G.9 Quadrilaterals Part 1 Parallelograms Modified by Lisa Palen

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

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.

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. That means that . So, if AB = 7, then _____ = 7?

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

Consecutive Angles Each angle is consecutive to two other angles. A is consecutive with B and 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 = _____? Consecutive INTERIOR Angles are Supplementary!

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

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!

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

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 = _____ .

Tests for Parallelograms 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 . 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 .

Theorem: Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. G H 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 . 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 is a parallelogram. y+2 6x = 4x + 8 2x = 8 x = 4 2y = y + 2 y = 2 6x 4x+8 2y Find the value of x and y that ensure the quadrilateral is a parallelogram. Example 2: 2x + 8 = 120 2x = 112 x = 56 5y + 120 = 180 5y = 60 y = 12 (2x + 8)° 5y° 120°

Part 3 Rectangles Lesson 6-3: Rectangles

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

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

Properties of Rectangles 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 E D C B A Lesson 6-3: Rectangles

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

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

Lesson 6-4: Rhombus & Square 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

Lesson 6-4: Rhombus & Square 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

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

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

Lesson 6-4: Rhombus & Square Rhombus Examples ….. Given: ABCD is a rhombus. Complete the following. If AB = 9, then AD = ______. If m<1 = 65, the m<2 = _____. m<3 = ______. If m<ADC = 80, the m<DAB = ______. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____. 9 units 65° 90° 100° 10 Lesson 6-4: Rhombus & Square

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.

Lesson 6-4: Rhombus & Square Squares – Examples…… Given: ABCD is a square. Complete the following. If AB = 10, then AD = _____ and DC = _____. If CE = 5, then DE = _____. m<ABC = _____. m<ACD = _____. m<AED = _____. 10 units 10 units 5 units 90° 45° 90° Lesson 6-4: Rhombus & Square

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

Lesson 6-5: Trapezoid & Kites 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 Trapezoid Leg Leg Base Lesson 6-5: Trapezoid & Kites

Lesson 6-5: Trapezoid & Kites 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. Median Lesson 6-5: Trapezoid & Kites

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

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

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

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