Question

In a rhombus, the difference of the measures of the two angles between a side and the diagonals is 32°. What are the measures of the angles of the rhombus?

Answer 1

Imagine the angles between a side and the diagonals of a rhombus.  They are part of a right triangle, with the hypotenuse being the side of the rhombus and the legs being part of the diagonals.

The sum of the angles is a+b+90=180, so a+b=90.
We know that a-b=32, so a=32+b.  Let's plug that into the equation above:
32+b+b=90, so 32+2b=90, so 2b=58, and b=29.
We can back solve for a: a+29=90, so a=61.

The measure of the small angles is 29 and 61.  However, these are not the angles of the rhombus.  We must double them: the angles of the rhombus are 122 and 58.

Answer 2

The measures of the angles are 122 degrees and 58 degrees

Assume the angles of the rhombus are A and B, such that:

• A = 2x
• B = 2y

So, the difference between the measures of the angles is given as:

$x - y = 32$

The angles are adjacent angles; and they add up to 90 degrees,

So, we have:

$x + y = 90$

Make x the subject in the first equation

$x = 32 + y$

Substitute 32 + y for x in the second equation

$32 + y + y = 90$

$32 + 2y = 90$

Subtract 32 from both sides

$2y = 58$

Divide both sides by 2

$y = 29$

Substitute 29 for y in $x = 32 + y$

$x = 32 + 29$

$x = 61$

Recall that:

$A = 2x$

$B = 2y$

So, we have:

$A=2\times 61 = 122$

$B=2\times 29 = 58$

Hence, the measures of the angles are 122 degrees and 58 degrees

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