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Parallelograms Key Ideas A parallelogram is a quadrilateral that has both pairs of opposite sides parallel. Besides having parallel sides, all parallelograms have the following properties:
Properties of Parallelograms Definition A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Besides having opposite sides parallel, parallelograms have several properties that can be proven using congruent triangles and parallel lines relationships. The Common Core Standards state that students should be able to prove theorems about parallelograms. Some of the theorem proofs that follow in this section are mentioned as examples. Your best strategy is to understand the basic approach and tools used in the proofs, not to memorize the proofs word for word. Many variations of these proofs are equally valid. Theorem Each diagonal in a parallelogram divides the parallelogram into two congruent triangles. Theorem Opposite sides of a parallelogram are congruent. Theorem Opposite angles in a parallelogram are congruent. The proof for these theorems follows. The strategy is to use the alternate interior angles formed by the diagonal to show the two triangles are congruent by ASA. Then the rest follows by CPCTC. The second pair of opposite angles can be proven congruent using the same method with the other diagonal. Given: parallelogram ABCD and diagonal Prove: △ABC △DCA
Theorem Consecutive angles in a parallelogram are supplementary. The proof that consecutive angles are supplementary is simply a statement of the same-side interior angle relationship for angles formed by parallel lines and a transversal. Given: parallelogram ABCD Prove: ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A are supplementary
Theorem The diagonals of a parallelogram bisect each other. Given: parallelogram ABCD, diagonals and intersect at E Prove: diagonals and bisect each other
A summary of the basic parallelogram properties is shown in this Figure. Angle and side relationships in parallelograms
If no figure is provided, be sure to sketch one. You can usually start labeling vertices anywhere on the parallelogram and can proceed in any direction as long as no vertices are skipped. The order of vertices must match the order listed in the name of the parallelogram.
The exception is parallelograms with one pair of parallel sides. The parallel sides on the figure must be those specified in the problem. Example 1 In parallelogram QRST, m∠Q is 18° more than twice m∠R. Find the measures of ∠Q and ∠R. Solution:
Example 2 In parallelogram HIKE, m∠H = (6x − 11)° and m∠K = (2x + 29)°. Find m∠H and m∠K. Solution:
Example 3 Given parallelogram ABCD, , m∠EFB = 70°, and m∠C = 65°. Find m∠DFB and m∠CBF.
Solution:
Example 4 In parallelogram ABCD, diagonals AC and BD intersect at E. If AE = (6x − 12) and EC = (2x + 28), find AC. Solution:
Proofs with Parallelograms Key Ideas A quadrilateral must be a parallelogram if any one of the following are true:
Requirements for Proving a Quadrilateral is a Parallelogram Any one of the following properties can be used to prove a quadrilateral is a parallelogram. You usually show one of the other properties first in the process of showing both pairs of triangles are congruent. One additional useful shortcut that is added to the list is “one pair of opposite sides is congruent and parallel.” Properties Sufficient to Prove a Quadrilateral Is a Parallelogram
Math Fact A kite is a counterexample that illustrates why one diagonal forming congruent triangles is not enough to prove that a quadrilateral is a parallelogram. Kites have two distinct pairs of consecutive sides congruent. Therefore the diagonal forms two congruent triangles.
In the kite shown, AB = BC, AD = CD, and △ABD ≅ △BCD. However, the figure is clearly not a parallelogram. Kites also have perpendicular diagonals.
A dart is similar to a kite except one of the angles is concave. Diagonal in dart MNOP lies outside the dart.
Parallelogram Proofs
When you are asked to prove a quadrilateral is a parallelogram, look for these common strategies:
Math Fact Mathematicians often use the words “sufficient” and “necessary” to describe the conditions needed for a statement to be true. A sufficient condition is all that it needed—no further proof is required. A necessary condition is required, but it is not enough to prove the statement is true.
For example, if given quadrilateral ABCD, two pairs of opposite sides being parallel is sufficient to prove that ABCD is a parallelogram. The first example is a proof of the sufficient condition “one pair of sides are congruent and parallel.” Example 1 Given: quadrilateral ABCD with , Prove: ABCD is a parallelogram
Example 2 Given: ∠E ≅ ∠G and ∠EDF ≅ ∠GFD Prove: DEFG is a parallelogram
Example 3 Given: quadrilateral JUMP with ∠J supplementary to ∠P and Prove: JUMP is a parallelogram Solution: sketch: because same side interior angles ∠J and ∠P are supplementary. Opposite sides and are both parallel and congruent, making JUMP a parallelogram. Example 4 Given: parallelogram AECF with extended to B, with extended to D, and Prove: ABCD is a parallelogram Solution: Strategy—Prove the triangles are congruent by SAS. Then use CPCTC and alternate interior angles.
