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FOURTH GRADE / MATHEMATICS /BASICS of GEOMETRY 1.3

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

   BASICS of GEOMETRY ARE
 ♦ 
POINTS, LINE SEGMENTS, RAYS, LINES (INTERSSECTING or PARALLEL) and ANGLES.
 ♦ THEY ARE OUR BASIC BLOCKS ON THE WAY TO UNDERSTAND, ANALYZE and SYNTHESIZE SHAPES

THERE ARE TWO GOLDEN RULES:
1. THEY ARE IDEALIZATIONS FROM OUR DAILY EXPERIENCES.
2. BEING THOUGHT OBJECTS THEY MAY LEAD AMOK; SO BE CAREFUL WITH THEM.
y
1. How big (in degrees) is the angle with the question mark in ?

!
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) 80
b) 160
c) 140
d) 120


A very simple equation indeed. Look at the ry1 .
y
2. Find the angles V and W using the data given in the figure below .

!
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) 25 and 75
b) 30 and 100
c) 25 and 100
d) 30 and 105
Look at the ry1 and see how easy two equations with two unknowns are .
3. Which of the following segments are of equal length ?

!


a) 1 and 2
b) 2 and 3
c) none
d) 3 and 4
Well this question is to check your eyes. The smallest difference between these segments (between 3 and 4) is about 1/4 ths of a millimeter. Some may have difficulties in differentiating. A single hair is about 60 μm. (a μm. is 1/1000 ths of a millimeter), about 1/4 ths of the minimal difference of our segments. Most of us see a sigle hair.
4. Which angle is the greatest of all ?

!


a) ∠c
b) ∠b
c) ∠a
d) none
∠a and ∠c are congruent .
5.
!


Which figure symbolizes best our method of adding ∠a and ∠b with a ruler and a compass ?

!


a) II
b) I
c) III
d) IV
The choice is obvious, the question is just for remembrance and the next question .
6. On which of the following facts is the method of adding angles with ruler and compass based ? (Def. of Radian)

a) Equal arc lengths correspond to equal angles
b) Equal arc lengths of congruent circles coresponds to equal angles
c) Equal arcs are equal
d) Arc lengths determine angles
Look at the definition of Radian and try to understand .
7. In the figure below we see a line passing through the non adjacent corners of a rectangle. We may just accept that the line bisects the rectangle and leave things there, not considering any of the consequences. This way we do not go anywhere, just stay still.
Considering consequences we begin to move, we begin our travel in the land of geometry which leads us to other lands.
Which of the followings will be a consequence of this acceptance made above ?

!


a) The sum of measures of inner (internal) angles of a right triangle is 180 degrees.
b) We get congruent figures with different colors.
c) We get similar figures.
d) We get triangles
Since the sum of inner angles of a rectangle is 360 degrees, the right triangle must have ... .
8. Based on the previous question the dark purple triangles I and II are supposed to have a total of 180 degrees for their inner angles.
Which fact makes this condition true for the whole dark purple triangle ?

!


a) Triangles are alike.
b) Triangles are congruent.
c) The adjacent angles at the bottom of I and II are both 90 degrees .
d) Triangles are similar.
Since the adjacent angles at the bottom of I and II are both 90 degrees , the remaining angles in both triangles should have a total of 90 degrees .
9. What do we in addition need to show that ∠1 and ∠a are congruent ?

!


a) red triangle is greater than blue
b) red and blue triangles are congruent
c) blue triangle is greater than red
d) nothing


All we know is still m(∠1 + ∠3) = m(∠d + ∠a) .
10. What we want to show you is that what we call plane geometry is a totality of truths which we divide and investigate in parts. These parts give us a set of facts which is redundant, i.e. some may be derived from the others. We axiomize some to be true and drive the others.
Previously we have assumed (axiomized) that the sum of the inner angles of a right triangle is 180 degrees and derived that alternate interior angles are congruent.
Alternate interior angles are formed when (at least) two parallel lines are cut by a transversal. So are alternate exteriors. Now we want to assume that alternate interior angles are congruent and deduce from this the internal angles of a triangle has a sum of 180 degrees.

!


How can we show that if alternete interiors are assumed to be congruent, alternate exteriors should be congruent ?

a) they are complements
b) they are right angles
c) they are acute anles
d) they are supplements
An alternate interior and its adjacent alternate exterior have a total measure of 180 degrees .
11. The sum of measures of ∠1 , ∠2 and ∠3 is 180 degrees is easily seen.
∠A and ∠2 are congruent because they are alternate interiors
Why are ∠B and ∠3 congruent ?

