Geometry: Given |AD|=12, |AC|=16, |AE|=6, |AB|=8, show angle ADE is congruent to angle ACB

[gamma_Juice]

 

[stapel]

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What are your thoughts? What have you tried? How far have you gotten?

Please re-read the "Read Before Posting" message, and reply with the requested information. Thank you!

Eliz.

 

[gamma_Juice]

I tried doing 4x^2+10=-2x^2-19x but got nowhere

 

[stapel]

I tried doing 4x^2+10=-2x^2-19x but got nowhere

One might set the two expressions equal to each other and solve, but only after one had proven that the angles are congruent.

What theorems/rules/etc have been covered recently, that they would generate the posted exercise? What have you tried with respect to answering the question, being the proving of the angles' congruency?

Please be complete. Thank you!

 

[The Highlander]

I tried doing 4x²+10 = -2x²-19x but got nowhere

No, doing so wouldn't be of much help; I suspect those expressions for the two angles' measures have just been included as a distraction.

Even if you were able to solve the equation you have set up (and it appears that was beyond your 'expertise' anyway) all you would find is that x is equal to -⅔ or -2½. Given the geometry of the figure x = -⅔ is unlikely as that would mean the measure of the angles was ≈ 11.78° whereas x = -2½ would mean their measure was 35° which is a much more 'sensible' result.

But you are not required to find the measure of the angles to answer this question! In fact, it is irrelevant to what you've been asked to do.

What do you know (or have been taught) about Similarity and, in particular, Similar Triangles?

Note, first of all, that the lengths of sides AC & AD are in the same ratio as the lengths of the sides AB & AE; you should be able to demonstrate this numerically (and may be required to do so as part of your answer?).
And the triangles (ΔABC & ΔADE) share the same vertex (at A) so what does that tell you about those two triangles and what else may be deduced from that conclusion?

Answering those two questions should set you on the right path.

Please come back and let us know how you got on from there...

Hope that helps. ?

PS: Do you know/understand what the symbol "≅" means? You might think it means "approximately equal to" (which might be confusing?) but that is not the case here! In this (geometrical) 'situation' it means that ∠ADE is congruent with/to ∠ACB. (I trust you know what "congruent" means? ?)

 

[Steven G]

I tried doing 4x^2+10=-2x^2-19x but got nowhere

Are you saying that you don't know how to solve the above equation? You will get a quadratic equation of the form 6x^2 + 19x + 10 = 0

What methods have you learned to solve quadratic equation? Try some/all of them!!!!

 

[Steven G]

View attachment 35849

One can argue that <ADE ≅ <ACB because it is given!!! Why is it given? Because both angles have the same number of slashes by them.

If I gave this problem on a Geometry test and a student wrote what I have above above then I would be thinking three things.
1) How did I give such a sloppy problem?
2) I must give this student full credit.
3) I need to tell my students that although their work may be correct they just needed to state that the conclusion, <ADE ≅ <ACB, was given.