Too Many Angles But Insufficient Information!

Geometry Level 2

An interior point $P$ is chosen in the rectangle $ABCD$ such that the sum of angles $APD$ and $BPC$ is $180^\circ$ . Find the sum of angles $\angle DAP$ and $\angle BCP$ in degrees.

The answer is 90.

2 solutions

Julian Poon
Oct 28, 2016

Let's clone the rectangle:

Since the sum of the blue angles are equal to 180°, through angle properties of circles, we can conclude that quadrilateral $APDP'$ is a cyclic quadrilateral. Redrawing the diagram:

Again through angle properties of circles, $\angle PAD=\angle PP'D$ (because the angles both intercept the same arc)

Considering triangle $P'OD$ , it can be seen that $\angle PAD+\angle PP'D=180°-\angle P'OD=180°-90°=90°$

Oh wow, very nice shape shifting!

Calvin Lin Staff - 4 years, 7 months ago

I agree this is very good! Not an approach I would have come up with. I do want to nitpick something small, though. You use the phrase "through angle properties of circles." It's faster to read if you just mention what you're actually using more specifically ("because the angles both intercept the same arc" for instance). Even someone who knows their properties very well has to search their mental bank and it takes just that small bit longer to read the solution.

Jason Dyer Staff - 4 years, 7 months ago

Alfa Claresta
Mar 10, 2017

Just using specialization, think rectangle as square and locate P on the center.

Too Many Angles But Insufficient Information!

The answer is 90.

2 solutions

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