Right angle

“ Heighths, by the Perpendicular or Plumbe: And Angles, by the square.] Contrariwise, an Oblique angle it is, when the one shanke standeth so upon another, that it inclineth, or leaneth more to one side, then it doth to the other: And one angle on the one side, is greater then that on the other.
Therefore,
14. All straight-shanked right angles are equall.
[That is, they are alike, and agreeable, or they doe fill the same place; as here are aio. and eio. And yet againe on the contrary: All straight shanked equall angles, are not right-angles.] ”

“ The perpendicularity of the one line, and the horizontality of the other, must not in any way be interfered with; in the sketch they should, in fact, meet and form a right angle, which should just be rounded off. ”

“ Again a right-angle is a perceived object which can be situated in many events; but, though rectangularity is posited by sense-awareness, the majority of geometrical relations are not so posited. Also rectangularity is in fact often not perceived when it can be proved to have been there for perception. Thus an object is often known merely as an abstract relation not directly posited in sense-awareness although it is there in nature. ”

“ If the angle of inclination be semirect, and the line of inclination be in the thicker medium, and the proportion of their densities be the same with that of the diagonal to the side of a square, and the separating superficies be plane, the refracted line will be in the separating superficies.—8. If a body be carried in a strait line upon another body, and do not penetrate the same, but be reflected from it, the angle of reflection will be equal to the angle of incidence.—9. The same happens in the generation of motion in the line of incidence. ”

“ If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then if each of the remaining angles be either less, or not less, than a right angle, or if one of them he a right angle, the triangles shall he equiangular to one another, and shall have those angles equal about touch the sides are proportionals. ”

“ The Visual Angle.—To examine objects carefully we usually bring them as close to the eye as possible, for the nearer to the eye the object is brought, the larger is the visual angle formed by it (see Fig. 387) , and the larger is its image upon the retina. The visual angle of an object is the angle at the eye lens between the rays that have come from the ends of the object. Consequently the more distant the object, the smaller is its visual angle. ”

“ On a given straight line to describe a segment of a circle, containing an angle equal to a given rectilineal angle.
Let AB be the given straight line, and C the given rectilineal angle: it is required to describe, on the given straight line AB, a segment of a circle containing an angle equal to the angle C.
First, let the angle G be a right angle. ”

“ How many pairs of vertical angles are formed by the intersection of two lines?
Read all the pairs of vertical angles formed by the lines l m, n p.
LESSON NINTH. KINDS OF ANGLES. RIGHT ANGLES. What do we call the angles a o c, c o b? (Diagram 9.)
Are they equal to each other?
Then they are called right angles.
A right angle is one of two adjacent angles that are equal to each other.
Are the adjacent angles c o b, b o d equal to each other?
Then what are they called?
Read the right angles below the line a b. On the left of c d.
Read three right angles whose vertices are at p.
24
Diagram 9. ”

“ The rectangular cross, though more positive and effective, is no more potential in securing this unity than the crossing of lines at a long angle. A series of right angles will in time arrive at the same point as the tangent, but less quickly. ”

“ The angle formed between any objects and our eye is the only means by which our sight estimates their size and distance, and as this angle is the simple effect of complex causes, the judgment we form does not distinguish between the several causes; we are compelled to be inaccurate. ”

“ A solid angle may be conceived two ways. First, for the aggregate of all the angles, which are made by the motion of a strait line, while one extreme point thereof remaining fixed, it is carried about any plain figure, in which the fixed point of the strait line is not contained. ”

“ Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of an individual ”

“ A space that is arched looks wider than it actually is, for the eye unconsciously follows the lines of the arch, and a distance or width effect is the result. The same space treated with a straight line is quickly bridged. The same space treated with lines that come to an angle looks narrower, for the reason that the eye becomes focused by the apex of the angle, and a height effect, not a breadth effect, is the result. ”

“ We perceive more distinctly the difference of direction presented at their intersection by the two sides of an acute or obtuse angle, than the deviation that exists between one of the sides and the perpendicular which we imagine placed on the other side, but which is not marked. ”

“ Nothing is contributed towards the quantity of an angle, neither by the length, nor by the equality, nor by the inequality of the lines which comprehend it. For the lines A B and A C comprehend the same angle which is comprehended by the lines A E and A F, or A B and A F. Nor is an angle either increased or diminished by the absolute quantity of the arch, which subtends the same; for both the greater arch B C and the lesser arch E F are subtended to the same angle. ”

“ Apollonius defined angle as "a contracting of a surface or a solid at one point under a broken line or surface." Another of the Greeks defined it as "a quantity, namely, a distance between the lines or surfaces containing it." Schotten [56] says that the definitions of angle generally fall into three groups:
a. An angle is the difference of direction between two lines that meet. This is no better than Euclid's, since "difference of direction" is as difficult to define as "inclination."
b. An angle is the amount of turning necessary to bring one side to the position of the other side. ”

“ They must, like us, have had to distinguish between right angles and oblique angles. They may have come to know, as some of the inhabitants of the earth came to know, four thousand years ago, that, in a right-angled triangle, the square on the larger side is equal to the sum of the squares on the other two sides. ”

“ Thirdly, that the bases be equall: which three conditions if they doe meete, it must needes be that both the angles shall bee equall: but if one of them be wanting, of necessity againe they must be unequall.]
This shall hereafter be declared and made plaine by many and sundry practises: and therefore here we bring no example of it.
12. An angle is either right or oblique.
Thus much of the Affections of an angle; the division into his kindes followeth. An angle is either Right or Oblique: as afore, at the 4 e ij. a line was right or straight, and oblique or crooked. [29]
13. ”

“ Why is m n g an acute angle?
What is an acute angle?
Here the attention is directed first to this particular angle; then this is taken as an example of its kind, and the idea generalized by describing the class. See also page 29.
Why are the lines e f and g h said to be parallel?
When are lines said to be parallel?
Many of the questions are intended to test the vividness of the pupil’s conception. See page 29.
Also page 78. If the circumference were divided into 360 equal parts, would each arc be large or small?
Many of the questions are intended to test the attention of the pupil. ”

“ In endeavouring, for example, to grasp the relation of the exterior angle to the two interior and opposite angles, the pupil may fail because he has not a clear knowledge of the equality of angles in connection with parallel lines. ”