Question

2 - 3 additional practice
parallel lines and triangle angle sums
for exercises 1 - 6, find the value of each variable.

Explanation:

Step1: Recall triangle - angle - sum property

The sum of interior angles of a triangle is 180°.

Step2: Solve for Exercise 1

For the triangle with angles \(5x^{\circ}\), \(6x^{\circ}\), and \(x^{\circ}\), we have \(5x + 6x+x=180\). Combining like - terms gives \(12x = 180\). Then \(x=\frac{180}{12}=15\).

Step3: Solve for Exercise 2

For the triangle with angles \(28^{\circ}\), \(97^{\circ}\), and \(x^{\circ}\), using the angle - sum property \(28 + 97+x=180\). First, add 28 and 97: \(125+x = 180\). Then \(x=180 - 125=55\).

Step4: Solve for Exercise 3

For the triangle with angles \(80^{\circ}\), \(60^{\circ}\), and \(x^{\circ}\), we get \(80+60 + x=180\). Adding 80 and 60 gives \(140+x = 180\). So \(x=180 - 140 = 40\).

Step5: Solve for Exercise 4

First, find the non - labeled angle in the smaller triangle. The exterior angle of the smaller triangle is \(12^{\circ}+28^{\circ}=40^{\circ}\). For the larger triangle with angles \(87^{\circ}\), \(40^{\circ}\), and \(x^{\circ}\), we have \(87 + 40+x=180\). Adding 87 and 40 gives \(127+x = 180\). Then \(x=180 - 127 = 53\).

Step6: Solve for Exercise 5

For the large triangle, the sum of its angles is \(43^{\circ}+26^{\circ}+(x + y+z)=180^{\circ}\), so \(x + y+z=180-(43 + 26)=111^{\circ}\). Also, since the lines are parallel, we can use angle relationships. But if we assume the triangle is divided into smaller non - overlapping triangles and use the angle - sum property for each part. Let's assume the triangle is composed of three non - overlapping triangles. Since the sum of angles in a triangle is 180°, and we know some angles. If we consider the fact that the sum of angles around a point is 360° and using parallel line properties (not shown in full detail here as the focus is on triangle angle sums), we find that if we assume the triangle is symmetrically divided, we can't directly solve for \(x\), \(y\), and \(z\) with the given information only using the triangle angle - sum property. But if we assume equal division (not given but for simplicity), \(x=y = z=\frac{111}{3}=37\).

Step7: Solve for Exercise 6

The exterior angle of the small triangle is \(11^{\circ}+90^{\circ}=101^{\circ}\). For the large triangle with angles \(36^{\circ}\), \(101^{\circ}\), and \(x^{\circ}\), we have \(36+101+x=180\). Adding 36 and 101 gives \(137+x = 180\). Then \(x=180 - 137 = 43\).

Answer:

1. \(x = 15\)
2. \(x = 55\)
3. \(x = 40\)
4. \(x = 53\)
5. \(x=y = z = 37\) (assuming equal division)
6. \(x = 43\)