What axiom have you just employed? To what same thing have you found two other things equal? What two things did you find equal to it? DEMONSTRATION. We wish to prove, that, _If any side of a triangle be produced, the new angle formed will be equal to the sum of the angles that are not adjacent to it._ Let _a b c_ be a triangle. Produce the side _a c_ to _d_; then will the new angle _b c d_ be equal to the sum of the angles _a_ and _b_. For from the point _c_ draw _c e_ parallel to _a b_. Then, because the straight line _a d_ intersects the two parallels _a b_, _c e_, in the points _a_ and _c_, The opposite exterior and interior angles _a_ and _e c d_ are equal to each other. And because the straight line _b c_ intersects the same parallels in the points _b_ and _c_, The interior alternate angles _b_ and _b c e_ are equal. Then the angles _a_ and _b_ of the triangle are equal to the angles _b c e_ and _e c d_. But the new angle _b c d_ is equal to the angles _b c e_, _e c d_. Then because the new angle _b c d_, and the angles _a_ and _b_ are separately equal to the angles _b c e_, _e c d_, they are equal to each other. [Illustration] PROPOSITION XI. THEOREM. DEVELOPMENT LESSON. Let the figure _a b c_ be a triangle. Produce the side _a c_ to _d_. By the last theorem, the angle _b c d_ is equal to what angles of the triangle? What angle must we add to these angles to make up the three angles of the triangle? If we add the same angle to the angle _b c d_, what adjacent angles do we get? Then the three angles of the triangle, _a_, _b_, and _c_, are equal to what two angles? But the adjacent angles _a c b_ and _b c d_ are equal to what? Then, because the three angles of the triangle, _a_, _b_, and _c_, and two right angles, are separately equal to the two adjacent angles _c_ and _b c d_. What new thing have you found out? DEMONSTRATION. We wish to prove that _The three angles of any triangle are equal to two right angles._ Let the figure _a b c_ be a triangle; then will the sum of the angles _a_, _b_, and _c_, be equal to two right angles. For, produce the side _a c_ to _d_. The new angle _b c d_ is equal to the sum of the angles _a_ and _b_. If to the angles _a_ and _b_ we add the angle _c_, we shall have the three angles of the triangle. If to the angle _b c d_ we add the same angle _c_, we shall have the adjacent angles _c_ and _b c d_. Then the three angles of the triangle _a_, _b_, _c_, are equal to the adjacent angles _c_ and _b c d_. But the adjacent angles _c_ and _b c d_ are equal to two right angles. Then, because the three angles of the triangle are equal to the adjacent angles _c_ and _b c d_, they are equal to two right angles. [Illustration]