Question

describing congruent angle relationships continued
use the angle relationships shown in each diagram to complete the equations.
12 = 135
x=
13 4x + 12=
x=
14 = 67
x=
15 what term refers to the pair of labeled angles? are these angles congruent? if so, find the value of x. if not, explain why not.

Explanation:

Step1: Solve problem 12

Since \(3x\) and \(135^{\circ}\) are vertical angles (vertical angles are equal), we have the equation \(3x = 135\).
Divide both sides by 3: \(x=\frac{135}{3}=45\).

Step2: Solve problem 13

Since \(4x + 12\) and \(112^{\circ}\) are corresponding angles (corresponding angles formed by parallel lines and a transversal are equal), we have the equation \(4x+12 = 112\).
Subtract 12 from both sides: \(4x=112 - 12=100\).
Divide both sides by 4: \(x=\frac{100}{4}=25\).

Step3: Solve problem 14

Since \(23x-2\) and \(67^{\circ}\) are alternate - exterior angles (alternate - exterior angles formed by parallel lines and a transversal are equal), we have the equation \(23x-2=67\).
Add 2 to both sides: \(23x=67 + 2=69\).
Divide both sides by 23: \(x=\frac{69}{23}=3\).

Step4: Solve problem 15

The pair of labeled angles \((10x - 19)^{\circ}\) and \((3x + 9)^{\circ}\) are vertical angles.
Set up the equation \(10x-19=3x + 9\).
Subtract \(3x\) from both sides: \(10x-3x-19=3x-3x + 9\), which simplifies to \(7x-19=9\).
Add 19 to both sides: \(7x-19 + 19=9+19\), so \(7x=28\).
Divide both sides by 7: \(x = 4\). Since vertical angles are congruent, these angles are congruent when \(x = 4\).

Answer:

1. \(3x\), \(x = 45\)
2. \(112\), \(x = 25\)
3. \(23x-2\), \(x = 3\)
4. The term is vertical angles. Yes, they are congruent. \(x = 4\)