...EAD will be equal to the angle KFor, draw the chord DE- Then the two triangles IKL and EAD, having the three sides of the one equal to the three sides of the other, each to each, the angle EAD will be equal -o the angle K (Bk- I- Th- viii)PROBLEM IXThrough a g1ven point to draw a line...

...AB. If it falls upon AB, the theorem is true by A. 7. Therefore, if two triangles, etc. THEOREM IX. If two triangles have the three sides of the one equal...three sides of the other, each to each, the triangles will be equal in all their parts. Let ABC and DEF be two triangles, having AB equal to DE, AC to DF,...

...similarity to those of the other. " — Campbells Rhet. This structure is very often neglected. Examples : "If two triangles have the three sides of the one...three sides of the other, each to each, the triangles will be equal in all their parts." The article before one here is improper, because the reference is...

...one triangle can be constructed so as to have its sides equal to three given lines, it is clear that if two triangles have the three sides of the one equal to the three sides of the other, these triangles must be identical, or be equal in all respects. And a similar remark may be made on...

...are therefore equal. And if the chord AB is given equal to the chord PQ, then the triangles AOB, POQ have the three sides of the one equal to the three sides of the other; and therefore the angle AOB=\he angle POQ; and therefore also the arc AB = the arc PQ. COR. 1. If the...

...than ABC. PROPOSITION XVII. — THEOREM. ELEMENTS OP GEOMETRY. Let ABC, DBF be two triangles, having the three sides of the one equal to the three sides of the other, eaeh to each, namely, AB to DE, AC toDF, andCB toEF; then their triangles will be equivalent. Let 0...

...angle c, opposite the side A B. I) B PROPOSITION XIX. THEOREM. DEMONSTRATION. We wish to prove that, If two triangles have the three sides of the one equal to the three sides of the other, each to each, they are equal in all their parts. Let the two triangles ABC, ADC, have the side AB of the one equal...

...remain ED = CF, (Ax. 3); but BE = AC, and AF= BD, (Th. 24); hence we have two A's, CAF and EBD, which have the three sides of the one equal to the three sides of the other, each to each; therefore, the two A's are equal, (Th. 21). If, from the whole figure ABDC, we take away the A CAF,...

...to it ; hence the angle A must be greater than the angle D. PROPOSITION XVIII. — THEOREM. BOOK I. Let the triangles ABC, DEF have the side AB equal to DE, AC to DF, and BC to EF ; then will the angle A be equal to D, the angle B to. the angle E, and the...

...called a direct, and the latter an indirect demonstration. THEOREM via. If two triangles have tlie three sides of the one equal to the three sides of the other, each to each, the three angles will aho be equal, each to each. Let the two triangles ABC, ABD, have the side AB equal...