"Erin & Antonio were shown this diagram & were asked to solve for measure of A in degrees."

[Ryuu]

"Erin and Antonio were shown the following diagram. They were asked to solve for the measure of angle A in degrees.​

Erin says the first step is to set 2x+7 equal to 5x-2.​

Antonio says the first step is to set 2x+7 plus 5x-2 equal to 180.​

Choose either Erin's or Antonio's statement and explain why it is correct or incorrect. If the statement is correct, explain how you know it's correct. If the statement is incorrect, explain why it's wrong.​

If you were to solve this problem, which steps would you take? Include any theorems, definitions, or reasons that explain the steps. Make sure you include all steps needed to solve for angle A."​

 

[stapel]

Erin and Antonio were shown the following diagram. They were asked to solve for the measure of angle A in degrees.

Erin says the first step is to set [imath]2x + 7[/imath] equal to [imath]5x - 2[/imath].
Antonio says the first step is to set [imath](2x + 7) + (5x - 2)[/imath] equal to [imath]180[/imath].

Choose either Erin's or Antonio's statement, and explain why it is correct or incorrect. If the statement is correct, explain how you know it's correct. If the statement is incorrect, explain why it's wrong.

If you were to solve this problem, which steps would you take? Include any theorems, definitions, or reasons that explain the steps. Make sure you include all steps needed to solve for angle A.

Okay; you've given us the exercise statement. Now we need the rest of the information.

Please re-read the "Read Before Posting" message, and reply with a clear listing of your thoughts and efforts so far. Thank you!

 

[Dr.Peterson]

"Erin and Antonio were shown the following diagram. They were asked to solve for the measure of angle A in degrees.​

Erin says the first step is to set 2x+7 equal to 5x-2.​

Antonio says the first step is to set 2x+7 plus 5x-2 equal to 180.​

Choose either Erin's or Antonio's statement and explain why it is correct or incorrect. If the statement is correct, explain how you know it's correct. If the statement is incorrect, explain why it's wrong.​

If you were to solve this problem, which steps would you take? Include any theorems, definitions, or reasons that explain the steps. Make sure you include all steps needed to solve for angle A."​

What theorems have you learned that might tell you how the angles at A and C are related?

 

[The Highlander]

Choose either Erin's or Antonio's statement and explain why it is correct or incorrect. If the statement is correct, explain how you know it's correct. If the statement is incorrect, explain why it's wrong.​

If you were to solve this problem, which steps would you take? Include any theorems, definitions, or reasons that explain the steps. Make sure you include all steps needed to solve for angle A."​

Hi @Ryuu,

Erin and Antonio have made suggestions that allow you to set up two equations...

Erin: 2x + 7 = 5x - 2​

and​

Antonio: 2x + 7 + 5x - 2 = 180​

Can you solve these equations to find a value for x in each case?

Once you have a value for x (in each case) you can then work out the measure of the angles at A & C (in each case) and that should help you to choose one of the statements and explain how it is right or wrong.

Unfortunately, (it seems to me) that the diagram does not provide enough information to solve for the measure of angle A.

The figure ADCE looks like a kite and the line segments CD & CE look like they are tangential to the circle. (If only they had marked those line segments as congruent (or provided some other piece of 'telling' information) then that would be different.)

Assuming that those "apparent facts" were true, would allow you to determine the measure of the angle at A but (in my opinion) that would be begging the question.

However, perhaps you are, indeed, expected to state just such assumptions as part of your answer to the final part of the question?

Hope that helps. ?

 

[skeeter]

Let's go ahead and assume the two external line segments are tangent to the circle at points D and E.
Connecting A to C creates two congruent right triangles.

What can you say about the sum of the two acute angles in any right triangle that would help you justify the sum of angles DAC and DCE ?

 

[pka]

Let's go ahead and assume the two external line segments are tangent to the circle at points D and E.
Connecting A to C creates two congruent right triangles.

What can you say about the sum of the two acute angles in any right triangle that would help you justify the sum of angles DAC and DCE ?


View attachment 36050

I tend to agree with @[SIZE=4]The Highlander[/SIZE] on this one. Although I am quite sure that you are correct about what was intended.
I do have one real dislike for the notation [imath](2x+7)^0~\&~(5x-2)^0[/imath], no real understanding as to its meaning.
I also think that this problem means to test the theorem that states: the measure of the angle formed by two such tangents is
one-half the difference in the measures of the intercepted major arc & minor arc.

[imath][/imath][imath][/imath]

 

[The Highlander]

I tend to agree with @[SIZE=4]The Highlander[/SIZE] on this one. Although I am quite sure that you are correct about what was intended.
I do have one real dislike for the notation [imath](2x+7)^0~\&~(5x-2)^0[/imath], no real understanding as to its meaning.
I also think that this problem means to test the theorem that states: the measure of the angle formed by two such tangents is
one-half the difference in the measures of the intercepted major arc & minor arc.

[imath][/imath][imath][/imath]

Indeed. (Though I have to admit I hadn't even heard of the theorem (about the arc measure difference) before ?, though I do believe it's much simpler than that, see reply to @skeeter, below.)

The question does not state that the line segments CD & CE are tangential to the circle or of equal length. It doesn't confirm that A is the centre of the circle, nor does it even say that the black/red figure is, in fact, a circle!

What if 2x + 7 + 5x - 2 = 180.7 (instead of 180)?
Then x = 25.1 and ∠A = 123.5° & ∠C = 57.2°

(Quadrilateral & Circle constructed in GeoGebra; final image (slightly) modified in Paint.net. ? )

​

Surely this could just as easily be what the picture in the OP's post actually represents? (Looks pretty much the same to me! ?)

ADCE is now a Trapezium (or Trapezoid, if you prefer) rather than the Kite (it looks like), both Erin & Antonio's suggestions are now wrong and nothing further can be reliably be deduced!

Let's go ahead and assume the two external line segments are tangent to the circle at points D and E.
Connecting A to C creates two congruent right triangles.

What can you say about the sum of the two acute angles in any right triangle that would help you justify the sum of angles DAC and DCE ?

Agreed, good approach, though I think you have missed a step and I believe there's an even simpler approach which (I suspect) is the one that the author(s) intended even though their question leaves much to be desired!

I think the best approach is...

First, Assume that A is the centre of a circle of radius AD,
then, assume that CD & CE are tangential to the circle.
(It may well be that the author(s) intended those to be "givens" anyway. ?‍)

Now the angles at D & E are fixed (and ADCE is without doubt a kite).

Thus, the sum of the internal angles* (in any quadrilateral) allows the use of one of the original "suggestions" to determine the value of x and, thence, the measure of the angle at A.

(On further consideration, my assumptions might be better stated (more elegantly) as: "Assuming that the line segments CD & CE are tangential to a circle of radius AD and centre A ".)

* (Theorem would require to be "stated" to comply with question's demands.)