Question

solve as directed.

1. if ∠b = 26°, what is the measure of ∠a?
2. if ∠c = 129°, what is the measure of ∠d?

use the drawing to answer questions 3 - 5.

1. name two pairs of vertical angles.
2. if ∠z measures 110°, what does ∠w measure?
3. ∠z = 110°. what does ∠x measure? how do you know?

use the drawing to answer questions 6 and 7.

1. name a supplementary angle to ∠mon.
2. what is the measure of ∠poq?

core connections: angle relationships
when one line crosses another line, vertical angles are formed. when one line crosses two parallel lines, the line, called a transversal, creates vertical angles in two places. these are called corresponding angles because they are in the same position on the parallel lines. corresponding angles are equal. if you know one angle, you can find the others.
example the measure of ∠4 is 40°. find all the other angles.

conclusion	reason	conclusion	reason
∠1 = 140°	∠1 and ∠2 are supplementary.	∠b = 40°	∠b and ∠d are vertical angles.
∠3 = 140°	∠1 and ∠3 are vertical.	∠a = 140°	∠1 and ∠a are corresponding angles.
		∠c = 140°	∠a and ∠c are vertical angles.

1. on the street map, ∠f = 137°. name three other angles with the same measure.
2. what is the measure of ∠a? how do you know you are correct?

Explanation:

Step1: Recall complementary - angle property

Complementary angles add up to 90°. Given $\angle b = 26^{\circ}$, and $\angle a+\angle b = 90^{\circ}$.
So, $\angle a=90^{\circ}-\angle b$.

Step2: Calculate $\angle a$

Substitute $\angle b = 26^{\circ}$ into the formula: $\angle a = 90^{\circ}-26^{\circ}=64^{\circ}$.

Step3: Recall supplementary - angle property for $\angle c$ and $\angle d$

Supplementary angles add up to 180°. Given $\angle c = 129^{\circ}$, and $\angle c+\angle d = 180^{\circ}$.
So, $\angle d=180^{\circ}-\angle c$.

Step4: Calculate $\angle d$

Substitute $\angle c = 129^{\circ}$ into the formula: $\angle d = 180^{\circ}-129^{\circ}=51^{\circ}$.

Step5: Identify vertical angles

Vertical angles are equal. For two intersecting lines, $\angle w$ and $\angle z$ are vertical angles, $\angle x$ and $\angle y$ are vertical angles.

Step6: Find $\angle w$ when $\angle z = 110^{\circ}$

Since vertical angles are equal, if $\angle z = 110^{\circ}$, then $\angle w=\angle z = 110^{\circ}$.

Step7: Find $\angle x$ when $\angle z = 110^{\circ}$

$\angle x$ and $\angle z$ are supplementary (a linear - pair). So $\angle x=180^{\circ}-\angle z$.
Substitute $\angle z = 110^{\circ}$: $\angle x = 180^{\circ}-110^{\circ}=70^{\circ}$.

Step8: Find supplementary angle to $\angle MON$

$\angle MON = 90^{\circ}+60^{\circ}=150^{\circ}$. A supplementary angle to $\angle MON$ is an angle that when added to $\angle MON$ gives 180°. So, $\angle NOQ$ is a supplementary angle to $\angle MON$ (since $\angle MON+\angle NOQ = 180^{\circ}$).

Step9: Calculate $\angle POQ$

$\angle POQ = 180^{\circ}-(90^{\circ}+60^{\circ})=30^{\circ}$.

Step10: For the street - map problem

If $\angle f = 137^{\circ}$, the angles with the same measure are the vertical angle to $\angle f$ and the corresponding angles to $\angle f$. Let's assume the vertical angle to $\angle f$ is $\angle e$, and the corresponding angles are $\angle c$ and $\angle b$.

Step11: Find $\angle a$ in the street - map problem

$\angle a$ and $\angle f$ are supplementary. So $\angle a=180^{\circ}-\angle f$.
Substitute $\angle f = 137^{\circ}$: $\angle a = 180^{\circ}-137^{\circ}=43^{\circ}$.

Answer:

1. $\angle a = 64^{\circ}$
2. $\angle d = 51^{\circ}$
3. Two pairs of vertical angles: $\angle w$ and $\angle z$, $\angle x$ and $\angle y$
4. $\angle w = 110^{\circ}$
5. $\angle x = 70^{\circ}$, because $\angle x$ and $\angle z$ are supplementary (form a linear - pair)
6. $\angle NOQ$
7. $\angle POQ = 30^{\circ}$
8. Three other angles with the same measure as $\angle f$: the vertical angle to $\angle f$ and two corresponding angles to $\angle f$
9. $\angle a = 43^{\circ}$, because $\angle a$ and $\angle f$ are supplementary.