PROPOSITIONS OF GEOMETRY Table 1c. Propositions of Geometry Machinery's Handbook, 31st Edition
63
In the figure, angle D is an exterior angle . The measure of an exterior angle is equal to the sum of the measures of the non-adjacent interior angles. Thus, A + B = D Intersecting lines form congruent vertical angles , that is, the angles opposite one another. In the figure, A and B are vertical angles and therefore congruent, and C and D are vertical angles and therefore congruent. That is,
A
D
B
D
B
A
A ≅ B C ≅ D
C
B
A
Corresponding angles formed when parallel lines are intersected by another line (a transversal ) have equal measure (are congruent). Lines l 1 and l 2 are intersected by s transversal, so all angles A are congruent, and all angles B are congruent. A quadrilateral is a four-sided polygon. In any quadrilateral, the sum of the measures of the interior angles is 360 degrees. Thus, A + B + C + D = 360 degrees A parallelogram is a quadrilateral in which each pair of opposite sides is parallel and thus congruent. Opposite angles are also congruent. Each diagonal divides the parallelogram into congruent triangles, and the diagonals bisect each other. In the figure, diagonal D bisects diagonal d at their midpoints. A rectangle is a parallelogram in which all four angles are right angles. In the figure, the rectangle’s base is marked b ; its height is marked h ; the parallelogram’s base is marked b 1 , and its height is marked h 1 . The area of a parallelogram is A = bh . If two parallelograms have a corresponding base and height of equal measure, their areas are equal. Thus, if b = b 1 and h and h 1 , then area A = area A 1 .
l 1
A
B
B
A
l 2
A
B
C
A
D
B
A
b
B
a
A
A 1
h
h 1
b
b 1
h
Triangles having equal base and equal height have equal area. Thus, if b = b 1 and h and h 1 , then area A = area A 1 .
A
A 1 h 1
b 1
b
90
If a diameter of a circle is perpendicular (at right angles) to a chord, then it bisects the chord, dividing it into two equal parts. In the figure, the chord has length c , so its parts have length c ⁄ 2 .
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