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For Euclid I. 30, see Appendix.
PROP. 30. THEOR. Straight lines which are parallel to the same straight line are parallel to one another. Given AB || CD, and EF || A
с To prove AB || EF. If AB is not || EF, they will E
F meet; then there will be two intersecting lines both || CD, which is impossible (Ax. 11).
:: AB is || EF. Q.E.D.
PROP. 31. PROB. Through a given point to draw a straight line parallel to a given straight line. Given st. line AB and pt. C.
F To draw through C a st.
line || AB. Take any point D in AB,
and join CD. At C in CD make the < ECD = CDB (I. 23). Produce EC. : ECD = _ CDB, and these are alternate angles, :. EF is || AB (I. 27), and has been drawn through
the given point C. Q.E.F.
73. Prove Prop. 30 by drawing a line to intersect the given lines.
74. If a straight line be parallel to two straight lines which meet at a point, these two shall be in the same straight line.
PROP. 32. THEOR. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are together equal to two right angles. Given A ABC having
2 Lis. Through C draw CE || AB (1. 31). :: AC meets the parallels AB, CE, :. LA LACE
(I. 29). :: BD meets the parallels AB, CE, :_ ECD = LB (I. 29).
:: LACD LA +_ B. To each of these equals add – ACB.
:: LACD + L ACB LA + B + - ACB. But ACD + LACB 2 Ls. (I. 13). :: LA + LB + L ACB
2 Lis. Q.E.D. 75. A straight line parallel to the base of an isosceles triangle and intersecting the sides, or the sides produced, forms with them an isosceles triangle.
76. Through a given point to draw a st. line to make with a given st. line an angle equal to a given angle. How many such lines are there?
77. If a straight line be drawn through the vertex of an isosceles triangle parallel to the base, it shall bisect the exterior angle at the vertex.
78. In a right-angled triangle find a point in the hypotenuse AB so that its distance from AC equals its distance from B.
79. In an isosceles triangle find two points in the equal sides equidistant from the extremities of the base and from one another.
Cor. 1.--The sum of the interior angles of any recti. lineal figure is equal to twice as many right angles as the figure has sides, minus four right angles. Take any rectilineal figure, as
B ABCDEF, and take G, any point within it. Join G with A, B, etc. (1.) There are as many triangles as
G the figure has sides.
(2.) The angles in each triangle amount to two right angles (I. 32). (3.) .. the angles in all the triangles
E together amount to twice as many right angles as the figure has sides (2n [:s., n=number of sides).
(4.) But the angles of all the triangles include the angles of the figure at A, B, C, etc., and the angles at G which amount to four right angles.
(5.) ... the angles of the figure amount to twice as many right' angles as the figure has sides, minus four right angles (2n _'.. — 4 _.s.)
Cor. 2.-The sum of the exterior angles of any convex rectilineal figure, formed by producing the sides all in one way, is four right angles.
Take ABCDEF, any convex rectilineal figure having the sides produced all in one way. To prove sum of exterior angles
A 2 four right angles.
(1.) Each exterior angle with its adjacent interior angle, as 21+22, equals two right angles.
(2.) :. all the exterior with all the interior amount to twice as many right angles as there are sides in the figure (2n L.s.)
(3.) But all the interior angles together with the angles at G amount to twice as many right angles as the figure has sides (Cor. 1) (2n L'.s.)
(4.) .. all the exterior angles with all the interior =all the interior angles with the angles at G.
(5.) .. all the exterior angles together= angles at G= four right angles (4 _'.s.)
Note.—When the rectilineal figure is not convex, but has a reflex angle or reflex angles, the result is modified as follows; the proof of which we leave to the student:21+ L 2+ 23+ 24+ 25-26=
80. The angles of a quadrilateral amount to four right angles.
81. Find the magnitude of the angle in a regular pentagon, hexagon, heptagon, octagon, etc. (Cor. 1).
82. Find the magnitude of the exterior angle of any regular polygon, as a pentagon, nonagon, decagon, or polygon of n sides.
83. The exterior angle of a regular polygon is ts of a rt. angle, how many sides has it ?
84. The interior angle of a regular polygon is 1f rt. angle, how many sides has it ?
85. Show that three regular hexagons can be placed so as to have a common angular point and to fill up
about it. 86. What other regular figures can be formed round common angular points in the same way?
87. Make a pattern composed of octagons and squares.
88. What other regular figures may be combined to fill up the
space round the common angular points? Make patterns of the combinations.
89. How many diagonals can be drawn in a pentagon ; in a hexagon; in a polygon of n sides ?
90. If all the sides of a polygon be produced in both ways, the angles formed by the meeting of every two alternate sides will together amount to twice as many right angles as the figure has sides, minus eight right angles (2n _'.s. -8 L'.s.)
91. The middle point of the hypotenuse of a right-angled triangle is equally distant from each of the three angles.
92. And conversely, If the middle point of one side of a triangle be equally distant from each of the three angles, the triangle shall be right-angled.
93. Given the perimeter of a triangle and the angles at the base, to construct the triangle.
PROP. 33. THEOR. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are also themselves equal and parallel. Given AB = and || CD, and
let them be joined towards
the same parts. To prove AC = and || BD. Join BC. .: BC meets the parallels AB, CD, :: L ABC
= _ BCD (I. 29), and AB CD (given), and BC is common, .:. Δ ABC
ABCD in every respect (I. 4)
- BD. Also – ACB = _ CBD, and .. AC is also || BD (I. 27).
Note.-Propositions have already occurred where the hypothesis and conclusion of one become the conclusion and hypothesis of another, as Props. 5 and 6; such propositions are converse the one of the other.
94. Mention some examples of converse propositions. Is the converse of a true proposition always a true proposition, e.g., the converse of "All squares are parallelograms ?” Enunciate a converse of Ex. 68, of Ex. 70, of Ex. 77, and of Prop. 33.
95. The straight lines joining the extremities of two equal and parallel straight lines towards the opposite parts bisect each other.
96. A square, rectangle, rhombus, and rhomboid are all parallelograms (Def. 29).
97. The straight lines joining the extremities of two lines which bisect each other form a parallelogram.
98. If lines are respectively parallel to, or respectively perpendicular to, the sides of a triangle, they shall form a triangle equiangular to the given triangle.