Question

Maths
-- find the angle which a diagonal makes with a side of aSquare​

Answer 1

Step-by-step explanation:

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What angle of the diagonal of a square makes with each of its sides?

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3 ANSWERS

, High School Student

It is 45 degrees.

For a proof involving trigonometry, all we have to do is notice that

tan(CAB)=1, so sin(CAB)cos(CAB)=1, implying that sin(CAB)=cos(CAB). The only angle for which sin and cos are equal between 0 and 90 degrees is 45 degrees. We can do the same for angles BCA, DCA, and DAC.

For a geometry proof, we can note that triangles ABC and ADC in the diagram are congruent by SSS (Side-Side-Side) Congruency, as shown above. Because of this congruency, all the corresponding angles between the two triangles are congruent - angle B is equal to angle D, and so on. Noting this we can see that angle DAC and BAC must be equal and add up to 90 degrees, so they must both be 45 degrees. We can do the same for angles BCA and DCA, making DAC and BAC 45–45–90 triangles.