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Geometry Theorems for MTH-5111

Geometry Theorems for MTH-5111. For theorems 1 to 55 check out http://paccadult.lbpsb.qc.ca “In the Classroom” under Math 436 (MTH-4111) titled ‘4111 Theorems’. l 1. l 2. l 3. 56. Two points define a straight line. 57. When two lines intersect, they share a common point.

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Geometry Theorems for MTH-5111

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  1. Geometry Theorems for MTH-5111 For theorems 1 to 55 check out http://paccadult.lbpsb.qc.ca“In the Classroom” under Math 436 (MTH-4111) titled ‘4111 Theorems’.

  2. l1 l2 l3 56. Two points define a straight line. 57. When two lines intersect, they share a common point. 58. Given a point NOT on a line, there is only one line that passes through that point that is parallel to the first line. 59. If 2 lines (l1 and l2) are parallel to a third line (l3 ), then they are parallel to each other.

  3. l3 l1 l1 l1 l2 l2 l2 l3 60. If 2 lines (l1 and l2) are perpendicular to a third line (l3 ), then they are parallel to each other. 61. If a line (l1 ) is perpendicular to one of 2 parallel lines (l2 ), then it is perpendicular to the second parallel (l3 ). 62. Given a point NOT on a line, there is only one line (l1 ) that passes through that point that is perpendicular to the first line (l2 ).

  4. A A < + < + < + B B C C > | - | > | - | > | - | 63. In any triangle the length of any side is less than the sum of the lengths of the other 2 sides. 64. In any triangle the length of any side is greater than than the difference of the lengths of the other 2 sides.

  5. A A D D C C B B E E F F 65. If an acute angle and a leg of a right triangle are congruent to an acute angle and the corresponding leg of another right triangle, then the triangles are congruent. 66. If the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent.

  6. 67. Three non-collinear points define a circle. 68. The diameter is the longest chord of a circle. 69. The diameter divides a circle into 2 congruent parts.

  7. 70. In one circle or in congruent circles, congruent arcs are subtended by congruent chords and vice versa. 71. Any diameter perpendicular to a chord divides that chord and each of the arcs that it subtends into 2 congruent parts. Conversely, any diameter that divides a chord or a subtended arc into 2 congruent parts is perpendicular to that chord. 72. In one circle or in congruent circles, 2congruent chords are equidistant from the centre and vice versa.

  8. 73. If a line is perpendicular to the radius of a circle at the endpoint of the radius of the circle, then the line is tangent to the circle. The converse is also true. 74. Two parallel lines be they secants or tangents, intercept congruent arcs of a circle.

  9. A P O B 75. If point P is located outside circle O, and if segments PA and PB are tangents at points A and B respectively, then OP bisects APB and segments PA and PB are congruent. 76. In a circle, the measure of a central angle is equal to the degree measure of its intercepted arc. 77. The measure of an inscribed angle is one-half the measure of its intercepted arc.

  10. 78. The measure of an angle formed by 2 chords intersecting inside a circle is one-half the sum of the measures of its arcs intercepted by its angle and its vertical angle. 79. The measure of an angle formed outside of a circle is one-half of the difference of the measures of its interceptedarcs.

  11. A B C D D B E A C 80. In any triangle, the bisector of an angle divides the opposite 2 segments whose lengths are proportional to those of the adjacent sides. 81. If 2 chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the other.

  12. B A P C D A P B C 82. If secants PAB and PCD of a circle have the same exterior point P then: 83. If tangent PA and secant PBC of a circle have the same exterior point P then:

  13. A A C C B B O O D D 84. In a circle, the ratio of the measures of 2 central angles is equal to the ratio of their intercepted arcs. 85. In a circle, the ratio of the areas of 2 sectors is equal to the ratio of the measures of their 2 central angles.

  14. 86. The circumferences of 2 circles have the same ratio as their radii. 87. The areas of 2 circles have the same ratio as the square of their radii. 88. The measures of similar arcs of 2 circles have the same ratio as the square of their radii.

  15. 89. The length of the leg of a right triangle is the geometric mean between the length of its projection on the hypotenuse and the hypotenuse. 90. The length of the altitude to the hypotenuse of a right triangle is the geometric mean between the lengths of the segments of the hypotenuse.

  16. 91. In a right triangle, the length of the hypotenuse multiplied by the length of the altitude to the hypotenuse is equal to the product of the 2 legs.

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