QUADRANGLES

Attention has hitherto been given to various properties of the plane figures formed by two or three straight lines. The figure that next presents itself is that formed by four lines each of which meets the next one in order.

138. Definitions. A plane figure formed by four line-segments that inclose a portion of the plane surface is called a quadrilateral figure, or a quadrangle.

These line-segments are called the sides, and their extremities the vertices of the quadrangle.

The angles formed by adjacent sides, and situated toward the interior of the boundary, are called the interior angles of the quadrangle, or briefly the angles.

The exterior angles conjunct to these will be called for brevity the conjunct angles.

A concave angle formed by one side and the prolongation of an adjacent side is called an exterior angle.

If all of the conjunct angles are convex (21), the quadrangle is called convex.

If one of the conjunct angles is concave, the figure is said to be concave at that angle.

In a convex quadrangle all the interior angles are concave, and no side when prolonged traverses the figure; but a concave quadrangle has one of the interior angles convex, and the sides of this angle traverse the figure when prolonged.

A line connecting two non-adjacent vertices is called a diagonal.

The sum of the sides is called the perimeter, and the sum of the angles the angle-sum.

In a convex quadrangle the sum of the exterior angles formed by prolonging each side one way, no two adjacent sides being prolonged through the same vertex, is called the exterior angle-sum.

The four sides and four angles are called the eight parts of the quadrangle.

Primary relations of parts.

139. RELATION 1.

The sum of any three sides is

greater than the fourth (89).

140. Cor. The sum of any two sides is greater than the difference of the other two.

141. RELATION 2. The angle-sum is equal to a perigon.

[Divide the quadrangle into two triangles by a diagonal and apply 129.]

142. Cor. I. A conjunct angle is equal to the sum of the three non-adjacent interior angles.

143. Cor. 2. Only one of the interior angles in a quadrangle can be convex.

144. Cor. 3. If two quadrangles have three angles of one equal respectively to three angles of the other, the remaining angles are equal.

Ex. 1. The sum of the four sides of a quadrangle is greater than the double of either diagonal, and greater than the sum of the diagonals.

Ex. 2. The sum of the four interior angles is equal to one third of the sum of the four conjunct angles.

SOME CONDITIONS OF EQUALITY

The following three theorems relate to the equality of two quadrangles under certain conditions.

Three sides and two included angles.

145. THEOREM 26. If two quadrangles have three sides and the two included angles of one equal to the corresponding parts in the other, the figures are equal.

Outline. Superpose the equal parts, and show that the coincidence of the remaining parts will follow as in 64.

Two adjacent sides and three angles.

146. THEOREM 27. If two quadrangles have two adjacent sides and any three angles of one equal to the corresponding parts in the other, the figures are equal.

Outline. The remaining angles are equal (144). Superpose the equal parts, and show that the remaining parts will coincide.

Two opposite sides and three angles.

147. THEOREM 28. If two quadrangles have two opposite sides and any three angles of one equal to the corresponding parts in the other, the figures are equal.

In the quadrangles ABCD and A'B'C'D', let AB equal A'B', CD equal c'D'. Also let angle 4 equal A', B equal B', C equal c', and consequently D equal D' (144).

To prove that the quadrangles are equal.

Prolong BC and AD to meet in P; also B'C' and 'D' to meet in P'.

The angle PCD equals P'C'D', and CDP equals C'D'P'.

Therefore the triangles PCD and P'C'D', having a side and the two adjacent angles in each equal, are themselves equal. For a similar reason the triangle PBA equals P'B'A'.

By subtraction of equals from equals the line BC equals B'C', and AD equals A'D'.

Hence the quadrangles are equal by the preceding theorem.

SPECIAL KINDS OF QUADRANGLES

148. Definitions. A quadrangle that has a pair of its opposite sides parallel, and the other pair not parallel, is called a trapezoid. One that has both pairs of opposite sides parallel is a parallelogram.

In contradistinction a quadrangle that has neither pair of sides parallel is called a trapezium.

A trapezoid whose opposite non-parallel sides are equal is said to be isosceles.

A parallelogram that has two adjacent sides equal is called a rhombus.

[It is shown later (154) that the four sides of a rhombus are equal. ]

A parallelogram that has one of its angles right is called a rectangle.

[It will appear (150) that all the angles of a rectangle are right angles.]

A rectangle that has two adjacent sides equal is a square. [It is shown later (154) that all the sides of a square are equal. A square is at once a rhombus and a rectangle.]

PARALLELOGRAMS AND TRAPEZOIDS

Theorems 29–36 with their corollaries establish the principal properties of parallelograms and trapezoids.

Angles of a parallelogram.

149. THEOREM 29. Any two consecutive angles of a parallelogram are supplemental; and any two opposite angles are equal.

[Apply 126 and corollaries, or prove independently by the method of that article.]