ANGLES OF A TRIANGLE. THEOREM XXII. 76. The sum of the three angles of a triangle equals two right To prove that a + 2 b+c=2 Ls. Produce AB, and let DB bell to AC. Then But and For Then Q. E. D. sm and n, substitute their equals s b and c. 77. COR. 1.-The sum of two angles of a triangle being given, the third can be found by subtracting their sum from two right angles. 78. COR. 2.-If two angles of a triangle are respectively equal to two angles of another, the third angles are also equal. 79. COR. 3.—In any triangle, there can be but one right angle, or but one obtuse angle. 80. COR. 4.-In any right-angled triangle, the sum of the acute angles equals a right angle; that is, they are complements of each other. 81. COR. 5.-In an equiangular triangle, each angle equals one-third of two right angles. THEOREM XXIII. 82. An exterior angle of a triangle equals the sum of the two interior non-adjacent angles. Let ABC be any ▲, and a an exterior . EQUALITY OF TRIANGLES. THEOREM XXIV. 83. Two triangles are equal in all their parts if two sides and the included angle of the one are respectively equal to two sides and the included angle of the other. 1= In the As ABC and DEF, let AB DE, AC A a To prove that ▲ ABC = = A DEF. Place the ABC upon DEF, applying F E a to its equal b, AB on its equal DE, and A Con its equal DF. The points C and B fall on the points F and E; THEOREM XXV. 84. Two triangles are equal in all their parts if two angles and the included side of the one are respectively equal to two angles and the included side of the other. In the As ABC and DEF, let a = d, Lb Le, and AB = DE. = Place the ▲ ABC upon the ▲ DEF, applying AB to its equal DE, the point A on D, and the point B on E. Since a = Ld, AC takes the direction DF, and C falls somewhere in DF or DF produced. Le, BC takes the direction EF, and C falls in EF or EF produced; .. the point C, falling in both DF and EF, falls at their intersection F; Δ ΑΒΓ = Δ DEF. (14) Q. E. D. 85. COR.-If a triangle has a side, its opposite angle, and one adjacent angle, respectively equal to the corresponding parts of another triangle, the triangles are equal. THEOREM XXVI. 86. Two triangles are equal in all their parts if the three sides of the one are respectively equal to the three sides of the other. = In the As ABC and DEF, let AC DF, BC=EF, and AB: = DE. Place ABC in the position DEG, AB in its equal DE, and the Zs a and c adjacent to the sb and d. (55) the points D and E are equally distant from F and G, and DE is to FG at its middle point. On DE as an axis, revolve DEG till it falls in the plane of DEF. Then the point G falls on F, since PG = PF; Δ ΑΒΓ = Δ DEF. (14) (Q. E. D.) 87. COR.-In equal triangles, the equal angles lie opposite the equal sides. 88. SCHOLIUM.-The statement, two triangles are equal, means that the six parts of the one are respectively equal to the six parts of the other. |