Unit 5 - Quadrilaterals MM1G3 d

Essential Quesitons

Parallelograms Examples

K L J H A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

There are 5 properties associated with parallelograms. 1)Both pairs of opposite sides are parallel. 2)Both pairs of opposite sides are congruent. 3)Both pairs of opposite angles are congruent. 4)Consecutive angles are supplementary. 5)The diagonals bisect each other.

K L J H Given: JKLH is a parallelogram Prove: Since diagonal KH is also a transversal, angle 1 and angle 2 are congruent as they are alternate interior angles. Likewise, angle 3 and angle 4 are congruent. Since KH is congruent to itself, triangle JKH and triangle LHK are congruent by ASA (By drawing diagonal JL, it can similarly be proven that triangle JKL and triangle LHJ are congruent and, thus,.)

Example 1: Find the perimeter of parallelogram. Solution: Opposite sides of a parallelogram are congruent. So, ZY = 12 cm and WZ = 8 cm. Therefore, P = 12cm + 8cm + 12cm + 8cm. P = 40 cm Z Y X W 8 cm 12 cm

Given: JKLH is a parallelogram Prove: K L J H

Given: JKLH is a parallelogram Prove: Diagonals KH and JL bisect each other Statements Reasons K L J H M 6 5

Example 2: is a parallelogram. What is the length, in units, of ? DC BA 2y 4x y+2 3x

Summary A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Properties of parallelograms: 1)Both pairs of opposite sides are parallel. 2)Both pairs of opposite sides are congruent. 3)Both pairs of opposite angles are congruent. 4)Consecutive angles are supplementary. 5)The diagonals bisect each other.

Investigate Parallelograms

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Rectangles Examples

A rectangle is a parallelogram with four right angles. Since ABCD is also a parallelogram, it has the following properties: D CB A

A BC D A rectangle has an additional property – the diagonals are congruent. In rectangle ABCD above,.

To investigate these properties, click below. s.html As you drag point A or point B, you should notice the values in the left margin changing. By clicking on various pieces of the rectangle, the measurement will be highlighted. You can verify the properties of the rectangle. Notice that as you drag point A or point B, the properties of the rectangle remain. For example, the diagonals are always congruent and bisect each other.

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Example 2:Solution:

Summary A rectangle is a parallelogram with four right angles. Properties of a rectangle: 1)Both pairs of opposite sides are parallel. 2)Both pairs of opposite sides are congruent. 3)Both pairs of opposite angles are congruent. 4)Consecutive angles are supplementary. 5)The diagonals bisect each other. 6)The diagonals are congruent

Investigate Rectangles

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Essential Questions

Rhombus Examples

A rhombus is a parallelogram with all sides congruent. DC BA Since ABCD is also a parallelogram, the following statements are true.

A rhombus has two additional properties. the diagonals are perpendicular the diagonals bisect opposite angles DC BA

To investigate these properties, click below. As you drag point B or point C, you should notice the measurements changing. However, you should see that the diagonals are always perpendicular. You should also see each of the angles are bisected, no matter the size of the rhombus. Notice that as you drag point B or point C, the rhombus will have the 5 properties for a parallelogram as well. For example, the diagonals bisect each other and the opposite angles are congruent.

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Example 2:

Summary A rhombus is a parallelogram with all sides congruent. Properties of a rhombus: 1)Both pairs of opposite sides are parallel. 2)Both pairs of opposite sides are congruent. 3)Both pairs of opposite angles are congruent. 4)Consecutive angles are supplementary. 5)The diagonals bisect each other. 6)The diagonals are perpendicular. 7)The diagonals bisect opposite angles.

Investigate Rhombi

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Squares Examples

A square is a parallelogram with four right angles and all sides congruent. Notice that a square meets the requirements for both a rectangle (four right angles) and a rhombus (all sides congruent). DC BA

To investigate the properties of a square, click below. tml As you drag point B, you should notice the measurements changing. However, you should see that the diagonals are always perpendicular and congruent. You should also see that the interior angles of the square are always Notice that as you drag point B, the square will have the 5 properties for a parallelogram as well. For example, the diagonals bisect each other and the consecutive angles are supplementary.

Example 1:

Example 2: B ZY XW The area of the square WXYZ is 100 square meters. Find BX.

Summary A square is a parallelogram with four right angles and all sides congruent. Properties of a square: 1)Both pairs of opposite sides are parallel. 2)Both pairs of opposite sides are congruent. 3)Both pairs of opposite angles are congruent. 4)Consecutive angles are supplementary. 5)The diagonals bisect each other. 6)The diagonals are congruent. 7)The diagonals are perpendicular. 8)The diagonals bisect opposite angles.

Investigate Squares!!

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Trapezoids

Trapezoids Examples

base leg AB CD A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. In the trapezoid at the right, the bases are The non-parallel sides are the legs. In the trapezoid above, the legs are A trapezoid has two pairs of base angles.

Example 1: X TW SR 87° 23° 93° 64°

 If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.  If a trapezoid is isosceles, then each pair of base angles is congruent. A B CD  If a trapezoid is isosceles, then the diagonals are congruent.

C ED F 115° 10 Example 2: CDEF is an isosceles trapezoid with CD = 10 and m ∠ E = 115°. Find EF, m ∠ C, m ∠ D, and m ∠ F.

Example 2: Trapezoid ABCD is an isosceles trapezoid. If BD = 2y + 3 and AC = 4y - 5, find BD and AC. A B CD

A B CD Example 3: ABCD is an isosceles trapezoid. Prove that  ACD is congruent to  BDC.

SUMMARY  A trapezoid is a quadrilateral with exactly one pair of parallel sides. These parallel sides are called the bases.  An isosceles trapezoid is a trapezoid where the non- parallel sides are congruent. These non-parallel sides are called the legs.  Each pair of base angles in an isosceles trapezoid are congruent.  The diagonals of an isosceles trapezoid are congruent.

Investigate Trapezoids

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KITES

Kites Examples

A kite is a quadrilateral with two distinct pairs of adjacent congruent sides. A B C D

Example 1: ABCD is a kite. If m ∠ D = 120°, and m ∠ A = 80°, what is m ∠ C? A B C D

The diagonals of a kite are perpendicular to each other. E D C B A 12

The longer diagonal of a kite bisects the shorter diagonal. E D C B A 12

Example 2: In kite ABCD, AE = 15 and BD = 40. Find AB. E

SUMMARY  A kite is a quadrilateral with two distinct pairs of adjacent congruent sides. Opposite sides are not congruent.  The diagonals of a kite are perpendicular to each other.  The longer diagonal of a kite bisects the shorter diagonal.

Investigate Kites!

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Classifying Quadrilaterals Activity

Math I; Unit 5 Test Review

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