Example. To construct a triangle having given one angle, the side opposite to it and one of the other sides.

Let ABC be the given angle, DE the length of the side opposite to ABC and FG the length of the other given side.

Construction. Make BH equal to FG.

With centre H and radius equal to DE describe the circle KLM,

cutting the line BC in the point K; join HK;

then HKB is the triangle required.

Proof. The triangle has the angle KBH, the given angle; the side HK is a radius of the circle whose radius is equal to DE, so that HK is equal to DE,

and the side BH was made equal to FG.

Wherefore the triangle BHK has been described as required. Q.E.F.

EXAMPLES XVI.

1. Make a triangle having two of its sides and the included angle equal respectively to two given finite straight lines and a given angle.

2. Make a triangle having given that two angles and the side adjacent to them are equal respectively to two given angles and a given finite straight line.

3. Construct a triangle having given that one angle is a right angle, one side containing the right angle is equal to a given finite straight line, and that the side opposite the right angle is double of the given finite straight line.

4. Shew that, when two sides of a triangle and the angle opposite one of them are given, there are in general two triangles which satisfy the given conditions.

5. Shew that when the side opposite the given angle in Question 4 is greater than the other given side there is always one triangle which can be drawn to satisfy the given conditions and that there is only one.

6. Describe a quadrilateral having each of its four sides equal to a given straight line and having one of its angles equal to a given rectilineal angle.

7. Construct a rhombus having its diagonals equal respectively to two given straight lines. [See Ex. XIV. 6.]

8. Construct a 'Kite' having given one of its sides and its two diagonals [See Ex. XIV. 7.]

SECTION V.

STRAIGHT ANGLES AND RIGHT ANGLES.

Straight Angles.

82. In considering the direction indicated by a finite straight line AB, if we consider a point C between A and B as the point from which the line is drawn, we have two lines drawn from C in different directions.

One of these lines is CA, the other is CB.

The directions of CA and CB are exactly opposite to each other; they are clearly different directions; therefore there is an angle at C.

83. DEF. The angle at any point in a finite straight line which the two parts of the line drawn from that point make with each other, is called a straight angle.

Thus, at every point C in a straight line AB there is a straight angle.

When another straight line is drawn from the point C the straight angle is divided into the two angles ACD, DCB.

Hence, when a straight line CD stands on another straight line BCA, the angles ACD, DCB are always together equal to a straight

angle.

L. E.

5

Proposition A.

84. All straight angles are equal to each other.

Let C and F be points in two straight lines ACB, DFE, so that ACB and DFE are two straight angles;

it is required to prove that the angles ACB, DEF are equal.

Let the straight line DE be applied to AB,

so that the point F is on C

and the point E on the straight line CB;
then the two straight lines DE and AB
have two points common; therefore

they coincide throughout their entire length. [Def., Art. 17.]
Hence the straight angle DEF at E
coincides with the straight angle ACB at C,
and is therefore equal to it.

Wherefore, all straight angles are equal to each other. Q.E.D.

85. Corollary. All right angles are equal to each other.

For a right angle is half a straight angle.

86. DEF. An angle less than one right angle is called an acute angle.

D

In the above figure ACB is an acute angle.

87. DEF. An angle greater than a right angle and less than a straight angle, is called an obtuse angle.

B

F

In the above figure ACB is an obtuse angle.

88. DEF. An angle greater than one straight angle and less than two straight angles, is called a reflex angle.

D

B

In the above figure the reflex angle ACB, is the amount of turning about C, which a line would have to go through, in turning from the direction CA to that of CB, in the manner indicated by the dotted

line.

89.

When any number of lines in one plane meet at a point, the angles at the point altogether make up four right angles.

For let CA, CB, CD, CE, CF be any number of lines all drawn from the point C. Produce any one of the lines AC through C to K. Then the angles at C make up the two straight angles one on each side of ACK.