Question

example 4
in the figure, m∠7 = 100°. find the measure of each angle.

1. ∠9 / m∠9 = 100°
2. ∠6 / m∠6 = 80°
3. ∠8 / m∠8 = 80°
4. ∠2 / m∠2 = 80°
5. ∠5 / m∠5 = 100°
6. ∠11 / m∠11 = 100°
7. ramps a parking garage ramp rises to connect two horizontal levels of a parking lot. the ramp makes a 10° angle with the horizontal. what is the measure of angle 1 in the figure? 10°+∠1 = 180°

∠1 = 180° - 10° = 170°

1. city engineering seventh avenue runs perpendicular to 1st and 2nd streets, which are parallel. however, maple avenue makes a 115° angle with 2nd street. what is the measure of angle 1?

∠1 = 180° - 115° = 65°
answer = 65°
example 5
find the value of the variables in each figure. explain your reasoning.
29.
30.

1. find the value of the variables in the figure.

mixed exercises
in the figure, m∠3 = 75° and m∠10 = 105°. find the measure of each angle.

1. ∠2
2. ∠5
3. ∠7
4. ∠15
5. ∠14
6. ∠9

Explanation:

Step1: Identify vertical - angle relationship for problem 29

Vertical angles are equal. So, $5x = 40$.
$5x=40$

Step2: Solve for $x$ in problem 29

Divide both sides of the equation by 5.
$x=\frac{40}{5}=8$

Step3: Identify vertical - angle relationship for $y$ in problem 29

$(3y - 1)=40$

Step4: Solve for $y$ in problem 29

Add 1 to both sides: $3y=40 + 1=41$, then divide by 3. $y=\frac{41}{3}$

Step5: Identify corresponding - angle and supplementary - angle relationships for problem 30

Corresponding angles are equal, so $7x=8x - 10$.
$7x=8x - 10$

Step6: Solve for $x$ in problem 30

Subtract $7x$ from both sides: $0=x - 10$, then $x = 10$.

Step7: Identify supplementary - angle relationship for $y$ in problem 30

$(6y + 20)+7x=180$. Substitute $x = 10$ into the equation.
$(6y + 20)+7\times10=180$
$(6y + 20)+70=180$
$6y+90 = 180$

Step8: Solve for $y$ in problem 30

Subtract 90 from both sides: $6y=180 - 90 = 90$, then divide by 6. $y = 15$

Step9: Identify corresponding - angle and supplementary - angle relationships for problem 31

Corresponding angles: $(3x + 17)=(4y+3)$. Supplementary angles: $(5x - 7)+(3x + 17)=180$.
First, simplify the supplementary - angle equation:
$5x-7+3x + 17=180$
$8x+10 = 180$

Step10: Solve for $x$ in problem 31

Subtract 10 from both sides: $8x=180 - 10=170$, then divide by 8. $x=\frac{170}{8}=\frac{85}{4}$

Step11: Substitute $x$ into the corresponding - angle equation to solve for $y$ in problem 31

Substitute $x=\frac{85}{4}$ into $3x + 17=4y+3$.
$3\times\frac{85}{4}+17=4y+3$
$\frac{255}{4}+17=4y+3$
$\frac{255 + 68}{4}=4y+3$
$\frac{323}{4}=4y+3$
Subtract 3 from both sides: $\frac{323}{4}-3=4y$
$\frac{323-12}{4}=4y$
$\frac{311}{4}=4y$
$y=\frac{311}{16}$

Answer:

For problem 29: $x = 8$, $y=\frac{41}{3}$
For problem 30: $x = 10$, $y = 15$
For problem 31: $x=\frac{85}{4}$, $y=\frac{311}{16}$