PROPOSITION XXXI. PROBLEM.

229. Two sides and the included angle of a triangle being given, to construct the triangle.

Let the two sides of the triangle be E and F, the included angle A.

and

It is required to construct a ▲ having two sides equal to E and F respectively, and their included

=

ZA.

Take HK equal to the side F.

At the point H draw the indefinite line HM,

PROPOSITION XXXII. PROBLEM.

230. A side and two adjacent angles of a triangle being given, to construct the triangle.

Let CE be the given side, A and B the given angles.

It is required to construct a ▲ having a side equal to CE. adjacent to that side equal to A and B respectively.

and two

231. SCHOLIUM. The problem is impossible when the two given angles are together equal to, or greater than, two right angles.

PROPOSITION XXXIII. PROBLEM.

232. The three sides of a triangle being given, to construct the triangle.

It is required to construct a ▲ having three sides respectively, equal to m, n, and o.

Draw A B equal to n.

From A as a centre, with a radius equal to o,

describe an arc;

and from B as a centre, with a radius equal to m,

describe an arc intersecting the former arc at C.

Draw CA and C B.

Then

ACAB is the ▲ required.

Q. E. F.

233. SCHOLIUM. The problem is impossible when one side

is equal to or greater than the sum of the other two.

PROPOSITION XXXIV. PROBLEM.

234. The hypotenuse and one side of a right triangle being given, to construct the triangle.

Let m be the given side, and o the hypotenuse.

It is required to construct a rt. A having the hypotenuse equal o and one side equal m.

Take A B equal to m.

At A erect a 1, A X.

From B as a centre, with a radius equal to o,

describe an arc cutting A X at C.

Draw C B.

Then

ACAB is the ▲ required.

Q. E. F.

PROPOSITION XXXV. PROBLEM.

235. The base, the altitude, and an angle at the base, of a triangle being given, to construct the triangle.

Let o equal the base, m the altitude, and C the angle at the base.

It is required to construct a ▲ having the base equal to o, the altitude equal to m, and an at the base equal to C.

Take A B equal to o.

At the point A, draw the indefinite line A R,

making the BAR = Z C.

At the point A, erect a LA X equal to m.

From X draw XS to AB,

and meeting the line AR at S.

Draw SB.

Then A ASB is the ▲ required.

Q. E. F.