1. If the oblique lines are on opposite sides of the perpendicular, show that the theorem is true.

2. Prove that the bisectors of two vertical angles are in the same straight line.

SUGGESTION. Show that the sum of the angles on one side of FE are equal to the sum of those on the other side.

3. Prove that the bisectors of two supplementary angles are perpendicular to each other.

EXERCISES.

SUGGESTION. Show that D.

LEBF=1⁄2 CBD=} a straight

angle.

4. If the angle ABD is 86° 14', how many degrees are there in each of the other angles formed at B?

5. If the angle ABD is two-thirds of the angle ABC, how many degrees are there in each of the other angles formed at B?

6. If the straight line CD is the shortest line that can be drawn from C without the line AB to AB, show that CD is perpendicular to AB.

7. If a perpendicular is erected at the middle point of a line, any point without the perpendicular is unequally distant from the extremities of the line.

That is, FA> FB.
SUGGESTION.

EA: =

EB; add EF to both

sides of this equation.

A

F

E

8. If any point be taken within a triangle, show that the sum of the lines joining the point to the vertices is less than the sum of the sides of the triangle.

PARALLEL LINES.

59. DEFINITION. Two straight lines are called Parallel when they lie in the same plane, and cannot meet nor approach one another, how

A

ever far they may be produced; as AB C

and CD.

B

-D

60. AXIOMS. 1. But one straight line can be drawn through

a given point parallel to a given straight line.

2. Since parallel lines cannot approach one another, they are every where equally distant from each other.

If a straight line EF cut two other straight lines AB and CD, it makes with those lines eight angles, to which particular names are given.

The angles 1, 4, 6, 7 are called Interior angles.

The angles 2, 3, 5, 8 are called Exterior angles.

The pairs of angles 1 and 7, 4 and 6, are called Alternateinterior angles.

The pairs of angles 2 and 8, 3 and 5, are called Alternateexterior angles.

The pairs of angles 1 and 5, 2 and 6, 4 and 8, 3 and 7, are called Exterior-interior angles.

The angles 2 and 6, 3 and 7, 4 and 8, 1 and 5, are called Corresponding angles.

PROPOSITION VIII. THEOREM.

61. Two straight lines perpendicular to the same straight line

two lines from this point of meeting perpendicular to the same line, which (by 52) is impossible.

Therefore CD and AB, if perpendicular to AC, are parallel.

EXERCISES.

1. Prove that two straight lines parallel to the same straight line are parallel to each other.

2. Prove that a straight line perpendicular to one of two parallels is also perpendicular to the other.

PROPOSITION IX. THEOREM.

62. If two parallel straight lines be cut by a third straight line, the alternate-interior angles are equal.

Let AB and GH be two parallel straight lines cut by the line EF at G and H.

The lines AB and CD, being parallel, have the same direction.

The lines EG and GH, being in one and the same straight line, are similarly directed.

Therefore the angles EGB and GHD have their sides similarly directed; that is, the differences of their directions are equal,

or

1. Prove that the alternate-exterior angles are equal,

LEGAL DHF, or ZEGB = ≤ CHF.

2. Prove that the sum of the two interior angles on the same side of the cutting line, or transversal, are equal to two right angles.

SUGGESTION. ZAGH=/ GHD; add / GHC.

3. When two straight lines are cut by a third straight line, if the exterior-interior angles be equal, these two straight lines are parallel.

SUGGESTION.

F

If AB is not parallel to CD, draw MN parallel to CD;

then apply 62 and 28.

PROPOSITION X. THEOREM.

63. Two angles whose sides are parallel each to each are either equal or supplementary.

Let AB be parallel to DH and BC to KF.

To prove that the angle ABC is equal to DEF and supplementary to DEK.

Let BC and DE intersect at G.

1. Since BC and KF are parallel and DH a cutting line (by 62),

Z DGC: = DEF.

Since AB and DH are parallel and BC a cutting line (by 62), Z ABC = Z DGC.

SCHOLIUM. Two parallels are said to be in the same direction, or in opposite directions, according as they lie on the same side or on opposite sides of the straight line joining the vertices.