if AB = 7, then _____ = 7?

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Opposite Angles One pair of opposite angles is A and C. The other pair is B and D.

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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 = ______.

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Consecutive Angles Each angle is consecutive to two other angles. A is consecutive with B and D.

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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

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Diagonals Diagonals are segments that join non- consecutive vertices. For example, in this diagram, the only two diagonals are.

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Diagonal Property When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal. So, P is the midpoint of. Therefore, they bisect each other; so and. But, the diagonals are not congruent!

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Diagonal Property Theorem The diagonals of a parallelogram bisect each other.

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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.

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Examples 1. Draw HKLP. 2. Complete: HK = _______ and HP = ________. 3. m

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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

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Part 2 Tests for Parallelograms

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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.

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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.

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Proving Quadrilaterals as Parallelograms If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 1: H G E F If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 2:

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Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 3: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 4: H G E F M then Quad. EFGH is a parallelogram. EM = GM and HM = FM

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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.

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Examples …… Find the values of x and y that ensures the quadrilateral is a parallelogram. Example 1: 6x 4x+8 y+2 2y 6x = 4x + 8 2x = 8 x = 4 2y = y + 2 y = 2 Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram. 120° 5y° (2x + 8)° 2x + 8 = 120 2x = 112 x = 56 5y + 120 = 180 5y = 60 y = 12

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Lesson 6-3: Rectangles23 Part 3 Rectangles

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Lesson 6-3: Rectangles24 Rectangles Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Definition:A rectangle is a quadrilateral with four right angles. Is a rectangle is a parallelogram? Thus a rectangle has all the properties of a parallelogram. Yes, since opposite angles are congruent.

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Lesson 6-3: Rectangles25 Properties of Rectangles Therefore, ∆ AEB, ∆ BEC, ∆ CED, and ∆ AED are isosceles triangles. If a parallelogram is a rectangle, then its diagonals are congruent. E D C B A Theorem: Converse: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

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Lesson 6-3: Rectangles26 Properties of Rectangles E D C B A Parallelogram Properties: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Plus: All angles are right angles. Diagonals are congruent. Also: ∆ AEB, ∆ BEC, ∆ CED, and ∆ AED are isosceles triangles

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Lesson 6-3: Rectangles27 Examples……. 1.If AE = 3x +2 and BE = 29, find the value of x. 2.If AC = 21, then BE = _______. 3. If m<1 = 4x and m<4 = 2x, find the value of x. 4.If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6. m<1=50, m<3=40, m<4=80, m<5=100, m<6=40 10.5 units x = 9 units x = 18 units 6 5 4 32 1 E D C B A

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Lesson 6-4: Rhombus & Square28 Part 4 Rhombi and Squares

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Lesson 6-4: Rhombus & Square29 Rhombus Definition:A rhombus is a quadrilateral with four congruent sides. 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. ≡ ≡ Is a rhombus a parallelogram? Yes, since opposite sides are congruent.

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Lesson 6-4: Rhombus & Square30 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.

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Lesson 6-4: Rhombus & Square31 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!

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Lesson 6-4: Rhombus & Square32 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! ≡ ≡

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Lesson 6-4: Rhombus & Square33 Rhombus Examples..... Given: ABCD is a rhombus. Complete the following. 1.If AB = 9, then AD = ______. 2.If m<1 = 65, the m<2 = _____. 3.m<3 = ______. 4.If m

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34 Square 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. 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.

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Lesson 6-4: Rhombus & Square35 Squares – Examples…... Given: ABCD is a square. Complete the following. 1.If AB = 10, then AD = _____ and DC = _____. 2.If CE = 5, then DE = _____. 3.m

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Lesson 6-5: Trapezoid & Kites36 Part 5 Trapezoids and Kites

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Lesson 6-5: Trapezoid & Kites37 Trapezoid A quadrilateral with exactly one pair of parallel sides. Definition: Base Leg Trapezoid The parallel sides are called bases and the non-parallel sides are called legs. Leg Base

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Lesson 6-5: Trapezoid & Kites38 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. Median Median of a Trapezoid

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Lesson 6-5: Trapezoid & Kites39 Isosceles Trapezoid A trapezoid with congruent legs. Definition: Isosceles trapezoid

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Lesson 6-5: Trapezoid & Kites40 Properties of Isosceles Trapezoid 2. The diagonals of an isosceles trapezoid are congruent. 1. Both pairs of base angles of an isosceles trapezoid are congruent. A B C D

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Lesson 6-5: Trapezoid & Kites41 Kite A quadrilateral with two distinct pairs of congruent adjacent sides. Definition: Theorem: Diagonals of a kite are perpendicular.

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Lesson 6-5: Trapezoid & Kites42 Isosceles Trapezoid Quadrilaterals Rectangle Parallelogram Rhombus Square Flow Chart Trapezoid Kite

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