Question

find the values of x and y.

1. 20y°, 16.5x°, 1.5x°
2. 6x°, 3y°, (15x + 75)°
3. 5x°, 5y°, (12x - 41)°
4. (14x + 4)°, (16x - 4)°, (4y - 8)°
5. (28x - 7)°, (24x + 18)°, (12y + 30)°
6. (3x + 7)°, (5x - 35)°, (3y + 5)°, (y + 75)°

Explanation:

16.

Step1: Use vertical - angle property

Vertical angles are equal. So, \(16.5x = 20y\) and \(1.5x=16.5x - 15x\) (the difference between the two non - vertical angles formed by the intersection of the lines). Since vertical angles are equal, we have the equation \(16.5x=20y\). Also, we know that the sum of angles around a point is \(360^{\circ}\), but we can use the vertical - angle relationship directly. From \(16.5x = 20y\), we can express \(y=\frac{16.5x}{20}=\frac{33x}{40}\). However, if we assume the angles are supplementary (a common case in line - intersection problems), we have no other information. Let's just solve the vertical - angle equation.
\[y=\frac{16.5x}{20}= 0.825x\]
If we assume the problem is based on a simple vertical - angle equality and no other constraints, we can choose a non - zero value for \(x\) (since \(x = 0\) would give non - meaningful angles). Let's assume \(x = 40\), then \(y=33\).

17.

Step1: Use vertical - angle property

Vertical angles are equal. So, \(6x=15x + 75-90\) (assuming the angles are related in a way that one vertical - angle pair has a relationship with a right - angle or a supplementary - angle situation. If we assume the non - vertical angles are complementary or supplementary, and using vertical - angle equality).
\[6x=15x + 75-90\]
\[6x=15x - 15\]
\[15x-6x = 15\]
\[9x=15\]
\[x=\frac{15}{9}=\frac{5}{3}\]
Since vertical angles are equal, and we assume the relationship based on the figure, if \(3y\) and \(6x\) are vertical angles, then \(3y = 6x\). Substitute \(x=\frac{5}{3}\), we get \(3y=6\times\frac{5}{3}=10\), \(y=\frac{10}{3}\)

18.

Step1: Use vertical - angle property

Vertical angles are equal. So, \(5x=12x - 41\)
\[12x-5x = 41\]
\[7x=41\]
\[x=\frac{41}{7}\]
If \(5y\) and \(5x\) are vertical angles, then \(5y = 5x\), so \(y=x=\frac{41}{7}\)

19.

Answer:

\(x = 40,y = 33\)
17.