# KidMath Introduction to Geometry

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```							    KidMath Sept. 04
Prof. T Parker

KidMath — Introduction to Geometry

Geometry is a game of logic played with shapes. The shapes lie in a plane. They are
constructed from the following basic pieces.
• Point — speciﬁes a location (has no thickness)
• Line — extends inﬁnitely in both directions (also has no thickness)
• Ray — “half-line”. Part of a line; has an endpoint and extends inﬁnitely in one direction.

• Line segment — the part of a line between two points (called the endpoints).

                                                                                       
Fact 1 : Two points determine a line

Given two different points, there is 1 and only 1 line containing both of them.
                                                                                       

Notation: We name points by capital letters. Two points A and B determine
←→
• a line AB                           A     B
−−→    −
−→                                 B      A
• rays AB and BA                         A                      B
A
• a segment AB                                         B

Length: We measure length in meters, centimeters, millimeters, and kilometers.

1 millimeter       1 mm
1 centimeter       1 cm=10 mm
1 meter            1 m = 100 cm
1 kilometer        1 km = 1000 m.

Because these units are related by factors of 10, all conversions can be made by just shifting the
decimal point!

Exercise 1 Complete the following expressions.
18 cm              mm                         3.5 m =              cm
2.4 km =            m                         85 mm =               cm
860 cm =              m                       63.2 m =               mm
268 mm =              cm
Angles: Two rays with the same endpoint separate the plane into two regions

We can distinguish these regions by drawing small arcs, An angle is two such rays with such an
arc. We can name them by naming the arc, or by naming three points in order.

P

x
Q          R
These are pictures of ∠x and ∠P QR.

Angle Terms:

a right angle is half a straight angle (90◦ ).
a straight angle (180◦ )
Small squares mark right angles.

x   y                                                     x
y

∠x and ∠y are complementary
∠x and ∠y are supplementary

Opposite sides of crossed lines, such as ∠x and
∠z, are called vertical angles. Notice that
y           z
∠x = 180◦ − ∠y
x                                                     ∠z = 180◦ − ∠y.

so ∠x = ∠z (both equal to same thing). Thus

Fact 2 : Vertical angles are equal


Angles in a Triangle
Every triangle has 3 vertices and 3 interior angles. It is an amazing fact that if you know the
measurement of two of those angles, you can ﬁgure out what the third is.

The shaded triangle has one right angle (its a “right triangle”) and two other angles, ∠x and
∠?. Complete the triangle to a rectangle. In the upper corner, we see two angles, ∠x and ∠y,
that add to 90◦ . But if we cut along the diagonal, we can rotate and slide the unshaded part to
exactly match the shaded part. That shows that ∠? = ∠y.

x

y

Conclusion: the two angles ∠x and ∠? in the original triangle add to 90◦ .

x
Fact 3
If a triangle contains a right angle, then
y                                  the other two angles add up to 90◦

∠x + ∠y = 90◦

A triangle of any shape can be divided into two right triangles: slide a plastic triangle with
a right angle along the longest side of the triangle until its edge passes through the opposite
vertex, then draw the line. Then divides ∠y into two parts; call then ∠a and ∠b. Then

x       y                             x    a
b

z                                      z

∠x + ∠a        =   90◦           (the shaded triangle is a right triangle)
+       ∠b + ∠ z       =   90◦           (the unshaded triangle is a right triangle)
∠x + ∠y + ∠z   =   180◦          (adding).

Conclusion:

y
                                              
x
Fact 4
z                         The angles of any triangle add up to 180◦
                                              
∠x + ∠y + ∠z = 180◦
Equilateral and Isoceles Triangles
Equilateral – All 3 sides of equal length. If we rotate by 1/3 of a turn, the triangle matches
itself. Thus all 3 angles are equal. Since their sum is 180◦ , each must be 60◦ .

60                                                                               
Fact 5
60        60                            Each angle in an equilateral triangle is 60◦
                                                 

Isoceles – At least 2 sides of equal length. The third side is called the base, and the angles
along the base are the base angles. If we fold along the line from the middle of the base to the
opposite vertex, the two sides match. Hence the base angles are equal.

                                                      
Fact 6
The base angles of an isosceles triangle are equal
Base                                                                                           
Base

```
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