Question

show or briefly describe how two copies of it can be composed into a parallelogram.

Answer:

To determine how two copies of a triangle can form a parallelogram, we can use the properties of triangles and parallelograms. Here's a step - by - step explanation:

Step 1: Recall the properties of a parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel and equal in length.

Step 2: Analyze the triangle copies

Let's take two congruent triangles (since we are using two copies of the same triangle, they are congruent).

1. Identify corresponding sides and angles: In congruent triangles, corresponding sides are equal and corresponding angles are equal.
2. Arrange the triangles: Place the two triangles such that a pair of their equal sides are adjacent to each other. For example, if we have triangle \(ABC\) and triangle \(A'B'C'\) (where \(ABC\cong A'B'C'\)), we can place side \(AB\) of triangle \(ABC\) adjacent to side \(A'B'\) of triangle \(A'B'C'\) in a way that the angles at \(B\) and \(B'\) (which are equal because the triangles are congruent) are supplementary (or form a linear pair) when combined.
3. Verify the parallelogram properties:

• After arranging the two triangles, we can see that the opposite sides of the resulting quadrilateral are equal. For instance, if we consider the sides formed by the non - adjacent sides of the triangles, since the triangles are congruent, these sides will be equal in length. Also, the opposite sides will be parallel. This is because the alternate interior angles formed by the transversal (the common side or a line parallel to the sides) will be equal (due to the congruence of the triangles and the way we arranged them), which satisfies the condition for parallel lines (if alternate interior angles are equal, the lines are parallel).

Geometric construction (optional but helpful for visualization)

1. Let's take a right - angled triangle for simplicity (the same logic applies to any triangle). Let the triangle have vertices \(A\), \(B\), and \(C\) with \(\angle B = 90^{\circ}\).
2. Take a second congruent triangle \(A'B'C'\) (where \(A'\) corresponds to \(A\), \(B'\) corresponds to \(B\), and \(C'\) corresponds to \(C\)).
3. Place the second triangle such that vertex \(B'\) is at vertex \(C\) of the first triangle, and side \(B'C'\) is along side \(BC\) (extended if necessary) and side \(A'B'\) is parallel to side \(AC\).
4. Now, the quadrilateral formed by \(A\), \(B\), \(C'\), \(A'\) (or the appropriate vertices) will be a parallelogram. The opposite sides \(AB\) and \(A'C'\) are equal and parallel, and the opposite sides \(AC\) and \(A'B'\) are equal and parallel.

In general, for any triangle, when we place two congruent triangles together by matching a pair of equal sides, the resulting figure is a parallelogram because the opposite sides of the quadrilateral formed are equal and parallel (by the congruence of the triangles and the properties of parallel lines and congruent triangles).

If we were to write a more formal proof:

Let \(\triangle ABC\) and \(\triangle A'B'C'\) be two congruent triangles, so \(AB = A'B'\), \(BC=B'C'\), \(AC = A'C'\) and \(\angle ABC=\angle A'B'C'\), \(\angle BAC=\angle B'A'C'\), \(\angle ACB=\angle A'C'B'\).

Arrange \(\triangle ABC\) and \(\triangle A'B'C'\) such that \(BC\) is adjacent to \(B'C'\) and \(B\) and \(B'\) are on opposite sides of \(AC\) (or a similar arrangement).

The quadrilateral \(ABA'C\) (the name of the quadrilateral may vary depending on the arrangement) has:

• \(AB = A'C\) (since \(AB = A'B'\) and \(A'B'=A'C\) as \(\triangle A'B'C'\cong\triangle ABC\)) and \(AB\parallel A'C\) (because \(\angle ABC=\angle A'C'B'\) and these are alternate interior angles for lines \(AB\) and \(A'C\) with transversal \(BC\)).
• \(AC = A'B\) (since \(AC = A'C'\) and \(A'C'=A'B\) as \(\triangle A'B'C'\cong\triangle ABC\)) and \(AC\parallel A'B\) (because \(\angle BAC=\angle A'BC\) and these are alternate interior angles for lines \(AC\) and \(A'B\) with transversal \(AB\)).

Since both pairs of opposite sides are equal and parallel, the quadrilateral is a parallelogram.