Chapter 5.4, Problem 4WE

To determine

To mark: the appropriate features of the given quadrilaterals in the table.

Explanation of Solution

Given Information: A feature of a quadrilateral and four types of quadrilaterals.

The four basic quadrilaterals are considered here which are Parallelogram, Rectangle, Rhombus and Square. The given feature is that, a diagonal forms two congruent triangles.

A parallelogram is a quadrilateral in which, the diagonals separate it into two triangles which are congruent to each other which can be represented as follows.

  

Two main features of the parallelogram make this possible.

1) The opposite sides of a parallelogram are parallel and congruent to each other. Hence equal in measure.

  $\overline{EF}\cong \overline{GH}$

2) The alternate interior angles formed by the two parallel lines crossed by a transversal are equal.

  $\begin{array}{l}\angle 1=\angle 2\\ \angle 3=\angle 4\end{array}$

Thus, by ASA congruence, the two triangles ΔEFG and ΔEGH are congruent.

Now, the remaining quadrilaterals which are rectangle, rhombus and square are parallelograms with their own specific features. and they satisfy all the properties of the parallelogram.

Thus, the feature is valid for all the quadrilaterals as marked below.

Property	Parallelogram	Rectangle	Rhombus	Square
4.A diag forms two $\cong$ Δs	✔	✔	✔	✔