Properties of Special Parallelograms Key Ideas The rhombus (plural is rhombi), rectangle, and square are special parallelograms. In addition to the parallelogram properties, the special parallelograms have the following properties.
Rectangles Definition A rectangle is a parallelogram with a right angle, as shown in Figure 9.2. Rectangle ABCD Properties of Rectangles
The Common Core Standards specify that students should be able to prove various theorems about parallelograms. As with the parallelogram property proofs, you should be familiar with the approach. However, don’t try to memorize the proof word for word.
The property of congruent diagonals can be proven by showing that the two overlapping triangles formed by the diagonals are congruent. Example 1 Given: rectangle ABCD Prove:
Example 2 In rectangle TRUE, diagonals and intersect at A. If TA = (10x − 12) and RA = (4x + 24), find the length of diagonal . Solution: Since diagonals in a rectangle are congruent and bisect each other, TA = AU = RA = AE. Example 3 In rectangle ABCD, diagonals and intersect at E. If m∠DAE = 68°, find m∠AEB, m∠ABE, and m∠CBE.
Rhombi Definition A rhombus (plural rhombi) is a parallelogram with a pair of consecutive congruent sides. Properties of a Rhombus
The first two properties of a rhombus are a direct result from the definition of the rhombus. The last two properties can be proven using the first two.
Figure shows a rhombus. Rhombus ABCD Example 1 In rhombus ABCD, m∠DCA = 27°. Find m∠BCA and m∠CBD. Solution:
Math Fact Since the diagonals of a rhombus are perpendicular, they will form right triangles. Be on the lookout for applications of the Pythagorean theorem when working with rhombi. Example 2 Rhombus ABCD has diagonals and intersecting at E. If EB = 3 and AE = 9, find the perimeter of ABCD. Express your answer in simplest radical form. Solution:Apply the Pythagorean theorem to △ABE
Squares Definition A square is a parallelogram with a right angle and a pair of consecutive congruent sides. Properties of a Square
Figure shows a square.
Square ABCD Math Fact Since the diagonals of a square are congruent, bisect each other, and are perpendicular, the four triangles they form are all isosceles right triangles with base angles measuring 45°. Example 1 RSTU is a square with diagonals and intersecting at W. Find m∠WUR and m∠TWS.
Example 2 The diagonal of a square measures 12 cm. What is the area of the square in cm2? Solution:Use the Pythagorean theorem with right triangle △ABC to find the side length of the square.
Trapezoids Key Ideas A trapezoid is a quadrilateral with exactly one pair of parallel sides. The same side interior angles formed by the two parallel bases are supplementary. An isosceles trapezoid has congruent legs, diagonals, and base angles. Properties and Definitions A trapezoid is a quadrilateral with at least one pair of parallel sides. A trapezoid with one pair of parallel sides is shown in Figure 9.5. The parallel sides are called the bases and the non-parallel sides are called the legs.
The same side interior angles form when the parallel bases are supplementary. Trapezoid with parallel bases and and with legs and Definition Isosceles trapezoid—a trapezoid with congruent legs. Properties of an Isosceles Trapezoid
Figure shows isosceles trapezoid ABCD with congruent legs and .
Isosceles trapezoid ABCD Example 1 In isosceles trapezoid ABCD, m∠A = 72° and m∠CDB = 28°. Find m∠1, m∠2, m∠3, and m∠4. Solution:
Example 2 In isosceles trapezoid ABCD, , AC = (5x – 10), BD = (2x + 2), m∠BAD = (3y − 12), and m∠ABC = (2y + 15). Find AC, m∠BAD, and m∠ADC.
Classifying Quadrilaterals and Proofs Involving Special Quadrilaterals All quadrilaterals can be classified according to their properties. The classifications we have seen are the general quadrilateral, trapezoid, parallelogram, rectangle, rhombus, and square. It is often helpful to consider a Venn diagram showing these classifications, as shown in Figure. Squares have all the properties of both rectangles and rhombi, so they are considered to be rectangles and rhombi as well as squares.
Venn diagram of quadrilateral classifications
For the special parallelograms, once you have shown that it is a parallelogram, any one of the additional properties is sufficient.
The following table shows a summary of the properties needed to prove each of the parallelogram and trapezoid types.
