...II. 4, which proves that the parallelograms about the diameter of a square are likewise squares. 5. Equal triangles on the same base, and on the same side of it, are between the same parallels. 6. If a straight line be bisected, and produced to any point, the square on the whole line thus produced,...

...figure has sides. If all the angles of an octagon are equal to one another, find the value of each. 3. Equal triangles on the same base, and on the same side of it, are between the same parallels. If two equal lines AB, CD cross one another, and AC is equal to ED, then shall AD be parallel to BC....

...upon two given straight lines is invariable. Under what conditions will this difference be /ею :' 2. Equal triangles on the same base, and on the same side of it, are between the same parallels. The sides .11! and .K'of a triangle are bisected in I) and E respectively, and RE, CD are produced...

...triangle having the same base. CONCLUSION. AB + AC<A'B + AC. PROOF. AA || BC. (253. INVERSE. Equivalent triangles on the same base, and on the same side of it, are between the same parallels.) Draw CND _L AA \ meeting BA produced in D. Join A'D. 4NAC = 4ACB, (168. If a transversal cuts two parallels,...

...equal to half the trapezium. PEOP. XXXIX.— THEOREM. Equal triangles (BAC, BDC) on the same bose (BC) and on the same side of it are between the same parallels. Dem. — Join AD. Then if AD be not parallel to BC, let AE be parallel to it, and let it cut BD in...

...having the same base. CONCLUSION. AB + AC < A'B + A'C. PROOF. A A' || BC. (253. INVERSE. Equivalent triangles on the same base, and on the same side of it, are between the same parallels.) Draw CND ± A A', meeting BA produced in D. Join A'D. 4NAC = £ACB, (168. If a transversal cuts two...

...AB" and for "the rectangle contained by the straight lines AJB, CD " 'is " rect. AR, CD." 1. If two triangles on the same base and on the same side of it have the sides which are terminated in an extremity of the base e*qual, those which are terminated...

...rectilineal angle. 3. Through a given point draw a right line parallel to a given right line. 4. Prove that equal triangles on the same base and on the same side of it are between the same parallels. 5. The sum of the four angles of any quadrilateral is equal to four right angles. 6. Prove that the...

...XY>PQ: ie PQ is the shortest chd. through O. THEOREM (3) — (Converse of iii. 21.) If any number of triangles, on the same base, and on the same side of it, have equal vertical angles, their •vertices all lie on the arc of a segment of a circle, of which...

...that the triangle formed by joining the centres of the circles is congruent with ABC. Ex. 424. — Two triangles on the same base and on the same side of it, have equal vertical angles. Show that the join of their vertices is parallel to the join of their orthocentres....