Question

The figure below shows a circle with center O. Segment PQ is tangent to the circle at P and segment RQ is a tangent to the circle at R.

Answer 1

Which statement and reason is most appropriate for box 3?

Answer 2

Segment OP is congruent to segment OR since they are congruent chords.

Angle OPQ is congruent to angle ORQ since they are corresponding sides of congruent triangles.



Segment OQ is congruent to segment OQ by the reflexive property of line segments.

Angle OPQ is congruent to angle ORQ since they are vertical angles.

Answer 3

@jim_thompson5910 May you please help me?

Answer 4

Segment OP is congruent to segment OR since they are congruent chords

is false because they are radii (not chords)

they are congruent though

Answer 5

@jim_thompson5910 , any reason I'm blocked?

Answer 6

Angle OPQ is congruent to angle ORQ since they are corresponding sides of congruent triangles.

that's partially true as well: they are congruent, but NOT for this reason (besides this was already proven)

Answer 7

I was thinking that it was the third choice

Answer 8

However I am not very good at proofs and I am not entirely sure

Answer 9

you are correct A$apGeometryQuestions

that's a true statement and it's useful in this case

Answer 10

Thank you so much for validating for me! @jim_thompson5910

Answer 11

You get a medal.

Answer 12

and Compassionate, I'm not sure how you got blocked

I may have blocked you a long time ago (don't remember doing so though)

I'll unblock you

Answer 13

yw

Answer 14

May you aid me with one last question please? @jim_thompson5910

Answer 15

ok

Answer 16

Give me some time to post it please. Maybe a minute.

Answer 17

Look at the figure shown below

Answer 18

A student made the table below to show the steps to prove that DC is equal to EC.

Answer 19

post a screenshot of the table

Answer 20

Here it is

Answer 21

@jim_thompson5910

Answer 22

ok notice how lines 4 and 6 have the same right side (after the equals sign)

Answer 23

what does that mean? what does that allow us to say?

Answer 24

They are supplementary angles?

Answer 25

no, nowhere does it say they add to 180

Answer 26

well the picture slightly suggests it, but I have no idea

Answer 27

this is a very counter-intuitive idea...but NEVER base any decisions solely on the drawing alone

Answer 28

the drawing may not be to scale

Answer 29

Yes, I understand; My fault.

Answer 30

no worries

Answer 31

they are corresponding? no?

Answer 32

ACD and DCE

Answer 33

ACE and BCD are equal to the same thing, so they must be ______

Answer 34

congruent/vertical?

Answer 35

let x = m<ACD + m<DCE

Answer 36

ok

Answer 37

so we can say that m<ACE = x and m<BCD = x

Answer 38

I'm just going to lines 4 and 6 and replacing m<ACD + m<DCE with x

Answer 39

This is true

Answer 40

m<ACD + m<DCE=x

Answer 41

yep

Answer 42

got it. Next step?

Answer 43

so if m<ACE = x and m<BCD = x, then what's their relationship?

Answer 44

they are congruent angles

Answer 45

yep

Answer 46

so their angle measures are equal

Answer 47

this is because both m∡ACD + m∡DCE equals them

Answer 48

and yes

Answer 49

so m<ACE = m<BCD

Answer 50

the reason is ______

Answer 51

their angle measures are equal

Answer 52

I'm not sure which postulate I would use to prove it

Answer 53

well if x = y and x = z, then y = z

what property are we using?

Answer 54

addition property of equality

Answer 55

no

Answer 56

Transitive property of equality

Answer 57

Im almost sure @jim_thompson5910

Answer 58

closer, but no

Answer 59

Last option: substitution property of equality, but I doubt it. :S

Answer 60

well I would just say substitution

Answer 61

the transitive property is the idea that if x = y and y = z, then x = z

so it's very similar

Answer 62

Ok.

Answer 63

substitution is the same idea basically

Answer 64

Yes, it is formally suggested as substitution property of equality in my class,

Answer 65

I don't think there's a "substitution property of equality" though

it's just known as "substitution"

Answer 66

hmm I guess "substitution property of equality" is a bit valid since it deals with equations

Answer 67

Yep, It doesn't really make a difference though. It's ok, either way.

Answer 68

yeah you could argue either way really

Answer 69

exactly. So would that be the justification used for the answer? "substitution"

Answer 70

yeah that's what I would ultimately go with: it's short and simple

Answer 71

So the final answer is: m<ACE = m<BCD ----> substitution

Answer 72

?

Answer 73

yep

Answer 74

I dont get why the proof would not work without that step

Answer 75

if I were grading this, I'd say it's correct

then again, the teacher may be looking for "substitution property of equality" or "Transitive property of equality"

but you could argue for either one really

Answer 76

well look at the next line

Answer 77

what is that line

Answer 78

The ASA pos. is saying that the two triangles are congruent

Answer 79

ok what does ASA stand for

Answer 80

Angle-side angle

Answer 81

ok so you need a pair of angles congruent

and you need a pair of sides congruent

the sides are between the angles

Answer 82

line 1 gives you the sides

line 2 gives you one pair of the angles

but which line gives you the other pair of angles?

Answer 83

the line we just solved

Answer 84

line 3 also does too i think

Answer 85

line 3 does not

Answer 86

if it did, why did we do all that work after line 3?

Answer 87

line 6 does though, that's the missing piece of info to complete and set up ASA

Answer 88

Oh snap lol. I'm foolish. So we then would have all angle measures

Answer 89

but what about line 7, AKA: the missing line

Answer 90

oh right, that's what I meant to say