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CHAPTER 15 Constructions
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15.14. Construct a triangle given (a) two sides and the median to one of them; (b) two sides and the altitude to one of them; (c) an angle, the angle bisector of the given angle, and a side adjacent to the given angle. (15.5) 15.15. Construct angles of measure 15 and 165 . 15.16. Given an angle with measure A, construct angles with measure (a) A 60 ; (b) A 30 ; (c) A (15.6) 120 . (15.6)
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15.17. Construct a parallelogram, given (a) two adjacent sides and an angle; (b) the diagonals and the acute angle at their intersection; (c) the diagonals and a side; (d) two adjacent sides and the altitude to one of them; (e) a side, an angle, and the altitude to the given side. (15.7) 15.18. Circumscribe a triangle about a given circle, if the points of tangency are given.
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(15.8)
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15.19. Secant AB passes through the center of circle O in Fig. 15-39. Circumscribe a quadrilateral about the circle so that A and B are opposite vertices. (15.8)
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Fig. 15-39
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15.20. Circumscribe and inscribe circles about (a) an acute triangle; (b) an obtuse triangle. 15.21. Circumscribe a circle about (a) a right triangle; (b) a rectangle; (c) a square. 15.22. Construct the inscribed and circumscribed circles of an equilateral triangle. 15.23. Locate the center of a circle drawn around the outside of a half-dollar piece.
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(15.9) (15.9) (15.9) (15.9)
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15.24. In a given circle, inscribe (a) a square; (b) a regular octagon; (c) a regular 16-gon; (d) a regular hexagon; (e) an equilateral triangle; (f) a regular dodecagon. 15.25. Construct a triangle similar to a given triangle with a base (a) three times as long; (b) half as long; (c) one and one-half times as long. (15.10)
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Proofs of Important Theorems
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16.1 Introduction
The theorems proved in this chapter are considered the most important in the logical sequence of geometry. They are as follows: 1. If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.) 2. The sum of the measures of the angles in a triangle equals 180 . 3. If two angles of a triangle are congruent, the sides opposite these angles are congruent. 4. Two right triangles are congruent if the hypotenuse and a leg of one are congruent to the corresponding parts of the other. 5. A diameter perpendicular to a chord bisects the chord and its arcs. 6. An angle inscribed in a circle is measured by one-half its intercepted arc. 7. An angle formed by two chords intersecting inside a circle is measured by one-half the sum of the intercepted arcs. 8a. An angle formed by two secants intersecting outside a circle is measured by one-half the difference of its intercepted arcs. 8b. An angle formed by a tangent and a secant intersecting outside a circle is measured by one-half the difference of its intercepted arcs. 8c. An angle formed by two tangents intersecting outside a circle is measured by one-half the difference of its intercepted arcs. 9. If three angles of one triangle are congruent to three angles of another triangle, the triangles are similar. 10. If the altitude is drawn to the hypotenuse of a right triangle, then (a) the two triangles thus formed are similar to the given triangle and to each other, and (b) each leg of the given triangle is the mean proportional between the hypotenuse and the projection of that leg upon the hypotenuse. 11. The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. 12. The area of a parallelogram equals the product of the length of one side and the length of the altitude to that side. 13. The area of a triangle is equal to one-half the product of the length of one side and the length of the altitude to that side. 14. The area of a trapezoid is equal to one-half the product of the length of the altitude and the sum of the lengths of the bases. 15. The area of a regular polygon is equal to one-half the product of its perimeter and the length of its apothem.
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