Of Surfaces.-Of Triangles. 14. A triangle is also half a parallelogram, having the same base and altitude. 15. The area of a triangle is equal to half the product of the base by the altitude; for, the base multiplied by the altitude gives a rectangle, which is double the triangle. Thus, D B the area of the triangle ABC, is equal to half the product of ABX CD. If the base of a triangle is 12, and the altitude 8 yards, the area will be 48 square yards. QUEST.-14. If a triangle and a parallelogram have the same base and altitude, how do they compare with each other? 15. What is the area of a triangle equal to? If the base of a triangle is 20 feet, and its altitude 5 feet, what is its area? If the base is 40 yards, and the altitude 6 yards, what is the area? If the base is 16 rods, and altitude 10 rods, what is the area? Properties of the Triangle. SECTION VI. PROPERTIES OF THE TRIANGLE. 1. If two triangles have two sides, and an included angle of the one, equal to two sides and the included angle of the other, each to each, the remaining parts will also be equal. AA BD That is, if we have the two triangles, ABC and DEF, having AC=DF, CB=FE, and angle C=F, then will 2. If two triangles have two angles, and the included side of the one equal to two angles and the included side of the other, the remaining parts will also be equal. ΔΔ QUEST.-1. Name the parts of one triangle, which being equal to the corresponding parts of another, will cause the remaining parts of the triangles also to be equal. 2. If two triangles have a side, and the adjacent angles in each equal, will the remaining parts also be equal? Properties of the Triangle. That is, if we have two tri angles ABC and DEF, having Angle A=D, angle B=E and AB=DE, then will AC-DF, CB=FE and A angle C-F. 3. The angles opposite the equal sides of an isosceles triangle are equal. Thus, if ABC be an isosceles triangle, the angle A=B. 4. A line drawn from the vertical angle, perpendicular to the base of an isosceles triangle, will divide the base into two equal parts. Thus, if A CD is perpendicular to AB, then AD=DB. B The perpendicular CD will also divide the vertical angle into two equal parts. QUEST.-3. What is an isosceles triangle (See § IV, Art. 12.)? Are the angles opposite the equal sides equal? 4. If a line be drawn from the vertical angle of an isosceles triangle, perpendicular to the base, how will it divide the base? How will it divide the vertical angle? Properties of the Triangle. 5. The greater side of every triangle is opposite the greater angle, and the greater angle opposite the greater side. Thus, if B is the greater angle, AC will be the greater side. QUEST.-5. If you know the greater side of a triangle, do you know the greater angle? Why? If you know the greater angle, do you know the greater side? Why? 6. Are the angles of an equilateral triangle equal to each other? 7. If one side of a triangle be produced out, what will the outward angle be equal to? 8. What is the sum of the three angles of any triangle equal to ? Properties of the Triangle. 9. In every right angled triangle, the sum of the two acute angles is equal to 90 degrees. Thus, B+C 90 degrees; This is evident, since A+B+C 180 degrees, and A=90 degrees. B 10. In every right angled triangle, the square described on the hypothenuse, is equal to the sum of the squares described on the other two sides. Thus, if ABC be a right angled triangle, right angled at C, then will the square D described on AB be equal to the sum of the cribed on the sides CB and AC. penter's theorem. D squares E, and F desThis is called the car QUEST.-9. In a right angled triangle, what is the sum of the two acute angles equal to ? 10. In a right angled triangle, what is the square on the hypothenuse equal to ? What theorem is this called? |
