Question

find the values of x and y.
a. x = 15, y = 17
b. x = 17, y = 15
c. x = 112, y = 68
d. x = 68, y = 112

in the rhombus, ∠1 = 130. what is the measure of ∠3?
130
50
25
75

Explanation:

Step1: Find the value of \(x\)

Vertical angles are equal. So \(7x + 7=112\)
Subtract \(7\) from both sides: \(7x=112 - 7\)
\(7x = 105\)
Divide both sides by \(7\): \(x=\frac{105}{7}=15\)

Step2: Find the value of \(y\)

Linear - pair angles sum to \(180^{\circ}\). So \(4y+112 = 180\)
Subtract \(112\) from both sides: \(4y=180 - 112\)
\(4y = 68\)
Divide both sides by \(4\): \(y=\frac{68}{4}=17\)

For the rhombus problem:

Step1: Properties of a rhombus

The diagonals of a rhombus bisect the vertex angles. The sum of adjacent angles in a rhombus is \(180^{\circ}\). But we can also use the angle - bisecting property.
Since the diagonals of a rhombus bisect the vertex angles, and \(\angle1 = 130^{\circ}\), the angle adjacent to \(\angle1\) is \(180 - 130=50^{\circ}\). And \(\angle3=\frac{1}{2}\times50^{\circ}\) (diagonal bisects the angle)
\(\angle3 = 25^{\circ}\)

Answer:

For the first problem: A. \(x = 15,y = 17\)
For the second problem: \(25\)