B will coincide with the point E. Therefore as the points A and B coincide with the points D and E, the line A B will coincide with the line D E. Hence the two triangles are identical, and have all the parts of the one equal to the corresponding parts of the other; namely, the side A B to the side D E, the angle A to the angle D, and the angle B to the angle E. Q. E. D.

THEOREM II.

In any two triangles A B C, D E F, if the angle A be equal to the angle D, the angle B to the angle E, and the sides A B and D E adjacent to these equal angles be also equal, the triangles will be identical, or equal in all respects.

For conceive the point A to be laid on the point C D, and the side A B on the side D E, then because these lines are equal, the point B will coincide with the point E. And because A B and D E coincide,

F

and the angle A is equal to the angle D, the side A BD E A C will fall on the side D F; and for a like reason B C will fall on E F. Therefore as A C falls on D F, and B C on E F, the point C must coincide with the point F; and consequently the two triangles are identical; having the other two sides A C, B C, equal to the two DF, EF, and the remaining angle C equal to the remaining angle F. Q. E.D.

THEOREM III.

In any isosceles triangle A B C, the angles A and B, opposite the equal sides, A C and R C, are equal.

For conceive the angle C to be bisected by the line C D ; then as the two triangles A C D and B C D, have A C equal to B C, C D common to both, and the angle A CD equal to the angle B C D, they are identical, (Theo. 1.); and therefore the angle A is equal to the angle B. Q.E.D.

C

AD B

Cor. 1. The line which bisects the angle included between the equal sides of an isosceles triangle, bisects the third side, and is also perpendicular to it.

Cor. 2. Every equilateral triangle is also an equiangular one.

THEOREM IV.

In any triangle AC B, if one angle as BAC be equal to another angle as A B C, the sides A C and B C opposite those equal angles will also be equal to each other.

D

For if A C and B C are not equal, suppose one of them as C AC, to be longer than the other, and conceive A D to be the part of AC which is equal to B C; and let D B be joined. Then because A D is equal to B C, AB common to both, and the angle D A B equal to the angle CAB; A

therefore the triangle A D B is equal to the triangle A B C, (Theo. 1.) the less to the greater, which is impossible. Hence A C and B C are not unequal, that is, they are equal. Q. E. D.

Cor. Every equiangular triangle is also an equilateral one.

THEOREM V.

If any two triangles A B C, DE F, have the sides A C and D F, A B and D E, and B C and E F, respectively equal, the triangles will be identical, and have the angles equal which are opposite to the equal sides.

C

B D

E

C

F

For let the point A be laid on the point D, and the line A B on the line DE; then because these lines are equal, the point B will coincide with the point E. Let the point C fall at G, the points G and F being on opposite sides of the line DE, and let FG be joined. Then as D F and DG are each equal to AC, they are equal to each other, and consequently the angles D F G and DG F are equal. In the same way E F G and E G F may be shewn to be equal; and consequently the angles DFE and D GE, or AC B, are equal. Hence, as the two sides A C, B C are respectively equal to the two sides D F, F E, and the angle AC B is also equal to the angle D F E, the triangles A B C and D F E are identical, (Theo. 1.) and have the angle B A C equal to the angle EDF; and the angle A B C equal to the angle D E F. Q. E. D.

THEOREM VI.

BD

E

The angles D BA and D B C which one straight line B D makes with another, A C, on the same side of it, are either two right angles, or together they are equal to two right angles.

E

D

For if DB is perpendicular to A C, then each of the angles D B A and D B C is a right angle. But if BD is not perpendicular to A C, let B E be perpendicular to it, then each of the angles A B E and E B C is a right angle,. and A B D exceeds a right angle by E B D, and D B C is less than a right angle by the same angle E B D; the two angles, A B D and D B C, are therefore together equal to two right angles. Q. E. D.

A B C

Cor. 1. All the angles that can be made at any point as B, by any number of lines, drawn on the same side of a line A C, are together equal to two right angles.

Cor. 2. As all the angles that can be made at the point B, on the other side of the line A C, are together equal to two right angles; all the angles that can be made, in the same plane, about any point, as B, are together equal to four right angles.

Cor. 3. If a straight line revolve in a plane about a point in which one of the extremities of the line remains fixed, the angles which the successive positions of the revolving line make with each other during a complete revolution, are together equal to four right angles; and the corresponding portions of the circumference of the circle, described by the other extremity of the revolving line, make together, the whole circumference. If, therefore, the circumference of a circle be assumed as the measure of four right angles; the arc intercepted between any two radii may be considered as the measure of the angle which these two radii make with each other.

Cor. 4. And hence we may further infer that, with the same radius, equal angles have equal measures, that the arc of a semicircle is the measure of two right angles, and the arc of a quadrant, the measure of one right angle.

THEOREM VII.

If two straight lines A B, B C, diverge from a point B in another right line BD, on opposite sides of it, so as to make the adjacent angles A B D and D B C together equal to two right angles, these two lines A B and B C are in the same straight line.

