Question

If AB is parallel to CD, EF bisects angle BEG, GF bisects angle EGD, find the number of degrees in angle EFG.

Answer 1

No figures were given?

Answer 2

no numbers

Answer 3

How it can be determined?

Answer 4

use : interior angles on same side are supplementary

Answer 5

they did not leave us with a single number or variable

Answer 6

I'm thinking :)

Answer 7

they only left us number answers...

Answer 8

all right then, what do we know about angle EGF and FGD?

Answer 9

40,60,90,120... is this a backsolving problem?

Answer 10

well...

Answer 11

FGD is part of the bisection?

Answer 12

yes, therefore...?

Answer 13

its angle measure is halved?

Answer 14

It's EGD that's halved, so what does that make FGD and EGF?

Answer 15

45 and 90?

Answer 16

45-45-90 right triangles?

Answer 17

no... remember, GF bisects EGD, therefore...?
(look back to the definition of bisect)

Answer 18

Dividing a geometric figure into two equal parts.

Answer 19

yes, so, that means FGD and EGF have equal measures, right?

Answer 20

right

Answer 21

67.5 and 67.5+45=180?

Answer 22

You better be sure, ok?
so what can you say about BEF and FEG as angles?

Answer 23

Let's not be hasty about the figures yet :)

Answer 24

k

Answer 25

so, What can you say about BEF and FEG?

Answer 26

They have a common vertex?

Answer 27

yeah, but other than that? something more useful to us :)
EF bisects BEG, work from there :)

Answer 28

BEF and FEG both have equal angle measures because of bisection?

Answer 29

Precisely :)

Answer 30

but, what are the exact angle measures?

Answer 31

We don't need them ;)
Trust me

Answer 32

hmm...

Answer 33

Okay, so what can you say about angles BEG and DGE?

Answer 34

Aren't they also going to have the same angle measures because of bisection?

Answer 35

no..., look closer, AB is parallel to CD, that makes BEG and DGE... what?

Answer 36

ALTERNATE INTERIOR ANGLES?

Answer 37

Alternate? look again, are they on the same side of the transverse or not?

Answer 38

ohhh.... same-side interior angles?

Answer 39

That's right, so, they must be...?

Answer 40

What is always true about same-side interior angles when the lines are parallel?

Answer 41

supplementary angles

Answer 42

That's right, so, we get our first figure

Answer 43

so they have to equal 180 degrees?

Answer 44

yes
so
BEG + DGE = ?

Answer 45

180!!

Answer 46

Correct, now, the tricky part ;)

Answer 47

k

Answer 48

What can you say about EGD and EGF?

Answer 49

supplementary to one another?

Answer 50

no..., remember what bisects EGD...

Answer 51

EFG?

Answer 52

An angle cannot bisect another angle, FG bisects EGD, right?

Answer 53

oh... I see the bisection now...

Answer 54

so that makes the measure of EGF ________ of the measure of EGD
fill in the blank

Answer 55

completmentary?

Answer 56

no..., EGF would be half of EGD

Answer 57

because of the bisection

Answer 58

k

Answer 59

and similarly
GEF is ________ of BEG
fill in the blank :)

Answer 60

half!!

Answer 61

that's right, so remember
BEG + DGE = 180, right?

Answer 62

right

Answer 63

Well, as we figured out
BEG = 2*GEF and
DGE = 2*EGF right?

Answer 64

yes

Answer 65

since GEF is half of BEG
and EGF is half of DGE

Answer 66

because the smaller ones are half the bigger ones

Answer 67

yes, now notice,
2*GEF + 2*EGF = 180, notice it?

Answer 68

yes

Answer 69

What do you notice then?

Answer 70

that those angles together are supplementary?

Answer 71

and the smaller angles are twice as small as the bigger ones?

Answer 72

ok, we can un-distribute the 2 and we get
2(GEF + EGF) = 180
Do you see it now? ;)

Answer 73

45+45+45+45=180

Answer 74

GEF=45, EGF=45

Answer 75

whoops
we DON'T KNOW what GEF or EGF are individually
but we do know that their sum is 90
keep that in mind

Answer 76

ok

Answer 77

so the sum of GEF and EGF twice is 180?

Answer 78

So, you do see the triangle FEG right?

Answer 79

yes, so ...?

Answer 80

Yes I do

Answer 81

the rule of 180 states that the angle measures of a triangle has to equal exactly 180 angle degrees

Answer 82

GEF and EGF, we don't know what they are exactly
but we DO know that they add up to 90, so what must be the measure of EFG?

Answer 83

90 is the third angle, so it has to equal 90 in order to fulfilll the 180 requirement

Answer 84

90
and that, mi amigo, is the answer :)
good job

Answer 85

finally!!!!!!!!!

Answer 86

lol

Answer 87

ty so much!

Answer 88

no problem