Proposition 29. Theorem.

119. If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle in the first triangle greater than the included angle in the second, then the third side of the first triangle is greater than the third side of the second.

Hyp. Let ABC, DEF be two As, having

but

Το prove

AB = DE, ACDF,

<BACEDF.

BC>EF.

Proof. Apply the ▲ ABC to the ▲ DEF so that AB shall coincide with DE, and the pt. C fall at H.

The whole is greater than any of its parts (Ax. 8).

But

The greater side is opposite the greater angle (117).

EH =

BC... BC > EF.

Q. E.D.

Proposition 30. Theorem.

120. Conversely, if two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first triangle greater than the third side of the second, then the included angle of the first triangle is greater than the included angle of the second.

Hyp. Let ABC, DEF be two As

ABDE, AC = DF,

<A> <D.

AA

having

To prove

Proof.

If A is not > D,

B

CE

F

having two sides and the included equal, each to each (104).

But this is contrary to the hypothesis.
Hence, since A can neither be = nor <<D,

NOTE.-Prop. 30 is here proved by an indirect method. This method is sometimes called the reductio ad absurdum. It consists in assuming that the conclusion to be proved is not true, and showing that this assumption leads to an absurdity, or to a result inconsistent with the hypothesis.

QUADRILATERALS.

121. A quadrilateral is a plane figure bounded by four straight lines, which are called its sides.

The straight lines which join opposite angles of a quadrilateral are called the diagonals.

122. A trapezium is a quadrilateral which has no two of its sides parallel.

123. A trapezoid is a quadrilateral which has two of its sides parallel.

The parallel sides of a trapezoid are called the bases, and the perpendicular distance between them is called the altitude. The line joining the middle. points of the non-parallel sides is called the middle parallel of the trapezoid.

124. A parallelogram is a quadrilateral which has its opposite sides parallel.

The bases of a parallelogram are the side on which it stands and the opposite side. The perpendicular distance between the bases is called the altitude.

125. A rectangle is a parallelogram whose angles are right angles.†

126. A square is a rectangle whose sides are all equal.‡

127. A rhomboid is a parallelogram whose angles are oblique and whose adjacent sides are unequal.

128. A rhombus, or lozenge, is a parallelogram whose sides are all equal. §

* Called also the median.

+ Called also a right-angled parallelogram.
Called also an equilateral rectangle.

§ Called also equilateral rhomboid.

Proposition 31. Theorem.

129. In every parallelogram, the opposite sides are equal, and the opposite angles are equal.

having a side and the two adjacent Ls equal, each to each (105).

130. COR. 1. A diagonal of a parallelogram divides it

into two equal triangles.

131. COR. 2. Two parallels included between two other parallels are equal.

Proposition 32. Theorem.

132. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

Hyp. Let ABCD be a quadrilat- D

1. If one angle of a parallelogram is a right angle, prove

that all its angles are right angles.

2. Prove that two parallels are everywhere equally distant. 3. If, in the figure of Prop. 32, BE be drawn parallel to AC and meeting DA produced to E, prove that the parallelogram EBCA will be equal to the parallelogram ABCD.