Example 1 A quadrilateral has perpendicular diagonals that bisect each other. The quadrilateral must be which of the following figures? rectangle rhombus square isosceles trapezoid Solution: Choice (2). The figure is a rhombus because it has one of the parallelogram properties (diagonals bisect each other) and one of the rhombus properties (perpendicular diagonals). Example 2 Which combination of properties could be used to prove a quadrilateral is a square? all four sides are congruent, and the diagonals bisect each other the diagonals are perpendicular and congruent the diagonals are congruent, perpendicular, and bisect each other both pairs of opposite sides are parallel, all angles are right angles, and the diagonals are congruent Solution: Choice (3). Diagonals that bisect each other implies a parallelogram. Perpendicular diagonals makes it a rhombus. Congruent diagonals makes it a rectangle. Therefore the figure is a square. Example 3 Quadrilateral JKLM has diagonals and that intersect at O. If JO = OL = KO = OM, what is the most specific classification that can be assigned to JKLM? rectangle rhombus square trapezoid Solution: Choice (1). A figure with congruent diagonals that bisect each other is a rectangle. Example 4 In parallelogram QRST, ∠QSR ≅ ∠QST, , and . The figure must be what type of quadrilateral? rectangle rhombus square kite Solution: Choice (2). Sides parallel and congruent is a parallel property, and diagonals that bisect the angles is a rhombus property. It is a good idea to write out the list of properties when doing a proof that involves parallelograms or special parallelograms. If you need to prove the figure is a particular type of parallelogram, make a plan by choosing one property from each of the required categories. When proving a figure is a rectangle, rhombus, or square, first prove the figure is a parallelogram. Then show that one of the special properties is true. Example 5 Given: , , △BEC ≅ △BFC,
∠BEC ≅ ∠DCE Prove: ABCD is a rectangle
Solution: Strategy—Mark the figure. Then prove ABCD is a parallelogram with opposite sides parallel. Then prove it is a rectangle with a right angle.
Example 6 Given: ∠BCA ≅ ∠DAC, ∠BAC ≅ ∠BCA, BC ≅ AD Prove: ABCD is a rhombus Solution:
Example 7 Given: ∠BAC ≅ ∠ACD, ∠CBD ≅ ∠ADB, △AED ≅ △CED, Prove: ABCD is a square
Parallelograms and Transformations Key Ideas Rigid motions can be applied to map a parallelogram onto itself. Mapping a Quadrilateral onto Itself The properties of parallelograms and trapezoids can be used to identify rigid motions that map the figures onto themselves. Right angles, congruent angles, congruent sides, bisected segments, and parallel lines are not only properties of the various parallelograms described in this guide; they are also defining features of the rigid motion transformations. When given a particular parallelogram, it is possible to identify a variety of transformations that map the figure onto itself.
Figure shows rectangle ABCD with diagonals intersecting at point E. A rotation of 180° about point E would map ABCD onto itself. The justification is based on the parallelogram properties. The diagonals bisect each other, so AE = EC. Points A and C are equidistant from the center of rotation and lie on a straight line through the center. Therefore A and B map to one another. The same justification explains why B and D map to one another. Rectangle ABCD under a 180° rotation
A rectangle also maps to itself after a reflection over a line through the midpoints of two opposite sides. In Figure, E and F are the midpoints of and . So AE = ED = BF = FC. is the perpendicular bisector of and . acts as the line of reflection that maps A to D and maps B to C.
. Rectangle ABCD reflected over a line through the midpoints of and
By applying the same reasoning as with the rectangle, we find that any parallelogram can be mapped onto itself with a 180° rotation. However, a reflection through the midpoints will not necessarily work. When a particular mapping is specified, keep corresponding parts in mind. We may need to have each point in the preimage map to a specific point in the image. A 180° rotation may not yield the desired mapping. In fact, a composition of two or more transformations may be needed. Often a translation is combined with a reflection. Example 1 Given rectangle ABCD, find a rigid motion or composition of rigid motions that will map rectangle ABCD onto itself, such that: the image of point A after the transformation is point B. the image of point A after the transformation is point C. the image of point A after the transformation is point D.
Solution: A translation that maps A to B followed by a reflection over maps ABCD onto itself with point A mapped to point B. A rotation of 180° about Point E maps ABCD onto itself with point A mapped to point C. A translation that maps A to D followed by a reflection over maps ABCD onto itself with point A mapped to point B. Example 2 Name 2 different rigid motions, or compositions of rigid motions, that will map parallelogram ABCD onto itself such that point A maps to point C? Solution: Rotate 180° about point E. Translate by followed by a rotation of 180° about C. Math Fact When mapping a parallelogram onto itself, a vertex can only be mapped to the opposite vertex unless the figure is a rectangle or square. The angles at the two vertices must be congruent to map onto each other. Opposite angles are congruent in a parallelogram, but consecutive angles are only congruent in rectangles and squares. Besides mapping the entire parallelogram, the triangles formed by the diagonals of parallelograms can be mapped from one to another or onto themselves. Example 3 Using the figure from the previous example, what rigid motion will map △AED to △CEB? Solution: A rotation of 180° about point E maps A to C and maps B to D. So it will map △AED to △CEB. Example 4 Using the figure from Example 2, explain why a reflection across diagonal will not map △AED to △CEB unless the parallelogram is also a rhombus. Solution: If the parallelogram is not a rhombus, the diagonals are not perpendicular. Since DE = EB but is not the perpendicular bisector of , reflecting over that line cannot map D to B.
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