Let the figure _a b c d_ be a parallelogram; then will the sides _a b_ and _c d_ be equal to each other; likewise the sides _a d_ and _b c_. For, draw the diagonal _b d_. Because the figure is a parallelogram, the sides _a b_ and _d c_ are parallel, and the interior alternate angles _n_ and _o_ are equal. Because the figure is a parallelogram, the interior alternate angles _r_ and _m_ are equal. Then the two triangles _a d b_, _b d c_, have two angles and the included side of the one equal to two angles and the included side of the other, each to each, and are therefore equal; And the side _a b_ opposite the angle _m_ is equal to the side _c d_ opposite the equal angle _r_; And the side _a d_ opposite the angle _n_ is equal to the side _b c_ opposite the equal angle _o_. TEST. Prove the same by drawing a diagonal from _a_ to _c_. [Illustration] PROPOSITION XVI. THEOREM. DEVELOPMENT LESSON. Suppose A B to be a straight line, and C any point out of it. From the point C draw a perpendicular C F to A B. Let us see if this perpendicular is not shorter than any other line we can draw from the same point to the same line. Draw any other line from C to A B as C E. Now, as C E is any line whatever other than a perpendicular, if we find that the perpendicular C F is shorter than it we must conclude that it is the shortest line that can be drawn from C to A B. Produce C F until F D is equal to C F, and then join E and D. In the triangles E F C, E F D, what two sides were drawn equal? What line is a side to each? How great an angle is C F E? What is a right angle? Then how do the angles C F E and E F D compare with each other? If the two triangles E F C, E F D, have the side C F of the one equal to the side F D of the other, the side E F common to both, and the included angle E F C of the one equal to the included angle E F D of the other, each to each, what do you infer? Then what third side of the one have you found equal to a third side of the other? C E is what part of the broken line C E D? C F is what part of the line C D? Which is shorter, the straight line C D, or the broken line C E D? Then how does the half of C D or C F compare with the half of C E D or C E? If C E is any line whatever other than a perpendicular, what may we now say of the perpendicular from the point C to the straight line A B? [Illustration] DEMONSTRATION. We wish to prove that _A perpendicular is the shortest distance from a point to a straight line._ Let A B be a straight line, and C A point out of it; then will the perpendicular C E be the shortest line that can be drawn from the point to the line. For draw any other line from C to A B, as C F.