11. Prove the interior opposite angles _a g h_, _g h c_, equal to two right angles, using the angle _g h d_. 12. Prove the same, using the angle _c h f_. 13. Prove the same, using the angle _a g e_. 14. Prove the interior opposite angles _b g h_, _g h d_, equal to two right angles, using the angle _a g h_. 15. Prove the same, using the angle _e g b_. 16. Prove the same, using the angle _f h d_. Compare the angles Yellow, Green, each with its exterior opposite angle, and see if you can prove that the exterior opposite angles _e g b_, _f h d_, are also equal to two right angles. [Illustration: Diagram 45.] PROPOSITION VII. THEOREM. DEVELOPMENT LESSON. Suppose we do not know whether the lines _a b_, _c d_, are parallel, or not; But, by measuring, we find that the interior angles Blue, Yellow, on the same side of the secant[4] line _e f_, are equal to two right angles: Footnote 4: “Secant” means “_cutting_.” The adjacent angles Blue, Red, are equal to what? Then, if the interior angles Blue, Yellow, are equal to two right angles, And the adjacent angles Blue, Red, are also equal to two right angles, What do you infer? From the interior angles Blue, Yellow, take away the angle Blue: what remains? From the adjacent angles Blue, Red, take away the same angle Blue: what remains? What do you know of the two remainders? The angle Red measures the direction of the line _g b_ from what line? The equal angle Yellow measures the direction of the line _h d_ from what line? Then if the lines _g b_, _h d_, have the same direction from the line _e f_, what do you call them? [Illustration: Diagram 46.] DEMONSTRATION. We wish to prove, that, _If a straight line intersects two other straight lines so that two interior angles on the same side of the intersecting line are equal to two right angles, the two lines are parallel._ Let the straight line _e f_ intersect the two straight lines _a b_, _c d_, in the points _g_ and _h_, so that the angles Red, Blue, are equal to two right angles. Then will the lines _a b_, _c d_, be parallel. For the angles Red, Blue, are supposed equal to two right angles. The adjacent angles Red, Green, are known to be also equal to two right angles. Then the interior angles Red, Blue, are equal to the adjacent angles Red, Green. If from the interior angles Red, Blue, we take away the angle Red, we have left the angle Blue. If from the adjacent angles Red, Green, we take the same angle Red, we shall have left the angle Green. Then the angle Blue is equal to the angle Green. But the angle Blue measures the direction of the line _h d_ from the line _e f_. And the angle Green measures the direction of the line _g b_ from the line _e f_.