Book I. Propositions 31 and 32
Back to Propositions 31, 32.
11. Solve the problem of Proposition 31:
Through a given point to draw straight line parallel to a given straight line.
To see the answer, pass your mouse over the colored area.
Let A be the given point, and BC the given straight line;
Choose any point D on BC, and draw AD;
the straight line AD, which meets the two straight lines EF, BC makes the alternate angles EAD, ADC equal. Therefore EF is parallel to BC. (I. 27)
12. a) State the hypothesis of Proposition 32.
A figure is a triangle and one side is extended.
2. b) State the conclusion.
The exterior angle is equal to the two opposite interior angles; and the three interior angles of a triangle are equal to two right angles.
2. c) Practice Proposition 32.
13. Prove Proposition 32 by drawing a straight line DE through A
3. This proof is attributed to Pythagoras, who lived some 250 years
Because DE is parallel to BC,
14. ABC is a circle with center D; ABD is a triangle; and ADC is a
14. straight line. Prove that angle BDC is double angle A.
AD is equal to DB because they are radii of the circle;
15. Prove: The acute angles of a right triangle are together equal to a
The three angles of a triangle are equal to two right angles,
16. In an isosceles right triangle, why is each acute angle half of a right
Since the triangle is isosceles, the base angles are equal.
17. Prove that if an acute angle of one right triangle is equal to an acute
That is, if angles B and E are right angles, and angle C equals
Angles A and C together equal one right angle,
18. According to the Corollary to I. 32, the four interior angles of any
19. In any five-sided rectilineal figure, the five angles are together equal
19. to how many right angles?
10. In the degree system of angular measurement, in which a right angle
10. is called 90°, how many degrees is each angle in a regular octagon?
10. (That is an eight-sided figure which is both equilateral and
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