Question

part 3 directions: solve for x. show all work! then find the measures of both angles.
10.
11.
part 4 directions: complete the proof.
given: lines ( a ) and ( b ) are parallel
cut by a transversal
prove: ( angle 2 cong angle 7 ) (alternate exterior angles
are congruent)
statements
lines ( a ) and ( b ) are parallel lines cut
by a transversal
( angle 2 cong angle 6 )
( angle 6 cong angle 7 )
( angle 2 cong angle 7 )
reasons
12.
13.
14.
15.

Explanation:

Step1: Solve for \(x\) in problem 10

Since the two angles \((2x - 60)^{\circ}\) and \((3x+30)^{\circ}\) are supplementary (they form a linear pair), we use the equation \(2x-60 + 3x + 30=180\).
Combine like terms: \(5x-30 = 180\).
Add \(30\) to both sides: \(5x=180 + 30=210\).
Divide both sides by \(5\): \(x=\frac{210}{5}=42\).

Step2: Find the measures of the angles in problem 10

For the first angle: \(2x-60=2\times42-60=84 - 60 = 24^{\circ}\).
For the second angle: \(3x + 30=3\times42+30=126+30 = 156^{\circ}\).

Step3: Solve for \(x\) in problem 11

Since the two angles \((2x + 53)^{\circ}\) and \((x + 60)^{\circ}\) are vertical angles (vertical angles are congruent), we use the equation \(2x+53=x + 60\).
Subtract \(x\) from both sides: \(2x-x+53=x-x + 60\), so \(x=60 - 53=7\).

Step4: Find the measures of the angles in problem 11

For the first angle: \(2x+53=2\times7+53=14 + 53=67^{\circ}\).
For the second angle: \(x + 60=7+60 = 67^{\circ}\).

Step5: Complete the proof (reasons)

1. Given (as stated in the problem).
2. If two parallel lines are cut by a transversal, then corresponding angles are congruent (\(\angle2\) and \(\angle6\) are corresponding angles).
3. Vertical angles are congruent (\(\angle6\) and \(\angle7\) are vertical angles).
4. Transitive property of congruence (if \(a\cong b\) and \(b\cong c\), then \(a\cong c\), here \(a = \angle2\), \(b=\angle6\), \(c=\angle7\)).

Answer:

• Problem 10: \(x = 42\), angles are \(24^{\circ}\) and \(156^{\circ}\).
• Problem 11: \(x = 7\), angles are \(67^{\circ}\) and \(67^{\circ}\).
• Proof Reasons:
• 12. Given.
• 13. Corresponding angles postulate.
• 14. Vertical angles theorem.
• 15. Transitive property of congruence.