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MathChapter 18: Polygons

Notes

Parallelograms Notes

Use parallelogram and rhombus properties, diagonal relationships, and perpendicular heights to solve angle, length, and area problems.

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A parallelogram problem is a property-selection problem. Translate every parallel mark, congruence tick, diagonal intersection, and right-angle marker before calculating.

Definition and anatomy

Parallelogram
A quadrilateral with two pairs of parallel opposite sides. For ABCD, \(AB\parallel CD\) and \(BC\parallel AD\).
Parallelogram anatomyParallelogram ABCD has both pairs of opposite sides explicitly marked parallel.
Parallelogram anatomyParallelogram ABCD has both pairs of opposite sides explicitly marked parallel.ABCDAB ∥ CD; BC ∥ AD
Parallelogram property summary
PropertyRelationshipHow it helps
Opposite sides\(AB=CD\), \(BC=AD\)Set side expressions equal.
Opposite angles\(\angle A=\angle C\), \(\angle B=\angle D\)Transfer an angle across the figure.
Consecutive anglesEach adjacent pair totals \(180^\circ\).Subtract a known angle from 180.
Diagonals\(AE=CE\), \(BE=DE\)Equate the two halves of the same diagonal.

Diagonals bisect each other

Parallelogram propertiesParallelogram ABCD has diagonals AC and BD meeting at E. Matching marks show that each diagonal is bisected, while parallel marks identify both pairs of parallel sides.
Parallelogram propertiesParallelogram ABCD has diagonals AC and BD meeting at E. Matching marks show that each diagonal is bisected, while parallel marks identify both pairs of parallel sides.EAE = CE; BE = DEABCD
Worked example

Side and diagonal algebra

In parallelogram ABCD, \(AB=4x+3\), \(CD=7x-18\), \(AE=2y+5\), and \(CE=5y-16\). Find x and y.

  1. Opposite sides

    Set \(4x+3=7x-18\), giving \(x=7\).

  2. Diagonal halves

    Set \(2y+5=5y-16\), giving \(y=7\).

\(x=7\) and \(y=7\).

Opposite versus consecutive angles

Choose the correct equation

Opposite

Set the measures equal.

Consecutive

Set their sum equal to \(180^\circ\).

Worked example

Algebraic consecutive angles

Consecutive angles measure \(3x+12\) and \(5x-8\) degrees. Find the smaller angle.

  1. Supplement

    \((3x+12)+(5x-8)=180\), so \(8x+4=180\) and \(x=22\).

  2. Evaluate

    The measures are \(78^\circ\) and \(102^\circ\).

The smaller angle is \(78^\circ\).

Rhombus properties

Rhombus
A parallelogram with four congruent sides. Its diagonals are perpendicular, and each diagonal bisects a pair of opposite angles.
Rhombus diagonalsRhombus ABCD has four congruent sides and perpendicular diagonals meeting at E. Matching arcs show that the diagonals bisect opposite angles.
Rhombus diagonalsRhombus ABCD has four congruent sides and perpendicular diagonals meeting at E. Matching arcs show that the diagonals bisect opposite angles.AC ⟂ BD; diagonals bisect opposite anglesABCDE

Area and perpendicular height

Parallelogram base and heightA slanted parallelogram shows base b and a perpendicular altitude h. The slanted side is labeled separately and is not used as the height.
Parallelogram base and heightA slanted parallelogram shows base b and a perpendicular altitude h. The slanted side is labeled separately and is not used as the height.hbslanted sideA = bhABCD
Parallelogram area
\[A=bh\]

Height is the perpendicular distance between the parallel base lines, not a slanted side.

Worked example

Height from a special triangle

A parallelogram has base 18 and a slanted side 10 making a \(30^\circ\) angle with the base. Find its area.

  1. Height

    The altitude is opposite 30 degrees in a 30-60-90 triangle, so \(h=10/2=5\).

  2. Area

    \(A=18(5)=90\).

The area is 90 square units.
Rhombus area from diagonalsRhombus diagonals d1 and d2 intersect perpendicularly and divide the rhombus into four right triangles.
Rhombus area from diagonalsRhombus diagonals d1 and d2 intersect perpendicularly and divide the rhombus into four right triangles.d₂d₁A = ½d₁d₂ABCDE
Rhombus area
\[A=\frac12d_1d_2\]

Use the full diagonal lengths. The factor one-half accounts for the four central right triangles.

Worked example

Half diagonals and area

A rhombus has half-diagonals 6 and 8. Find its side and area.

  1. Side

    A central right triangle gives \(s=\sqrt{6^2+8^2}=10\).

  2. Full diagonals

    The diagonals are 12 and 16.

  3. Area

    \(A=\tfrac12(12)(16)=96\).

Side 10; area 96 square units.
Mini check

Property check

One parallelogram angle is \(64^\circ\). What are the other three?

  1. 64, 116, 116
  2. 116, 64, 116
  3. 64, 64, 64
Show answer and explanation

Answer: \(116^\circ,64^\circ,116^\circ\) in consecutive order.

Opposite angles match; consecutive angles total 180 degrees.

Key takeaways

Key takeaways

  • Opposite sides and opposite angles are congruent; consecutive angles are supplementary.
  • Parallelogram diagonals bisect each other, but need not be congruent or perpendicular.
  • Rhombus diagonals are perpendicular and bisect opposite angles.
  • Use perpendicular height in \(A=bh\), and full diagonals in \(A=\tfrac12d_1d_2\).
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.