Answer:
1) x = 55°
2) x = 30°
3) x = 68°
Step-by-step explanation:
In the given set of questions where it requires to find the unknown value of x to prove that v ||w:
Definition of Consecutive Interior Angles Theorem:
The Consecutive Interior Angles Theorem states that for any given two parallel lines cut by a transversal, the measures of a pair of consecutive interior angles have a sum of 180°. Hence, the pair of consecutive interior angles are supplementary.
Question 1
1.) In the given diagram, the two consective interior angles are m ∠70° and m ∠ 2x°. As defined by the Consecutive Interior Angles Theorem:
m ∠70° + m ∠ 2x° = 180°
Solve for x algebraically:
70° + 2x° = 180°
Subtract 70° from both sides:
70° - 70° + 2x° = 180° - 70°
2x° = 110°
Divide both sides by 2 to solve for x:
[tex]\LARGE\mathsf{\frac {2x^{\circ}}{2}\:=\:\frac{110^{\circ}}{2}}[/tex]
x = 55°
Verify whether the value for x is valid:
70° + 2x° = 180°
70° + 2(55)° = 180°
70° + 110° = 180°
180° = 180° (True statement).
Question 2:
2) In the given diagram, the two consective interior angles are m ∠2x° and m ∠ 4x°. As defined by the Consecutive Interior Angles Theorem:
m ∠2x° + m ∠ 4x° = 180°
Solve for x algebraically:
2x° + 4x° = 180°
Add like terms:
6x° = 180°
Divide both sides by 6 to solve for x:
[tex]\LARGE\mathsf{\frac {6x^{\circ}}{6}\:=\:\frac{180^{\circ}}{6}}[/tex]
x = 30°
Question 3:
The Consecutive Interior Angles Theorem still applies in this question. The two consective interior angles are m ∠90° and m ∠ (x + 22)°. As defined by the Consecutive Interior Angles Theorem:
m ∠90° + m ∠ (x + 22)° = 180°
Solve for x algebraically:
90° + x° + 22 = 180°
Combine like terms:
112° + x° = 180°
Subtract 112° from both sides to isolate x:
112° - 112° + x° = 180° - 112°
x = 68°
Verify whether the value for x is valid:
90° + x° + 22 = 180°
90° + 68° + 22 = 180°
180° = 180° (True statement).
