which has one of its angles an obtuse angle. All other triangles are called acute-angled triangles.

DEF. 32. A triangle is sometimes regarded as standing on a selected side which is then called its base, and the intersection of the other two sides is called the vertex. When two of the sides of a triangle have been mentioned, the remaining side is often called the base.

DEF. 33. The side of a right-angled triangle which is opposite to the right angle is called the hypotenuse.

DEF. 34. Figures that may be made by superposition to coincide with one another are said to be identically equal; and every part of one being equal to a corresponding part of the other, they are said to be equal in all respects.

THEOR. 5. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by these sides equal, then the triangles are identically equal, and of the angles those are equal which are opposite to the equal sides. [By Superposition.]*

THEOR. 6. The angles at the base of an isosceles triangle are equal to one another. [By a single application of Theor. 5, or directly by Superposition.]

COR. If a triangle is equilateral, it is also equiangular. THEOR. 7. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the arms common to these angles equal,

* Throughout this Syllabus a method of proof has been indicated wherever it was felt that this would make the principles upon which the Syllabus is drawn up more readily understood.

then the triangles are identically equal, and of the sides those are equal which are opposite to the

equal angles. [By Superposition.]

THEOR. 8. If the angles at the base of a triangle are equal to one another, the triangle is isosceles. [By Theor. 7,

or directly by Superposition.]

COR. If a triangle is equiangular, it is also equilateral. THEOR. 9. If any side of a triangle is produced, the exterior angle is greater than either of the interior opposite

angles.

THEOR. 10. The greater side of every triangle has the greater angle opposite to it.

THEOR. 11. The greater angle of every triangle has the greater side opposite to it.

THEOR. 12. Any two sides of a triangle are together greater than the third side.

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

THEOR. 13. If from the ends of the side of a triangle two straight lines are drawn to a point within the triangle, these are together less than the two other sides of the

triangle, but contain a greater angle.

THEOR. 14. If two triangles have two sides of the one equal to two sides of the other, each to each, but the included angles unequal, then the bases are unequal, the base of that which has the greater angle being greater than the base of the other.

THEOR. 15. If two triangles have the three sides of the one equal to the three sides of the other, each to each, then the triangles are identically equal, and of the angles. those are equal which are opposite to equal sides.

[Alternative proofs, (i) by Theors. 14 and 5. (ii) By Theors. 6 and 5.]

THEOR. 16. If two triangles have two sides of the one equal to two sides of the other, each to each, but the bases unequal, then the included angles are unequal, the angle of that which has the greater base being

greater than the angle of the other. [By Rule of Conversion.]

THEOR. 17. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides opposite to one pair of equal angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to equal angles. [By Superposition and Theor. 9.]

THEOR. 18. Any two angles of a triangle are together less than two right angles.

COR. I. If a triangle has one right angle or obtuse angle, its remaining angles are acute.

COR. 2. From a given point outside a given straight line, only one perpendicular can be drawn to that line.

THEOR. 19. Of all the straight lines that can be drawn to a given straight line from a given point outside it,

the perpendicular is the shortest; and of the others, those which make equal angles with the perpendicular are equal; and that which makes a greater angle with the perpendicular is greater than that which makes a less angle.

COR. Not more than two equal straight lines can be drawn from a given point to a given straight line.

THEOR. 20. If two triangles have two sides of the one equal to

two sides of the other, each to each, and have likewise the angles opposite to one of the equal sides in each equal, then the angles opposite to the other two equal sides are either equal or supplementary, and in the former case the triangles are identically equal. [By Superposition.]

COR. Two such triangles are identically equal

(1) If the two angles given equal are right angles or obtuse angles.

(2) If the angles opposite to the other two equal sides are both acute, or both obtuse, or if one of them is a right angle.

(3) If the side opposite the given angle in each triangle is not less than the other given side.

SECTION 3.

PARALLELS AND PARALLELOGRAMS.

DEF. 35. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet.

AXIOM 5. Two straight lines that intersect one another cannot both be parallel to the same straight line.

DEF. 36. A trapezium is a quadrilateral that has only one pair of opposite sides parallel.

This figure is sometimes called a trapezoid.

DEF. 37. A parallelogram is a quadrilateral whose opposite sides are parallel.

DEF. 38. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to one another.

and 5 are called alternate

angles; lastly, 1 and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles.

DEF. 39. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted

between perpendiculars let fall on it from the extremities of the former.

THEOR. 21. If a straight line intersects two other straight lines and makes the alternate angles equal, the straight lines are parallel. [Contrapositive of Theor. 9.]

THEOR. 22. If two straight lines are parallel, and are intersected by a third straight line, the alternate angles are equal. [By Rule of Identity, using Ax. 5.]

THEOR. 23. If a straight line intersects two other straight lines and makes either a pair of alternate angles equal, or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary.

THEOR. 24. Straight lines that are parallel to the same straight line are parallel to one another. [Contrapositive of Ax. 5.]

THEOR. 25. If a side of a triangle is produced, the exterior angle is equal to the two interior opposite angles; and the three interior angles of a triangle are together equal to two right angles.