“To what extent am I to open them?”
“That is quite immaterial; you may draw your circle of any magnitude you please, provided it cuts both the lines of the angles we are about to measure. All circles, of whatever dimensions, are supposed to be divided into 360 parts, called degrees; the size, but not the number, of such degrees will therefore increase with the magnitude of the circle. And since the opening of an angle is necessarily a portion of a circle, it must embrace a certain number of degrees; and two angles are, accordingly, said to be equal, when they contain an equal number of them.”
“Now I understand it,” said Louisa: “as the dimensions of an angle depend upon the number of degrees contained between its lines, it evidently must be the opening, and not the length of the lines, that determines the measure of the angle.”
“Say, rather, the value of the angle, for that is the usual expression: but I perceive you understand me; tell me, therefore, how many degrees are contained in each of the two angles formed by one line falling perpendicularly on another, as in the above figure.”
“I perceive that the two angles together are just equal to half the circle; and, since you say that the whole circle is divided into 360 degrees, each angle must measure 90 of them, or the two together make up 180.”
“You are quite right, and I beg you to remember, that an angle of 90 degrees, is called a right angle, and that, when one line is perpendicular to another, it will always form, as you have just seen, a right angle on either side.”
“I now understand,” said Louisa, “what is meant BY lines being at right angles to each other: But, papa,” continued she, “what are obtuse and acute angles, of which I have so often heard you speak?”
Mr. Seymour replied, that he could better explain their nature by a drawing, than by any verbal description. “Here,” said he, “is an acute angle, A; and here an obtuse one, B: the former, you perceive, is one that contains less than 90 degrees; the latter, one which contains more, and is consequently greater than a right angle.”
Louisa fully comprehended the explanation, and observed, that she should remember, whenever an angle measured less than a right angle, that it was acute, and when more, obtuse. “But you have not yet explained to me,” she continued, “the meaning of a triangle.”