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87. DEF. A Right triangle is one which has one of the angles a right angle.
88. Der. The side opposite the right angle is called the Hypotenuse.
89. DEF. An Obtuse triangle is one which has one of the angles an obtuse angle.
90. DEF. An Acute triangle is one which has all the angles acute.
91. Def. An Equiangular triangle is one which has all the angles equal.
92. Def. In any triangle, the angle opposite the base is called the Vertical angle, and its vertex is called the Vertex of the triangle.
93. Def. The Altitude of a triangle is the perpendicular distance from the vertex to the base, or the base produced.
94. DEF. The Exterior angle of a triangle is the angle included between a side and an adjacent side produced, as Z CBD.
95. Def. The two angles of a triangle which are opposite the exterior angle, are called the two opposite interior angles, as Is A and C.
96. Any side of a triangle is less than the sum of the other two sides.
Since a straight line is the shortest distance between two points,
AC<A B + B C.
97. Any side of a triangle is greater than the difference of the other two sides.
In the inequality A C< A B + BC,
transpose A B. Then
AC – AB<BC'; or
BC > AC – A B.
Ex. 1. Show that the sum of the distances of any point in a triangle from the three angles of the triangle is greater than half the sum of the sides of the triangle.
2. Show that the locus of all the points at a given distance from a given straight line A B consists of two parallel lines, drawn on opposite sides of A B, and at the given distance from it.
3. Show that the two equal straight lines drawn from a point to a straight line make equal acute angles with that line.
4. Show that, if two angles have their sides perpendicular, each to each, they are either equal or supplementary.
PROPOSITION XXI. THEOREM.
98. The sum of the three angles of a triangle is equal to two right angles.
Let A B C be a triangle.
We are to prove < B + Z BCA + LA = two rt. s.
Draw C E ll to A B, and prolong A C. Then < ECF +ECB +_ BCA = 2 rt. 4, § 34 (the sum of all the As about a point on the same side of a straight line
= 2 rt. ).
§ 68 (being alt.-int. £). Substitute for ZEC Fand Z BC E their equal Ł, A and B.
L A + 2B +_ BCA = 2 rt. 4.
Q. E. D.
99. COROLLARY 1. If the sum of two angles of a triangle be known, the third angle can be found by taking this sum from two right angles.
100. Cor. 2. If two triangles have two angles of the one equal to two angles of the other, the third angles will be equal.
101. Cor. 3. If two right triangles have an acute angle of the one equal to an acute angle of the other, the other acute angles will be equal.
102. Cor. 4. In a triangle there can be but one right angle, or one obtuse angle.
103. Cor. 5. In a right triangle the two acute angles are complements of each other.
104. Cor. 6. In an equiangular triangle, each angle is one third of two right angles, or two thirds of one right angle.
PROPOSITION XXII. THEOREM. 105. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Let BC H be an exterior angle of the triangle A B C. We are to prove ZBCH= L A + B.
Z BCH + LAC B= 2 rt. &, § 34
(being sup.-adj. €). LA + 2B + LACB= 2 rt. $ $ 98
(three & of a A = two rt. 6). ..L BCH + LACB=LA +B+LACB. Ax. 1.
Take away from each of these equals the common ZACB; then ZBCH=2 A+ Z B.
Q. E. D.
· Proposition XXIII. THEOREM. 106. Two triangles are equal in all respects when two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.
In the triangles A B C and A' B'C', let A B = A' B',
A C= A' C', ZA= A'.
Take up the A A BC and place it upon the A A' B'C' SO that A B shall coincide with A' B'.
Then AC will take the direction of A' C',
(for ZA= Z A', by hyp.),
(for A C = A' C', by hyp.);
Q. E. D.