Interpreting Angles/Right Triangles/Rectangles

Question

I don't want to ask the entire question as I still want to answer it myself. However, there is something (a step) I would like to confirm that I have encountered in the question.

Suppose there is a figure ABCD that very much looks like a rectangle. We are given: ANGLE BAD is 90 degrees; ANGLE BCD is also 90 degrees. Can I assume that the figure is a rectangle?

My take: I think not. What if ANGLE ABC and ANGLE ADC are not 90 degrees each? I have no way to know. Am I right? (There is also a diagonal drawn, from B to D, and I know the value of ONE of the angles (say, DBC) inside the resultant triangle BCD, but I am not even going there yet.) Based on this information that I have, is my conclusion right? I would be grateful for any insights.

Kindly shed some light. My attempt at the figure, below:

B_______________________________C
| |
| |
| |
|_____________ __________________|
A D

mindful · Answer

Whoops. The figure didn't quite come out the way I intended it to (though when I constructed it, it looked fine.) Hope you can figure out the labelling anyway.

theCEO · Answer

mindful wrote:I don't want to ask the entire question as I still want to answer it myself. However, there is something (a step) I would like to confirm that I have encountered in the question.

Suppose there is a figure ABCD that very much looks like a rectangle. We are given: ANGLE BAD is 90 degrees; ANGLE BCD is also 90 degrees. Can I assume that the figure is a rectangle?

No you cannot assume that the figure is a rectangle. See pic.

mindful · Answer

Thought so! Thank you!

Now what if BD was 2; and DC was 1. Does that change things at all?

My guess: No. All it tells me is that BC is Root 3, and the angles in Triangle BDC are 30-60-90 (with Angle DBC being 30, and so on). Can I deduce anything about angles in Triangle ADB, except that the sum of ANGLE ABD and ANGLE BDA is 90 degrees? Kindly enlighten.

theCEO · Answer

mindful wrote:Thought so! Thank you! Now what if BD was 2; and DC was 1. Does that change things at all?

My guess: No. All it tells me is that BC is Root 3, and the angles in Triangle BDC are 30-60-90 (with Angle DBC being 30, and so on). Can I deduce anything about angles in Triangle ADB, except that the sum of ANGLE ABD and ANGLE BDA is 90 degrees? Kindly enlighten.

Here's some visual confirmation of your suspicion: as you can see, all you're doing is grafting another right triangle on the hypotenuse of a 30-60-90.

Matt@VeritasPrep · Answer

mindful wrote:Thought so! Thank you! Now what if BD was 2; and DC was 1. Does that change things at all?

My guess: No. All it tells me is that BC is Root 3, and the angles in Triangle BDC are 30-60-90 (with Angle DBC being 30, and so on). Can I deduce anything about angles in Triangle ADB, except that the sum of ANGLE ABD and ANGLE BDA is 90 degrees? Kindly enlighten.

Here's some visual confirmation of your suspicion: as you can see, all you're doing is grafting another right triangle on the hypotenuse of a 30-60-90.

mindful · Answer

Thanks everyone! The figures really clarify it all.

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