CHAPTER III TRIANGLES: PROOF BY SUPERPOSITION 57. Triangles classified as to sides. - Triangles are classified according to the lengths of their sides, as follows: A scalene triangle is one that has no two sides equal. SCALENE ISOSCELES EQUILATERAL In an isosceles triangle the vertex formed by the intersection of the two equal sides is called the vertex of the triangle, and the side opposite that vertex is called the base. Point out the vertex and the base of the above isosceles triangle. EXERCISES 1. A triangle may be constructed with sides equal to three given linesegments, a, b, and c, as follows: Draw a line-segment AB equal to a. With center at A and radius equal to b, draw an arc. With center at B and radius equal to c, draw a second arc, meeting the first arc at a point C. Draw AC and BC. Then ▲ ABC has its sides equal to a, b, and c by the construction. Draw any three line-segments, then construct a triangle with sides equal to these three segments. 2. Construct an isosceles triangle with base a equal to a given line-segment x and each of the equal sides equal to a given line-segment y. 3. Construct an isosceles triangle each of whose equal sides is twice as long as the base. 4. Construct an equilateral triangle with each side equal to a given line-segment m. 58. Triangles classified as to angles. - Triangles are classified according to their angles, as follows: An acute triangle is a triangle all of whose angles are acute. An obtuse triangle is a triangle one of whose angles is obtuse. A right triangle is a triangle one of whose angles is a right angle. An equiangular triangle is a triangle all of whose angles are equal. ADAA ACUTE OBTUSE RIGHT EQUIANGULAR In a right triangle the side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. Point out the hypotenuse of the above right triangle. EXERCISES a b 1. A triangle may be constructed with two sides and the included angle (angle formed by the two sides) equal to two given line-segments a and b and the given angle x, respectively, as follows: Draw a line-segment AB equal to a. Construct BAC equal to ≤ x as in § 14. Mark off AC equal to b. Draw BC. Then ▲ ABC has two sides and the included angle equal to a, b, and ≤ x, respectively, by construction. Draw any two line-segments and any angle, then construct a triangle with two sides and the included angle equal to these line-segments and angle, respectively. x B 2. Construct a right triangle with the legs equal to two given linesegments m and n. For the method of drawing the right angle, see the construction in § 23. 3. Construct an isosceles right triangle with each leg equal to a given line-segment s. 4. Construct a right triangle with one leg and the hypotenuse equal to two given line-segments a and b, respectively. Note that b must be longer than a. 5. Construct a right triangle with the hypotenuse twice as long as one of the legs. 6. Construct a triangle with two angles and the included side (the side connecting the vertices of those angles) equal to two given angles x and y and a given line-segment a, respectively. C E 59. Altitudes and medians of a triangle. -- A perpendicular to any side of a triangle from the opposite vertex is called an altitude of the triangle. How many altitudes has a triangle? Name the altitudes in A ABC, and tell how each is drawn. A line-segment joining any vertex of a triangle to the middle point of the opposite side is called a median of the triangle. How many medians has a triangle? Name the medians in A MNO, and tell how each is drawn. A F B M R N EXERCISES 1. Draw any triangle ABC, then draw the altitude from the vertex C to the side AB. See § 24 for the construction. 2. Draw an obtuse triangle. Draw the altitude from the vertex of one of the acute angles. Where does it meet the opposite side? 3. Draw a right triangle. Draw the altitude from the vertex of one of the acute angles. Where does it fall? 4. If an altitude of a triangle meets the opposite side produced through a vertex, what kind of triangle is it? 5. If an altitude of a triangle coincides with one of the sides, what kind of triangle is it? 6. If all altitudes of a triangle lie within the triangle, what kind of triangle is it? 7. By using the construction in § 11, draw a median of a triangle. 60. Congruence of figures. -If one of two figures may be placed upon the other so that they coincide throughout, the figures are called congruent. It is evident that if two figures are congruent, for each part of one there is an equal part of the other. The equal parts of two congruent figures are called corresponding parts. In written work the words "is congruent to" are usually expressed by the symbol . Thus the fact that ▲ ABC is congruent to A DEF is written ▲ ABC A DEF. 61. Testing congruence of triangles. Draw on paper (cardboard would be better if it were at hand) a large triangle A B E = ABC. Then construct accurately a second triangle DEF so that DE AB, DF AC, and D=A. The method of = ≤ construction is indicated in the above diagram. Explain it. Having constructed A DEF, cut it out with shears by cutting along the three sides. Now place A DEF upon ▲ ABC so that DE coincides with its equal AB, D falling at A and E at B. When this is done, why does DF fall along AC? Since DF falls along AC, where does the vertex F fall? Why must it? Where does the side EF fall? Why must it? It is thus seen that the triangles are congruent, because they coincide. -- 62. Proof by superposition. Instead of testing the congruence of figures by cutting one out and applying it to the other to see if they can be made to coincide, as in § 61, the congruence may be proved by merely imagining one figure placed upon the other and reasoning out that they must coincide. The imaginary placing of one figure upon the other is called superposition. The following theorems show how triangles are proved congruent by the method of superposition. 63. Theorem. If two sides and the included angle of one triangle are equal respectively to two sides and the included angle of another, the triangles are congruent. A B D F E Hypothesis. In ▲ ABC and ▲ DEF, AB = DE, AC = DF, and A=LD. Conclusion. A ABC A DEF. Proof. 1. AB DE. = Hyp. 2. .. ▲ ABC may be placed upon ▲ DEF so that AB coin cides with DE, A falling upon D and B upon E. 3. LA=LD. 4. . AC must fall along DF. 5. ACDF. 6. .. C must fall upon F. 7. .. CB must coincide with FE. 8. .. ΔΑΒ0~ Δ DEF. § 10 Нур. § 13 Hyp. § 10 § 9, II Def. Congruence Draw two triangles accurately with the parts equal as given in the hypothesis, and write out the complete proof without reference to the book. |