Exercise 3 Page 367

To explore the properties of a parallelogram, we start by considering a general parallelogram. To draw it, we can use a dynamic geometry software. Remember that a parallelogram is a quadrilateral with both pairs of opposite sides parallel.

To find the properties, first we will find the side lengths, then the angle measures, and finally we will use the diagonals of the parallelogram.

Side Lengths

With a ruler, let's measure each side of the parallelogram drawn above.

Measuring the sides of parallelogram ABCD

As we can see, we found two pairs of congruent sides. Thus, we write the first property.

The opposite sides of a parallelogram are congruent.

Angle Measures

Since

\overline{A D} ∥ \overline{B C}

and

\overline{A B}

and

\overline{C D}

are transversals, by the Consecutive Interior Angles Theorem, we get two pairs of supplementary angles.

{m ∠ A + m ∠ B = 18 0^{\circ} m ∠ C + m ∠ D = 18 0^{\circ} (I) (II)

Similarly, since

\overline{A B} ∥ \overline{C D}

and

\overline{A D}

and

\overline{B C}

are transversals, we can get two more pairs of supplementary angles.

{m ∠ A + m ∠ D = 18 0^{\circ} m ∠ B + m ∠ C = 18 0^{\circ} (III) (IV)

From the four equations written above, we obtain a second property of parallelograms.

The consecutive angles of a parallelogram are supplementary.

Additionally, let's subtract Equation

(III)

from Equation

(I) .

m ∠ A + m ∠ B^{-} m ∠ A + m ∠ D m ∠ B - m ∠ D = 18 0^{\circ} = 18 0^{\circ} = 0^{\circ}

From the above, we conclude that

∠ B

and

∠ D

have the same measure, so they are congruent. Similarly, by subtracting Equations

(I)

and

(IV)

we get that

∠ A

and

∠ C

are congruent.

These latter relations lead us to write a third property of parallelograms.

The opposite angles of a parallelogram are congruent.

Diagonals

Let's draw the diagonals of the parallelogram, and let's label its intersection point.

Since $\overline{A D} ∥ \overline{B C}$ and $\overline{B D}$ and $\overline{A C}$ are transversals, by the Alternate Interior Angles Theorem, we get that $∠ B D A ≅ ∠ D B C$ and $∠ C A D ≅ ∠ A C B .$

By the first property we found, we have that

\overline{A D} ≅ \overline{B C} .

∠ B D A ≅ ∠ D B C \overline{A D} ≅ \overline{B C} ∠ C A D ≅ ∠ A C B A ngle S ide A ngle

The Angle-Side-Angle (ASA) Congruence Theorem gives us that

△ A D M ≅ △ C B M .

This implies that

\overline{B M} ≅ \overline{D M}

and

\overline{A M} ≅ \overline{C M},

so

M

is the midpoint of both diagonals.

The latter fact leads us to write the last property of parallelograms.

The diagonals of a parallelogram bisect each other.