4.1 Parallelograms
Polygons are many sided, coplanar figures. A triangle is an example of a three-sided plane figure and a quadrilateral is a four-sided plane figure. Under the heading of quadrilateral, there are a number of different categories that we shall look at. There are parallelograms, trapezoids, and then quadrilaterals which are neither of these. Under the category of parallelograms, there are rectangles, rhombuses, and squares.
- Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
There are a number of properties of parallelograms that we now want to establish. It is important to distinguish between these properties of something known to be a parallelogram and the definition given above. The first of these properties we did as an example in the last chapter, but we repeat it here.
- Theorem 4.1:
Opposite sides of a parallelogram are congruent.
Given: $ABCD$ is a
parallelogram. Prove: $\overline{AB}\cong\overline{CD}$ and $\overline{AD}\cong\overline{BC}$ |
Proof: Given that $ABCD$ is a parallelogram, introduce diagonal $\overline{AC}$.
By the definition of a parallelogram, $\overline{AD}\parallel\overline{BC}$ and $\overline{AB}\parallel\overline{CD}$. Because of P$\rightarrow$AIC, $\angle 1\cong\angle 2$ and $\angle 3\cong\angle 4$. Since $\overline{AC}\cong\overline{AC}$, we have $\triangle ADC\cong\triangle CBA$ because of ASA. This makes the corresponding sides congruent, so $\overline{AB}\cong\overline{CD}$ and $\overline{AD}\cong\overline{BC}$.
Using the same approach — congruent triangles — we can also prove the following theorem.
- Theorem 4.2:
Opposite angles of a parallelogram are congruent.
The next theorem is proved in essentially the same way, through congruent triangles. It's just that the pair of triangles is different.
- Theorem 4.3: The diagonals of a parallelogram bisect each other.
Given: $ABCD$ is a
parallelogram. Prove: $\overline{AP}\cong\overline{PC}$ and $\overline{DP}\cong\overline{PB}$ |
Proof: By Theorem 4.1, $\overline{AD}\cong\overline{BC}$; by the definition of a parallelogram, $\overline{AD}\parallel\overline{BC}$, By P$\rightarrow$AIC, $\angle 1\cong \angle 2$ and $\angle 3\cong \angle 4$. Therefore, by ASA $\triangle APD\cong\triangle CPB$, which makes $\overline{AP}\cong\overline{PC}$ and $\overline{DP}\cong\overline{PB}$.
Since opposite sides of a parallelogram are parallel and since same-side interior angles formed by a transversal cutting parallel lines are supplementary, it is easy to prove the following theorem.
- Theorem 4.4:
Consecutive angles of a parallelogram are supplementary.
The proof is left for the exercises.
We collect here the facts from this section about a quadrilateral known to be a parallelogram.
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