For if they are not in the same straight line, let A B produced, be in the direction BE. Then the angles ABD and DBE together, are equal to two right

angles; and as A B D and D B C together are also A B equal to two right angles; if the common angle ABD

D

E

be omitted from each sum, the remaining angles DBE and DBC must be equal; a part to the whole, which is impossible. The lines A B and B C are therefore not otherwise than in the same straight line, that is they form one straight line. Q. E. D.

THEOREM VIII.

If two straight lines A B, D E, intersect each other in C, any two vertical or opposite angles as A CD and B C E are equal.

E

B

For as A C meets D E in C, the angles A C D and AC E are together equal to two right angles; and for A a like reason the angles BCE and A C E are equal to two right angles. Hence the angles A CD and ACE together are equal to the angles B C E and ACE together, and by omitting from each sum the common angle A C E, we have the remaining angles A C D and B C E equal to each other. Q. E. D.

D

THEOREM IX.

If any side, as A B, of a triangle A B C be produced, as to D, the outward angle C B D is greater than either of the inward and opposite angles A and C.

For let BC be bisected in E, join A E, and produce it till E F is equal to A E, and join B F, then because A E is equal to E F, and B E to E C, and the angle A E C to the angle F E B, (Theo. 9.) therefore A the angle FBE is equal to the angle AC E, (Theo.

G

1.) and consequently the whole angle DBC is greater than the angle C. In the same way by producing C B, and bisecting A B, it may be shewn that the angle A B G, which is equal to CBD, is greater than BA C. Q.E.D.

THEOREM X.

Any two angles, as A B C and B C A, of a triangle A B C, are together less than two right angles.

For let A B be produced to D, then the angles A B C and CBD are together equal to two right angles; but the angle CBD is greater than the angle A CB; consequently the angles A B C and A C B together are less▲ than two right angles. Q. E. D.

THEOREM XI.

In any triangle A B C if the side A B be greater than the side A C, the angle AC B, opposite the greater side, is greater than the angle A B C opposite the less.

DB

For let A D be the part of A B which is equal to A C, and join DC; then because A C and AD are equal, the angles A CD and ADC are equal, (Theo. 3.) But the angle ADC is greater than the angle DBC or A ABC, (Theo. 9.); whence A CD is also greater than ABC. Much more therefore is the whole angle A C D greater than A BC. Q. E. D. Cor. In any triangle, the greatest side has the greatest angle opposite to it, and the least side has the least angle opposite to it.

THEOREM XII.

In any triangle A B C, if one angle, as C, be greater than another angle, as A, the side A B, opposite the greater angle, is greater than the side B C opposite the less.

For if A B is not greater than B C it must be either equal to it or less; AB cannot be equal to B C, for then the angle ACB would be equal to the angle B A C, (Theo. 3.) which it is not, by condition; neither can it be less, for then the angle A C B would A

B

be less than BA C, (Theo. 11.) which it is not, by condition. Hence as A B is neither equal to BC nor less than it, it must be greater, Q. E. D.

THEOREM XIII.

If two trianges A B C, DE F, have two sides A B, B C, of the one respectively equal to D E, E F, two sides of the other, but the angle A B C included by the two sides of the one, greater than the angle D E F included by the corresponding sides of the other; then the side AC will be greater than the side D F.

B

E

Let ABG be the part of the angle ABC which is equal to D E F, and let B G be equal to E For BC. Then the triangles A B G, DEF are identical, (Theo. 1.) and have the sides A G and D F equal to each other. And as B C and A BG are equal, the angles B C G and B G C are equal, (Theo. 3.); hence B G C is greater than AC G, and much more is A G C greater than A C G; therefore (Theo.12) AC is greater than A G, or than its equal D F. Q. E. D.

THEOREM XIV.

C

G

D F

If two triangles A B C, D E F (see the last figure) have two sides A B, BC of the one respectively equal to D E, E F, two sides of the other, but the third side A C of the one greater than the corresponding side DF of the other, then the angle A B C will be greater than the angle D E F.

If A B C is not greater than DEF, it must either be equal to it, or less. But if ABC and D E F were equal, then A C and D F would be equal, (Theo. 1.) which they are not; and if ABC were less than D E F, then A C would be less than D F, (Theo, 13.) which Hence as A C is neither equal to D F nor less, it must be Q. E. D.

it is not.

greater.

THEOREM XV.

In any triangle ABC, the sum of any two of its sides, as AC and C B, is greater than the remaining side A B.

For let A C be produced till C D is equal to C B, and join D B; then because C D is equal to C B, the angle CDB is equal to the angle C BD, (Theo. 3.); therefore the whole angle A B D is greater than the angle CDB or AD B; and consequently the side A D, or A the sum of A C and C B, is greater than A B. Q. E.D.

THEOREM XVI.

In any triangle ABC, the difference of any two of its sides, as A B and A C, is less than the remaining side B C.

For let A D be the part of A B which is equal to A C, then D B will be the difference of A B and A C; and as AC and B C together are greater than A B, (Theo. 1,5.) or than AD and DB together; if the equal parts ACÁ

DB