SOLUTION: If WXYZ is a rhombus and measure of angle YWZ = 43 degrees, find the measure of angle ZXW.

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Question 1134178: If WXYZ is a rhombus and measure of angle YWZ = 43 degrees, find the measure of angle ZXW.
Found 3 solutions by Theo, greenestamps, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the opposite angles of a rhombus are equal and the adjacent angles of a rhombus are supplementary.

i don't really understand your lettering scheme, but if you are talking about opposite angles, then, if one of them is 43, the opposite angle is also 43.

the adjacent angles to these angles would be 180 - 43 = 137 degrees.

those adjacent angles are opposite to each other, therefore equal to each other.

you have 2 angles of 43 degrees each and 2 angles of 137 degrees each for a total of 360 degrees.

since a rhombus is a special kind of quadrilateral, the sum of the interior angles must be 360 degrees, as it is.

here is a reference on the properties of a rhombus.

https://socratic.org/questions/what-are-all-of-the-properties-of-a-rhombus

i think you should check your lettering scheme.

once you have that down, make sure a referenced angle has the letter for the appropriate corner as the middle letter in the designation.

the angle referenced as YWZ should have the vertex at corner W with the adjacent sides of YW and WZ.

the angle referenced as ZXW whould have the vertex at corner X with the adjacent sides of ZX and XW.

the lettering of the rhombus usually goes from one corner clockwise around the figure.

if i assume this convention, then a rhombus lettered WXYZ can't have an angle lettered YWZ.

it could have angles lettered as WXY or XYZ or YZW or ZWX.
they could also be lettered as YXW or ZYX or WZY or XWZ.

your angle lettering is not one of these.

bottom line is i don't believe your lettering of the rhombus is accurate based on lettering conventions for figures and angles.

check your lettering convention to see that you got it right.

keep in mind that the angles are either opposite each other or adjacent to each other.

if opposite, they are equal to each other.
if adjacent they are supplementary to each other.

if you have an angle labeled as YWZ, then the opposite angle has to be YXZ and the adjacent angles have to be WYX or WZX.

if you have an angle labeled as YWZ, you can't have an angle labeled as ZXW.







Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


With the given lettering scheme, angle YWZ is an angle formed by a diagonal and one side of the rhombus.

The diagonals of a rhombus divide the rhombus into four congruent right triangles.

That means angle YWX is also 43 degrees.

And that means angle ZXW is 90-43 = 47 degrees.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
It is very easy to solve.


You are given that the angle YWZ is 43 degrees.


Hence, the angle ZWX has the measure which is TWO TIMES the measure of the angle YWZ, i.e. 86 degrees 
(since the diagonal WZY of the rhombus WXYZ bisects its angle W).



Next, since the angle ZWX (which is exactly the angle W at the vertex W of the rhombus) is 86 degrees,

then the angle WXY of the rhombus (which is the angle at the vertex X of the rhombus) is 180 - 86 = 94 degrees

It is because any two consecutive (= adjacent) interior angles of each rhombus are supplementary.



The last step is to notice that the angle ZXW is HALF of the angle WXY (diagonals of the rhombus bisect its angles).

Hence, the angle ZXW measure is 94/2 = 47 degrees.     ANSWER

Completed and Solved.

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On properties of rhombis, see the lessons
    - Diagonals of a rhombus are perpendicular
    - Diagonals of a rhombus bisect its angles
    - The length of diagonals of a rhombus
    - A circle inscribed in the rhombus
    - HOW TO solve problems on the rhombus sides and diagonals measures - Examples
    - The length of diagonals of a rhombus
    - PROPERTIES OF RHOMBIS
in this site.

Also,  you have this free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lessons are the part of this textbook under the topic "Properties of rhombis".


Save the link to this online textbook together with its description

Free of charge online textbook in GEOMETRY
https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson

to your archive and use it when it is needed.


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