Question

GeometryQuestion 5Find the measure of the numbered angles in each rhombus.(Find each angle)

Answer 1

Hello there. To solve this question, we'll have to remember some properties about quadrilaterals.

Given the following rhombus with numbered angles:

We want to determine the measure of each of the numbered angles.

In this case, notice the rhombus is a parallelogram that has parallel sides with the same measure. This means the sides colored with the same color in the above image have the same measure.

This means that the angles 1 and 4 have to be congruent to each other, as well as the angles 2 and 3.

In fact, these angles are the same and we can determine them by showing that the missing angle at the other corner of the rhombus is also 116º:

We simply have to add the angles inside the triangles composing and this sum will be equal to 360º (the total sum of the internal angles in a quadrilateral)

Hence we'll get

`\begin{gathered} 2\cdot116^(\circ)+m\angle1+m\angle2+m\angle3+m\angle4=360^(\circ) \\ \\ \text{ But since }m\angle1=m\angle4\text{ and }m\angle2=m\angle3 \\ \\ 232^(\circ)+2\cdot m\angle1+2\cdot m\angle2=360^(\circ) \\ \\ m\angle1+m\angle2=64^(\circ) \\ \\ \\ \end{gathered}`

Notice also that the line passing through the rhombus divides it into two equal triangles, so we say it bisects the rhombus.

Hence the measures of the angles 1 and 2 are also equal, that means

`\begin{gathered} 2\cdot m\angle1=64^(\circ) \\ \\ \Rightarrow m\angle1=32^(\circ) \end{gathered}`

And by the last equality, we get

`m\angle1=m\angle2=m\angle3=m\angle4=32^(\circ)`

This is the answer to this question.