In a rhombus, each diagonal bisects a pair of opposite angles.
109, 71, 109 and 71
Practice makes perfect
We are told that the length of each side of the rhombus is 16 and the longer diagonal has a length of 26. We can model this in the following diagram.
We want to find the measures of its interior angles. In a rhombus the diagonals are perpendicular lines, so they create 4 right triangles. To help with the process of solving, we will label the interior angles and we will draw the shorter diagonal.
Looking at the above diagram, we can see that the hypotenuses of all these triangles are the same length as the side of the rhombus. The legs of these triangles are the same length as half the lengths of the diagonals. We will use the sine ratio to find m∠ 2 in the triangle.
sin (m∠ 2) = Length of leg opposite tom∠ 2/Length of hypotenuse
In our triangle, we have that the length of the opposite leg to m∠ 2 and hypotenuse are 13 and 16.
sin (m∠ 2) = Opposite/hypotenuse=13/16
The sine of the angle is 1316. Now, to isolate m∠ 2 we will use the inverse function of sin.
sin (m∠ 2)=13/16 ⇔ m∠ 2=sin ^(- 1)(13/16)
Let's use a calculator to find the value of sin ^(- 1)( 1316). First, we will set our calculator into degree mode. To do so, push MODE, select Degree instead of Radian in the third row, and push ENTER. Next, we push 2ND followed by SIN, introduce the value 1316, and press ENTER.
Therefore, m∠ 2 correct to the nearest tenth degree is about 54.3^(∘). Now, we see that ∠ 1, the right angle and ∠ 2 are the interior angles of the triangle. By the Triangle Angle-Sum Theorem, we conclude that their sum is 180. Let's find ∠ 1.
m∠ 1+ 90 +m∠ 2=180
We will substitute 54.3 for m∠ 2 in our equation and solve for m∠ 1.
Once we have the values of m∠ 1 and m∠ 2, we will find the values of interior angles. To find m∠ A, we know that the diagonals bisect the angles of the rhombus, so m∠ A of the rhombus is twice m∠ 1.
m∠ A&= 2(m∠ 1)
&⇕
&=2(35.7)
&=71.4
Therefore, m∠ A correct to the nearest degree is about 71^(∘). Now, we will find the m∠ B in a similar way. The m∠ B is twice m∠ 2.
m∠ B&= 2(m∠ 2)
&⇕
&=2(54.3)
&=108.6
We know that m∠ B correct to the nearest degree is about 109^(∘). Now we will find m∠ C and m∠ D. To do so, let's recall that in a rhombus the opposite angles are congruent.
m∠ C&=m∠ A=71^(∘)
m∠ D&=m∠ B=109^(∘)
