(d) Measure the angle that each of these lines makes with MN. How do they compare with Z PON? As the point A moves away from 0, what happens to the size of the <?

How many lines can you draw from PL to MN? (e) What is the shortest line that you can draw from a point to a given line?

(f) The distance from a point to a line always means the shortest distance.

(g) What is the distance of P from MN?

(h) Do you believe the following?

(1) The shortest distance from a point to a line is the perpendicular drawn from the point

to the line.

(2) One and only one perpendicular can be drawn to a line at the same point.

(3) If you believe the above statement No. (2), examine your cube or oblong.

How many lines are 1 to DB at point B?

Can there be more than two lines

DB at B?

to

How many can there be, if we consider only one face or plane surface at a time? Would statement No. (2) be correct if we

limit it to lines in the same plane? In the same plane, one and only one perpendicular can be drawn to a line at the same point.

(4) Is this statement true?

In the same plane, one and only one perpendicular can be drawn from a given point to a line.

(a) Suppose AB is the line to be bisected (cut in two equal parts).

Open the compass a little more than half of the
line AB. Put the sharp point on A and draw
one arc above and another below the line.
Then from B make two arcs intersecting the
others, being careful not
not to
to change the
opening of the compass. Call the points
of intersection C and D, and draw a line
between them, which cuts AB at some point,
as M. Then M is the midpoint of AB, or
AB is bisected at M.

(b) With compass show that AM

=

MB.

(c) With compass and protractor show that CD is the perpendicular bisector of AB.

2. To bisect an angle.

(a) By folding.

You have already folded a paper twice to make rt. 4. Fold such a paper again. What is the size of the new angle? Fold again. What is

the size now? Cut out an angle of any size and bisect it by folding.

(b) By using a compass.

Suppose ZAOB is to be bisected. Put the compass point on O and draw two arcs cutting the sides of the angle at X and Y. Put the compass point

on X and then on Y and, without changing the opening, draw two arcs intersecting at P. Draw a line from O through P. What does OP do to ZAOB? Cut out the ZAOB and show by folding that it is bisected.

CHAPTER FIVE

FURTHER STUDY OF TRIANGLES

A. IMPORTANT LINES IN TRIANGLES

I. In Isosceles Triangles

1. Draw and cut out a number of isosceles triangles of different sizes and shapes.

(a) Fold each through the vertex angle so that the two edges lie one on the other.

(b) Do the base angles exactly fit?

(c) Does this test verify the measure with your protractor?

(d) Unfold one or more of these triangles.

Measure the two parts of the base.

In what two ways can you show that the fold bisects the base?

(e) What kind of angles does the fold make as it meets the base?

In two ways prove your answer correct.

What name do you give to a line that makes right angles with another line?

(f) With your protractor measure the two parts of the angle at the vertex.

When the isosceles triangle is folded, do these two angles exactly fit?

What else besides the base of the triangle is bisected by the fold?

2. Draw three isosceles triangles exactly equal.

(a) In ▲ A, bisect the vertex Z.

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(b) In ▲ B, draw the bisector of the base.

What does the perpendicular bisector of the base do to the angle at the vertex? Test carefully. (c) In ▲ C, draw a from the vertex to the base. What does this L to the base do to the base? What does it do to the at the vertex? with compass and protractor.

II. In Scalene Triangles

Test

1. Draw and cut out three scalene triangles.

(a) In one of these triangles

make three folds.

(1) An altitude

(2) A bisector of the vertex angle

(3) A perpendicular bi

sector of the base

(4) Are these folds the

same or are they three separate lines?