Question

3-2 angles formed by parallel lines and transversals
find each angle measure.

1. 135° x°
2. (15x - 7)° (19x - 15)°
3. (54x + 14)° (43x + 36)°

3-3 proving lines parallel
use the given information and the theorems and postulates you have learned to show that a || b.

1. m∠8=(13x + 20)°,m∠6=(7x + 38)°,x = 3
2. ∠1≅∠5
3. m∠8 + m∠7 = 180°
4. m∠8 = m∠4
5. the tower shown is supported by guy - wires such that m∠1=(3x + 12)°,m∠2=(4x - 2)°, and x = 14. show that the guy - wires are parallel.

3-4 perpendicular lines

1. write a two - column proof.

given: ∠1≅∠2,ℓ⊥n
prove: ℓ⊥p

Explanation:

Step1: Analyze question 9

The angle of $135^{\circ}$ and $x^{\circ}$ are supplementary (linear - pair of angles).
$x + 135=180$

Step2: Solve for $x$

Subtract 135 from both sides of the equation.
$x=180 - 135$
$x = 45$

Step3: Analyze question 10

The angles $(15x - 7)^{\circ}$ and $(19x-15)^{\circ}$ are corresponding angles (assuming parallel lines and a transversal), so they are equal.
$15x-7=19x - 15$

Step4: Solve for $x$

First, move the $x$ - terms to one side and the constants to the other side.
$15 - 7=19x-15x$
$4x = 8$
$x = 2$

Step5: Analyze question 11

The angles $(54x + 14)^{\circ}$ and $(43x + 36)^{\circ}$ are vertical angles, so they are equal.
$54x+14=43x + 36$

Step6: Solve for $x$

Move the $x$ - terms to one side and the constants to the other side.
$54x-43x=36 - 14$
$11x=22$
$x = 2$

Step7: Analyze question 12

Given $m\angle8=(13x + 20)^{\circ}$, $m\angle6=(7x + 38)^{\circ}$, and $x = 3$.
First, find $m\angle8$:
$m\angle8=13\times3+20=39 + 20=59^{\circ}$
Then, find $m\angle6$:
$m\angle6=7\times3+38=21 + 38=59^{\circ}$
Since $m\angle8=m\angle6$, by the alternate - interior angles theorem, $a\parallel b$.

Step8: Analyze question 14

Given $m\angle8 + m\angle7=180^{\circ}$.
By the same - side interior angles postulate, if the sum of two same - side interior angles is $180^{\circ}$, then the two lines are parallel. So $a\parallel b$.

Step9: Analyze question 15

Given $m\angle8=m\angle4$.
By the corresponding angles postulate, if corresponding angles are equal, then the two lines are parallel. So $a\parallel b$.

Step10: Analyze question 16

Given $m\angle1=(3x + 12)^{\circ}$, $m\angle2=(4x - 2)^{\circ}$, and $x = 14$.
First, find $m\angle1$:
$m\angle1=3\times14+12=42 + 12=54^{\circ}$
Then, find $m\angle2$:
$m\angle2=4\times14-2=56 - 2=54^{\circ}$
Since $m\angle1=m\angle2$, by the corresponding angles postulate, the guy - wires are parallel.

Step11: Analyze question 17

Two - column proof:

Statements	Reasons
2. $m\angle1=m\angle2$	Definition of congruent angles
3. $m\angle2 = 90^{\circ}$ (since $\ell\perp n$, so $m\angle1 = 90^{\circ}$)	Definition of perpendicular lines
4. $\ell\perp p$	Definition of perpendicular lines (because $m\angle2=90^{\circ}$)

Answer:

Question 9: $x = 45$
Question 10: $x = 2$
Question 11: $x = 2$
Question 12: $a\parallel b$ because $m\angle8=m\angle6 = 59^{\circ}$
Question 14: $a\parallel b$ because $m\angle8 + m\angle7=180^{\circ}$
Question 15: $a\parallel b$ because $m\angle8=m\angle4$
Question 16: The guy - wires are parallel because $m\angle1=m\angle2 = 54^{\circ}$
Question 17: See the two - column proof above.