Then the lines _g b_, _h d_, have the same direction, and are parallel. TEST LESSON. 1. Prove the same without the colors. 2. Prove the same, using the angle _f h d_. 3. Prove the same, supposing the angles _a g h_, _g h c_, equal to two right angles, and using the angle _a g e_. 4. Prove the same, using the angle _c h f_. See Note E, Appendix. PROPOSITION VIII. THEOREM. The following demonstration is very easy. Read it once, and see if you can go through it without a second reading:— DEMONSTRATION. [Illustration] We wish to prove that _The sum of any two sides of a triangle is greater than the third side._ Let the figure _a b c_ be a triangle, then will the sum of any two sides, as _a c_, _c b_, be greater than the third side _a b_. For the straight line _a b_ is the shortest distance between the two points _a_ and _b_, and is therefore less than the broken line _a c b_. PROPOSITION IX. PROBLEM. The following solution is so easy that you will understand it at once:— We wish _To construct an equilateral triangle on a given straight line._ [Illustration] SOLUTION. Let _a b_ be the given line. With the point _a_ as a centre, and _a b_ as a radius, draw the circumference of the circle, or a part of one. With the point _b_ as a centre, and the same radius _a b_, draw another circumference, or a part of one. From the point _c_, in which the circumferences or arcs intersect, draw the straight lines _a c_ and _b c_. Now, because the lines _a b_ and _a c_ are radii of the same circle, they are equal. And, because the lines _a b_ and _b c_ are radii of the same circle, they are also equal. Then, because the two lines _a c_, _b c_, are separately equal to the line _a b_, they are equal to each other, and the triangle is equilateral. [Illustration] PROPOSITION X. THEOREM. DEVELOPMENT LESSON. Let the figure _a b c_ be a triangle. Produce the side _a c_ to _d_. We have now another angle, _b c d_, and we wish to find out if it is equal to any of the angles of the triangle. From the point _c_ draw the line _c e_ parallel to _a b_. Because the straight line _a d_ intersects the two parallels _a b_, _c e_, the angle _a_ is equal to what other angle? Because the straight line _b c_ intersects the two parallels _a b_, _c e_, the angle _b_ is equal to what other angle? Then the angles _a_ and _b_ are equal to what two angles? How does the angle _b c d_ compare with the angles _b c e_, _e c d_? Then, if the angles _a_ and _b_, on the one hand, and the angle _b c d_, on the other, are separately equal to the angles _b c e_, _e c d_, What have you found out?