In a rhombus 𝐹𝐺𝐻𝐽, the measure of angle 𝐽𝐹𝐾 is three 𝑦 plus six degrees and the measure of angle 𝐾𝐹𝐺 is seven 𝑦 minus 14 degrees. Find the value of 𝑦.
So we’ve been given a diagram of a rhombus 𝐹𝐺𝐻𝐽 and the measures of two angles. These two angles are the angles in one corner of the rhombus, where the diagonal 𝐹𝐻 has divided this vertex up into two parts.
We need to recall the key facts about the diagonals of a rhombus. Here is a key fact. In a rhombus, each diagonal bisects a pair of opposite angles. Remember, bisects means to cut in half. So the diagonal 𝐹𝐻 highlighted in pink here bisects the two angles at the opposite side of the rhombus, angle 𝐺𝐹𝐽 and angle 𝐽𝐻𝐺.
This means then that the angle marked in green and the angle marked in orange are equal to each other. And therefore so are the two expressions. We can express this as an equation. Seven 𝑦 minus 14 is equal to three 𝑦 plus six.
Now that we have an equation, this question has become an algebraic problem. We need to solve this equation in order to find the value of 𝑦. As the returns involving 𝑦 on both sides, I’m going to begin by subtracting three 𝑦 from each side of the equation. Doing so eliminates the three 𝑦 on the right-hand side and gives four 𝑦 minus 14 on the left-hand side. So we have four 𝑦 minus 14 is equal to six.
Next I’m going to add 14 to both sides of the equation. This cancels out the negative 14 on the left-hand side and gives four 𝑦 is equal to 20. The final step in solving this equation is to divide both sides by four. This gives 𝑦 is equal to five.
So we have our solution to the problem. The value of 𝑦 is five. Remember, the key fact we used in this question is that, in a rhombus, each diagonal bisects a pair of opposite angles.