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QUESTIONS ON SECTION I.
1. What is meant by the Elements of Plane Geometry?
2. Explain the terms axiom, theorem, converse, contrapositive, giving examples of each.
3. State the Geometrical Axioms.
4. What is meant by the axiom of the rule of Identity?
5. State the fact that “all geese have two legs” in the form of a theorem, with hypothesis and conclusion; and write down its obverse, converse, and contrapositive theorems.
6. Define a plane surface, and give the test by which a surface is ascertained to be or not to be plane.
7. On what does the magnitude of an angle depend? Shew that its magnitude does not depend on the length of the arms.
8. What is meant by saying that two points determine a straight line?
9. What are adjacent angles; supplementary angles ; reflex angles ?
10. Shew how to find the sum and difference of two straight lines : and prove that their sum and difference together are double of the greater of the two straight lines.
II. Given the sum and difference of two straight lines; find the lengths of the straight lines.
12. Enunciate and prove the obverse and converse of Theorem 4.
Def. 30. An isosceles triangle is that which has two sides equal.
Def. 31. A right-angled triangle is that which has one of its angles a right angle. An obtuse-angled triangle is that 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.
THEOREM 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.
Part. En. Let the two triangles BAC, EDF have two sides of the one equal to two sides of the other, each to each, and likewise the included angles equal, viz.
BA = ED,
AC=DF, and the included angle BAC = the included angle EDF; it is required to prove that the triangles are equal in all respects, viz. the base BC equal to the base EF, and the angle B to the angle E, and the angle C to the angle F, and the area ABC to the area DEF.
Proof. If the point A be placed on the point D, and the line AB were placed along DE, then because the angle BAC = the angle EDF, (Hyp.) therefore the line AC would lie along DF. And because AB=DE,
(Hyp.) therefore the point B would coincide with the point E. And because AC=DF;
(Hyp.) therefore the point C would coincide with the point F.
Therefore BC would coincide with EF, (Ax. 2.) and therefore BC= EF:
(Ax. 1.) and the angles B and C respectively coincide with and are equal to the angles E and F, and the area of the triangle BAC coincides with and is equal to the area of the triangle EDF*.
Q. E. D. * Eucl. I. 4.
THEOREM 6. The angles at the base of an isosceles triangle are equal to one another *.
Part. En. Let ABC be an isosceles triangle, having the side AB equal to the side AC; it is required to prove that the angle B is equal to the angle C.
Proof. Let AX be the bisector of the angle BAC,
(Ax. 4.) meeting the base BC in X. Then in the triangles BAX, CAX we have BA = AC,
(Hyp.)) AX common, and the included angle BAX=the included angle CAX.
(Hyp.)' Therefore the triangles are equal in all respects, (Th. 5.) that is, the angle at B = the angle at C.
Q. E. D. COR. 1. If the equal sides be produced the angles on the other side of the base will be equal.
CÓR. 2. If a triangle is equilateral it is also equiangular.
* Eucl. I. 5.
THEOREM 7. If two triangles have one side of the one equal to one side of the other, and the angles at the extremities of those sides equal, each to each, then the triangles are equal in all respects, those sides being equal which are opposite to the equal angles *.
Part. En. Let the triangles ABC, DEF have
BC=EF, the angle B= the angle E, and the angle C= the angle F; it is required to prove that the triangles are equal in all respects.
Proof. For if the point B were placed on the point E, and the line BC along the line EF; then because BC = EF,
(Hyp.) therefore the point C would fall on F.
And because the angle B = the angle E, (Hyp.) therefore the line BA would fall along the line ED.
And because the angle C=the angle F, (Hyp.) therefore the line CA would fall along the line FD: therefore the point A would fall on the point D, since two straight lines can intersect in one point only;
(Ax. 2.) and therefore the triangles coincide and are equal in all respects, AB being equal to DE, AC to DF, and the
* Eucl. 1. 26. Part 1.