107. An exterior angle of a triangle is an angle formed outside the triangle, between one side of the triangle and another side prolonged. [< ABX.] The angles within the triangle, at the other vertices are the opposite c interior angles. [44 and 4 c.] B X 108. THEOREM. An exterior angle of a triangle is equal to the sum of the opposite interior angles. 109. THEOREM. An exterior angle of a triangle is greater than either of the opposite interior angles. (See Ax. 5.) 110. THEOREM. The sum of the angles of any triangle is two right BCX= 2 A + ZB (?) (108). ..ZA+ZB+ ACB = 2 rt. 4=180° (Ax. 6). Q.E.D. 111. COR. The sum of any two angles of a triangle is less than two right angles. (See Ax. 5.) 112. COR. A triangle cannot have more than one right angle 01 more than one obtuse angle. 113. COR. Two angles of every triangle are acute. (See 112.) 114. THEOREM. The acute angles of a right triangle are complementary. Proof Their sum = 1 rt. 2 (110 and Ax. 2). Hence they are complementary. (See 20.) 115. COR. Each angle of an equiangular triangle is 60°. 116. THEOREM. If two right triangles have an acute angle of one equal to an acute angle of the other, the remaining acute angles are equal. (See 114 and 48.) 117. THEOREM. If two triangles have two angles of the one equal to two angles of the other, the third angle of the first is equal to the third angle of the second. (See 110 and Ax. 2.) 118. THEOREM. Two triangles are equal if a side and any two angles of the one are equal respectively to a homologous side and the two homologous angles of the other. Proof: The third .. the A are = (54). of one ▲ = third of other ▲ (117). 119. THEOREM. Two right triangles are equal if a leg and the opposite acute angle of one are equal respectively to a leg and the opposite acute angle of the other. (See 118.) = = Ex. 1. In the figure of 107, if ▲ A 40° and C 70°, degrees are there in Z ABX? in 4 ABC? how many Ex. 2. If each of the equal angles of an isosceles triangle is 50°, how many degrees are there in the third angle? Ex. 3. If one of the acute angles of a right triangle is 25°, how many degrees are there in the other? Ex. 4. State and prove the converse of 114. 120. THEOREM. If two angles of a triangle are equal, the triangle is isosceles. [Converse of 55.] Given: AABC; ZA=ZC. To Prove: AB = BC. Proof: Suppose BX drawn L to AC. In rt. A ABX and CBX, BX = BX (?); ▲ A = 2¢ (?). X .. ▲ ABX = ▲ CBX (?) (119). .. AB=BC (?). Q.E.D. 121. THEOREM. An equiangular triangle is equilateral. 122. THEOREM. If two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side. Given: A ABC; AB>AC. To Prove: ≤ ACB > ≤ B. Proof On AB take AR = AC. [We may, because AB > AC.] Draw CR and let Z ARC = x. B ARC is an ext. ≤ of ▲ CBR (?). .. <x> ≤ B (109). Also, ▲ ACR = Z ARC = Zx (?) (55). Again, ▲ ACB> ≤ x (?) (Ax. 5). ..Z ACB > ≤ B (Ax. 11). Q.E.D. 123. THEOREM. If two angles of a triangle are unequal, the side opposite the greater angle is greater than the side opposite the less angle. Given: A ABC; LACB><B. To Prove: AB > AC. Proof: In ACB, suppose BCR constructed = LB. Also AR+CR > AC (?). R .. AR+BR > AC (Ax. 6). That is, AB > AC. B Q.E.D. 124. THEOREM. The hypotenuse is the longest side of a right tri angle. (See 123.) QUADRILATERALS 125. A quadrilateral is a portion of a plane bounded by four straight lines. These four lines are called the sides. The vertices of a quadrilateral are the four points at which the sides intersect. The angles of a quadrilateral are the four angles at the four vertices. The diagonal of a rectilinear figure is a line joining two vertices, not in the same side. 126. A trapezium is a quadrilateral having no two sides parallel. A trapezoid is a quadrilateral having two and only two sides parallel. A parallelogram is a quadrilateral having its opposite sides parallel (□). TRAPEZOID PARALLELOGRAM SQUARE RECTANGLE 127. A rectangle is a parallelogram whose angles are right angles. A rhomboid is a parallelogram whose angles are not right angles. 128. A square is an equilateral rectangle. A rhombus is an equilateral rhomboid. 129. The side upon which a figure appears to stand is called its base. A trapezoid and all kinds of parallelograms are said to have two bases, the actual base and the side parallel to it. The non-parallel sides of a trapezoid are sometimes called the legs. An isosceles trapezoid is a trapezoid whose legs are equal. The median of a trapezoid is the line connecting the midpoints of the legs. The altitude of a trapezoid and of all kinds of parallelograms is the perpendicular distance between the bases. 130. THEOREM. The opposite sides of a parallelogram are equal. 131. COR. Parallel lines included between parallel lines are equal. (See 130.) 132. COR. The diagonal of a parallelogram divides it into two equal triangles. 133. COR. The opposite angles of a parallelogram are equal. (See 27.) 134. THEOREM. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. [Converse of 130.] = A || AD = BC (?). .. ▲ ABD = ▲ CBD (?) (58). Q. E.D. |