!


a) vertical of alternate interior
b) alternate interior
c) alternate exterior
d) vertical of alternate exterior
Alternate interior of B is the vertical of 3 .
12. Now we have gone both way. The situation reminds of the possibility of a new set of basic elements of geometry (other than points, lines and angles) that directly produce non redundant axioms.
Do you think it is worth researching ?

a) Yes
b) No
c) Perhaps
d) Sure
It is supposed to lead to an equivalent geometry . The one we have is simple enough.
Equivalency left aside, maybe due to being conditioned, I can not imagine any candidate. And I can not show the impossibility. So it remains open.
13. We define terms by terms. That this process can not continue ad infinitum forces us to have some basic terms that are not defined.
As for definitions, which of the following properties should a definition have ?

I- Name the term being defined.
II- Places the "defined" in the class (smallest possible) it belongs to.
III- Distinguishes it from other class members.

a) I and III
b) I and II
c) II and III
d) all
All .
14. How many angles are formed by three mutually noncollinear (no two of them are collinear) lines ?

!
a) 3
b) 12
c) 9
d) 6
3vial (trivial). Just a preparation.
15. How many angles are formed by four mutually noncollinear lines ? (Link/q.8)

!


a) 24
b) 6
c) 12
d) 18
It should be as many as the intersection points multiplied by 4.Look at question 8 of test 1 of this group via the Link.
16. That 6 intersection points and 6x4 angles are formed by 4 mutually noncollinear lines implies that at the most 2 triangles can be formed ?

!


a) it may be
b) yes
c) no, some angles and vertexes may be shared
d) sure
The first 3 form a triangle and for the fourth there are 3 possibilities: It crosses the triangle formed as the red, it passes from the upper side or lower side of it as the green or blue rays. In case it intersects the triangle formed by the first 3 ...
17. What can be said about the measures of ∠1 and ∠2 ( sides are perpendicular to each other's) ?

!


a) m(∠1) > m(∠2)
b) equal
c) m(∠1) < m(∠2)
d) m(∠1) = 90
See the next question's tip .
18. Why are ∠1 and ∠2 of question 17 of equal measure ?

!


a) right angles
b) right and vertical angles
c) vertical angles
d) same angles
One of the angles of the triangles are right angles, one of the others are vertical .
19. 1/60 ths of a degree is called a minute and 1/60 ths of a minute is called a second
27 878/3600 degrees make 27 degrees (°) ... minutes (') and ... seconds ('') ?

a) 14 and 38
b) 18 and 34
c) 16 and 38
d) 16 and 34
Divide 878 by 60 to find 14 with a remainder of 38 .
20. What is the difference of the measures of the angles with measures 42° 30' 20'' and 36° 20' 30'' ?

a) 6° 09' 50''
b) 7°
c) 6° 09' 50''
d) 6° 10' 50''
Remember borrow .
21. Add the measures 20° 40' 30'' and 30° 30' 10'' ?

a) 51° 10 ' 50''
b) 52° 10 ' 40''
c) 51° 10 ' 40''
d) 52°
Remember carry
y
22. r1 and r2 are parallel ; red segments are bisectors of the angles in which they are.
What is the equation between v and w (the measures of the angles) ?
(Link/q.22)

In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) w = v
b) v = 2*w
c) w=2*v
d) w = v/2


Have a look at question 22 of test 1 of this section via the Link given.
23. How many angles do you see in the yellow colored region ?



a) 9
b) 10
c) 11
d) 12
Be careful about the top corner.
24. What will be the measure of the angle obtained by subtracting angles of measure 88 degrees and 24 degrees from a straight angle ?

a) 72
b) 80
c) 68
d) 78
180 - 88 - 24 = ?
25. Which of the angles below has the smallest measure ?

a) acute
b) right
c) obtuse
d) straight
3vial answer .
This is the feedback!

THANK YOU FOR YOUR VISIT; HOPE WE COULD BE OF USE.
FOR MUTUAL BENEFIT YOU MAY USE OUR FORUM.
© 2011 , Ä°smail GERMAN. ALL RIGHTS RESERVED
PLEASE PREFER GIVING LINK IF THAT SUFFICES

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