XVII AXIOMS-THEOREMS CONNECTED WITH CIRCLES THERE are some statements in Geometry which are so easy to understand, that nobody can deny their truth. Such a statement is called an AXIOM. Here are two Axioms: Things which are equal to the same thing, are equal to one another. All right angles are equal. Both these Axioms are so obviously true, that you would laugh at anyone who said they were not true. To prove 1. Take 3 pennies. Call one A, the next B and the next C. Now, since A is 1d. and B is 1d. A is equal to B. And, since A is 1d. and C is ld. A is equal to C. And, since B is 1d. and C' is 1d. B is equal to C. So, A, B and C are all equal. To prove the next axiom, measure any number of right angles with your protractor. They all contain 90 degrees. So they must be equal. Axioms are often referred to in Geometry, when you want to prove a proposition. You will remember, a proposition which tells you to construct something, is called a Problem, andhas the letters Q.E.F. put at the end of it. But a Proposition which tells you to prove something is called a Theorem, and has the letters Q.E.D. (quod erat demonstrandum, which was to be proved) put at the end of it. You may use the sign (two eyes and a nose) for the word "because," and the same sign upside down for the word "therefore,' .. when you are writing out the proof of a proposition. A Theorem consists of two clauses. The first clause tells us what we are to assume, and is called the Hypothesis. The second tells us what it is required to prove and is called the Conclusion. If the hypothesis and conclusion are interchanged, a new Theorem is made, which is called a Converse Theorem. There are some Theorems, connected with angles, which can be proved, but certain names and definitions must first be known. An Angle is the inclination of two lines to one another, which meet together, but are not in the same straight line. The Arms of the angle are the lines which form the angle. The Vertex of the angle is the point at which the lines meet. Adjacent angles are those which lie on either side of a common arm. Vertically opposite angles are those made when two straight lines cross one another. B In the above diagram you have 4 angles AXC, CXB, BXD, DXA. If you take AB as the common arm, the Adjacent Angles will be AXC and CXB on one side of AB, and AXD, DXB on the other side of AB. If you measure these angles, you will find that they are together equal to two right angles. You can find the adjacent angles if you take CD as the common arm. As AB and CD cross each other at the point X, the vertically opposite angles will be :— 1. AXC and BXD. 2. AXD and CXB. If you measure them you will find they are equal. We have proved by actual measurement : 1. The Adjacent Angles which one straight line makes with another straight line, on one side of it, are together equal to two right angles. 2. If two straight lines intersect one another then the vertically opposite angles are equal. XVIII ADJACENT ANGLES ARE EQUAL TO TWO RIGHT ANGLES. VERTICALLY OPPOSITE ANGLES ARE EQUAL Theorem (Book I, Prop. 13). THE adjacent angles which one straight line makes with another straight line, on one side of it, are together equal to two right angles. 8 Given. Let the straight line CO make, with the straight line AB the adjacent s AOC, COB. Required. To prove that the 28 AOC, COB are together equal to 2 Ls. Construction. Draw OD at Ls to AB. Proof. the LS AOC, COB together the S AOC, COD, DOB. And the LS AOD, DOB together LS AOC, COD, DOB. = = the thes AOC, COB together the s AOD, DOB. = But the SAOD, DOB are Ls. = 2 Ls. Q.E.D. You will note, in the above proposition, Euclid has made use of some axioms. (Things which are equal to the same thing are equal to one another. All right angles are equal.) He has also made use of an old proposition in the construction, for he has drawn OD at Ls to AB. This construction gives the key to the proof. Axiom. If equals be taken from equals the remainders are equal. Theorem (Book I, Prop. 15). If two straight lines intersect one another, then the vertically opposite angles shall be equal. Given. Let the straight lines AB and CD intersect at E. Required. To prove that the AEC = AB meets the straight line CD in E, . adj. s AEC, CEB = 2 Ls. |