...angles. PROPOSITION XXI. THEOREM. If two straight lines intersect a third straight line, making the sum of the interior angles on the same side less than two right anqlcs, the two lines will meet if sufficiently produced. Let the two lines CD, IL, meet the line EF,...

...three sides in their plane, at their middle point*. Any two of these perpendiculars, as DL, EM, will meet if sufficiently produced ; for, if they do not meet, they are parallel (40). Then, AB, which by hypothesis is perpendicular to DL, is also perpendicular to KM parallel to...

...contained by the other two. COR. — If a straight line meeting two other straight lines make the two interior angles on the same side less than two right angles, the two straight lines, if produced, will meet on that side on which are the angles less than two right angles....

...two right angles; but those straight lines which, with another straight line falling upon them, make the interior angles on the same side less than two right angles, do meet together if continually "oduced ; therefore the straight lines AB, CD, if oduced far enough,...

...angles. PROPOSITION XXI. THEOREM. // two straight lines intersect a third straight line, making the sum of the interior angles on the same side less than...than two right angles: then will IL and CD meet if suff1ciently produced. For, if they do not meet, they must be parallel (D. 16). But, if they were parallel,...

...parallels, is also bisected. PROPOSITION XX. THEOREM 127. If two lines are cut by a transversal, making the sum of the interior angles on the same side less than two right angles, the lines will meet if sufficiently produced. Let AB and CD be cut by EF, making Z 1 + Z 2 < 2 KA's. To...

...parallels, is also bisected. PROPOSITION XX. THEOREM 127. If two lines are cut by a transversal, making the sum of the. interior angles on the same side less than two right angles, the lines will meet if sufficiently produced. f Let AB and CD be cut by EF, making Z 1 + Z 2 < 2 RA's....

...parallels, viz. , that — "those straight lines which, with another straight line falling upon them, make the interior angles on the same side less than two right angles, do meet if continually produced." This is exactly the point to be proved, for it is only one form of...

...impossible. "Similarly (2) we can show that the straight line falling on the parallels does not make the interior angles on the same side less than two right angles. " But (3), if it makes them neither greater nor less than two right angles, it can only make the interior...

...right angles are equal to one another. 5. That if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two...