Ex. 307. The difference of any two sides of a triangle is less than the third side.

Ex. 308. A diagonal of a quadrilateral is less than one half the perimeter of the figure.

PROPOSITION XXIX. THEOREM

128. If two sides of a triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side.

Ex. 309. If in ▲ ABC, AB= 7, BC 8, CA = 6, which is the greatest angle of the triangle? which the smallest ?

Ex. 310. If one arm AC of an isosceles triangle

ABC is produced to D, then

LABD> <ADB.

Ex. 311. If in ▲ ABC, AB> AC, and ▲ B = 60°, prove that

Ex. 312. If in AABC, AB> AC, and A= 60°,

which is the greatest angle of the triangle ?

Ex. 313. If in quadrilateral ABCD, AB> BC and AD> CD, then <C> <A.

129. NOTE.

The sides of a triangle are often designated by italics which correspond with the letters of the opposite vertices. Thus, in ▲ ABC, AB = c, BC = a, and CA = b.

PROPOSITION XXX. THEOREM

130. If two angles of a triangle are unequal, the sides opposite are unequal, and the greater side is opposite the greater angle.

131. COR. The perpendicular is the shortest line that can be drawn from a point to a given line. (115.)

NOTE. The method used in the above proof is known as the indirect method or reductio ad absurdum. Instead of showing that a certain conclusion is true, we examine all conclusions which contradict the one to be proved, and demonstrate that these are false. Thus, in order to demonstrate a=b, we simply prove that the statement ab is false. Or to prove m>n, we disprove the only contradictory conclusions, viz. m = n, and m<n.

The indirect proof is often used to prove converses, especially those that establish inequalities. (Compare Props. XII and XVII.)

Ex. 314. Which is the greatest side of a right triangle? of an obtuse triangle ?

Ex. 315. If two angles of a triangle are 50° and 60° respectively, which is the greater of the two opposite sides? which is the greatest side of the triangle? which is the shortest?

Ex. 316. If in ▲ ABC, AB> AC and ≤ B = 60°, which is the greatest side of the triangle ? which is the smallest ?

Ex. 317. If in ▲ ABC, AB > AC, and ZA

est side of the triangle?

=

60°, which is the great

Ex. 318. Of two lines drawn from a point in a perpendicular to a given line, cutting off on the given line unequal segments from the foot of the perpendicular, the more remote is the greater.

Ex. 319. If ZA of ▲ ABC equals two thirds of a right angle, and the exterior angle DBC equals 110°, which is the greatest side of the triangle? which is the shortest?

Ex. 320. If in ▲ ABC AB=AC, prove that DC > DB.

Ex. 321. In triangle ABC, AC > BC. If the bisectors of angles A and B meet in D, prove AD > BD.

Ex. 322. In the diagram given here, AC> BC, AD LAC, and B DB1 BC. Prove BD > AD.

Ex. 323. Prove Prop. XXX directly.

HINT. At B draw angle CBD = 2 C.

A

Ex. 320

Ex. 322

B

Ex. 324. The sum of the altitudes of a triangle is less than the perimeter of the triangle.

[See practical problems, p. 289, No. 21.]

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

Ex. 326. If in triangle ACB the median AD be drawn forming the acute angle ADB, prove

Ex. 327. If in ▲ ABC, A = Z B, and a point D in AB be taken so that ACD > DCB, then AD>DR.

PROPOSITION XXXII. THEOREM

133. If two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second,