A more practical use for it is made by machinists to determine whether a casting is a true semicircle. Taking a carpenter's square as here shown, if the vertex touches the curve at every point as the square slides around, it is a true semicircle. By a similar method a circle may be described by sliding a draftsman's triangle so that two sides touch two tacks driven in a board. [Illustration] Another interesting application of this corollary may be seen by taking an ordinary paper protractor _ACB_, and fastening a plumb line at _B_. If the protractor is so held that the plumb line cuts the semicircle at _C_, then _AC_ is level because it is perpendicular to the vertical line _BC_. Thus, if a class wishes to determine the horizontal line _AC_, while sighting up a hill in the direction _AB_, this is easily determined without a spirit level. It follows from this corollary, as the pupil has already found, that the mid-point of the hypotenuse of a right triangle is equidistant from the three vertices. This is useful in outdoor measuring, forming the basis of one of the best methods of letting fall a perpendicular from an external point to a line. [Illustration] Suppose _XY_ to be the edge of a sidewalk, and _P_ a point in the street from which we wish to lay a gas pipe perpendicular to the walk. From _P_ swing a cord or tape, say 60 feet long, until it meets _XY_ at _A_. Then take _M_, the mid-point of _PA_, and swing _MP_ about _M_, to meet _XY_ at _B_. Then _B_ is the foot of the perpendicular, since [L]_PBA_ can be inscribed in a semicircle. 2. _Angles inscribed in the same segment are equal._ [Illustration] By driving two nails in a board, at _A_ and _B_, and taking an angle _P_ made of rigid material (in particular, as already stated, a carpenter's square), a pencil placed at _P_ will generate an arc of a circle if the arms slide along _A_ and _B_. This is an interesting exercise for pupils. THEOREM. _An angle formed by two chords intersecting within the circle is measured by half the sum of the intercepted arcs._ THEOREM. _An angle formed by a tangent and a chord drawn from the point of tangency is measured by half the intercepted arc._ THEOREM. _An angle formed by two secants, a secant and a tangent, or two tangents, drawn to a circle from an external point, is measured by half the difference of the intercepted arcs._