Ex. 422. If the arm CA of isosceles triangle ABC be produced so that AD equals the base AB, then C = 180° - 4 (≤ D). Ex. 423. If the base AB of isosceles triangle ABC is produced to D, and CD is drawn, then LBCDLA-LD. Ex. 424. If in isosceles triangle ABC, one end (B) of the base AB is connected with a point D in the op posite arm, then LA = (2 CDB + 2 CBD). B Ex. 425. If the side AC of triangle ABC is produced to D so that Ex. 426. If AD is the altitude and AE the bisector of the angle BAC of the triangle ABC, prove ZDAE = (< B-2C). B DE с Ex. 427. The sum of three angles of a quadrilateral, diminished by the fourth exterior angle, is equal to a straight angle. Ex. 428. The bisectors of two exterior angles of a triangle include an angle equal to one half the third exterior angle. HINT. Express every angle in terms of ZA and LB. MISCELLANEOUS EXERCISES Ex. 429. What are the different tests by which to determine the congruence of triangles ? Ex. 430. State some of the properties of all triangles. Ex. 431. Ex. 432. State some of the special properties of isosceles triangles. Ex. 433. What additional properties does a rhombus have? Ex. 434. The bisectors of supplementary adjacent angles are perpendicular to each other. Ex. 435. If the bisectors of two adjacent angles are perpendicular to each other, the exterior sides of these angles form a straight line (i.e. they form a straight angle). H Ex. 436. The bisectors of vertical angles are in a straight line. Ex. 439. The altitude upon the hypotenuse of a right triangle divides the figure into two triangles which are mutually equiangular. Ex. 440. If through any point D on the bisector of an angle A a parallel be drawn to one of the sides, to meet the other side in B, then AB = BD. Ex. 441. If from a point in the bisector of an angle, lines are drawn parallel to the sides of the angle, either a square or a rhombus is formed. Ex. 442. If the vertex angles of two isosceles triangles are supplementary, the base angles are complementary. Ex. 443. If an arm of an isosceles triangle is produced by its own length through the vertex, and the end of the prolongation is joined to the nearest end of the base, the line joining is perpendicular to the base. Ex. 444. Homologous medians of equal triangles are equal. Ex. 445. Homologous altitudes of equal triangles are equal. Ex. 446. Two isosceles triangles are equal if the vertex angle and the altitude upon an arm of the one are respectively equal to the vertex angle and the homologous altitude of the other. Ex. 447. If in the pentagon ABCDE, B Ex. 448. Two equilateral triangles are equal if the altitude of one equals the altitude of the other. Ex. 449. If the opposite sides of a hexagon are parallel, and two of opposite sides are equal, all opposite sides are equal. Ex. 450. If from the ends of the base BC of an isosceles triangle ABC, equal parts, BD and CE, be laid off on one arm and the prolongation of the other, the line joining D and E is bisected by the base. Ex. 451. If two lines are intersected by a transversal, and the bisectors of the interior angles on the same side of the transversal are perpendicular to each other, these lines are parallel. B Ex. 452. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram. Ex. 453. In the diagram given here, if AB || ED, 21+22+3 = 4 rt. 4. Ex. 454. State and prove the converse of the preceding proposition. Ex. 455. Find the number of diagonals in a polygon of 5 sides; of 8 sides; of 10 sides; of n sides. Ex. 456. How many sides has a polygon, the sum of whose interior angles equals three times the sum of the exterior angles? (I.e. one ext. at each vertex.) Ex. 457. How many sides has a polygon the sum of whose interior angles is equal to n times the sum of the exterior angles ? Ex. 458. How many sides has a polygon, the sum of whose interior angles is equal to three times the sum of the angles of a hexagon? Ex. 459. How many sides has an equiangular polygon whose exterior angle equals the interior angle of an equilateral triangle ? Ex. 460. If the upper base of an isosceles trapezoid is equal to one of the arms, the diagonals bisect the angles at the lower base. Ex. 461. If a perpendicular be dropped from the vertex to the base of a triangle, each segment of the base will be shorter than the adjacent side of the triangle. Ex. 462. If the vertices of a triangle lie in the sides of another triangle, the perimeter of the first is less than the perimeter of the second. * Ex. 463. The perpendiculars from two vertices of a triangle upon the median drawn from the third vertex are equal. 622769A Ex. 464. The lines joining the mid-points of the sides of a rectangle, taken in order, inclose a rhombus. Ex. 465. The lines joining the mid-points of opposite sides of any quadrilateral bisect each other. (Ex. 372.) Ex. 466. The mid-points of two opposite sides of a quadrilateral and the mid-points of the diagonals determine the vertices of a parallelogram. (Ex. 375.) Ex. 467. A line from the vertex of an isosceles triangle to any point in the base is shorter than either arm. D Ex. 468. If in the triangle ABC AB> AC, and D is a point in the prolongation of BA, then DB > DC. Ex. 469. Lines joining the mid-points of two opposite sides of a parallelogram to the ends of a diagonal trisect the other diagonal. * Ex. 470. If a point D in a side BC of triangle ABC is joined to A, and AC BC, AB = AD = DC, then ≤ C = 36°. = * Ex. 471. If any point E in the median CE is joined to A and B and <B><A, prove that <2><1 (annexed diagram). Ex. 472. From a given point without a line, to draw a line forming with the given line an angle equal to half a right angle. Ex. 473. From a given point without a line, to draw a line forming with the given line an angle of 60°. How many such lines can be drawn? Ex. 474. From a given point without a line to draw a line, making a given angle with the given line. Ex. 475. Construct a line terminating in the sides of a given angle and equal and parallel to a given line. Ex. 476. The diagonals of an isosceles trapezoid are equal. * Ex. 477. If the diagonals of a trapezoid are equal, the trapezoid is isosceles. 171. METHOD XII. In order to prove that the sum of two lines, a and b, equals a third line, c, either (a) Construct the sum of a and b, and prove the line so obtained is equal to c, or (b) Lay off a (or b) on c, and prove that the line representing the difference equals b or (a). Ex. 478. The sum of the perpendiculars dropped from any point in the base of an isosceles triangle to the arms is equal to the altitude upon one of the arms. Ex. 479. In ▲ ABC, if E F Ex. 480. If through a point D in the base AB of isosceles triangle ABC, parallels are drawn to the arms meeting the arms in E and F respectively, then Ex. 481. The sum of the three perpendiculars dropped from any point of an equilateral triangle upon the sides is constant, and equal to the altitude of the triangle. (Ex. 478.) * Ex. 482. If the altitude BD of A ABC is intersected by another altitude in G, and EH and HF are perpendicularbisectors, prove BG = 2(HE), and AG = 2(HF). Ex. 483. The line joining the point of intersection of the altitudes of a triangle and the point of intersection of the three perpendicular-bisectors cuts off one third of the corresponding median. (Ex. 482.) B H *Ex. 484. The points of intersection of the altitudes, medians, and perpendicular-bisectors of a triangle lie in a straight line. |