Question

example 4 given m∠12 = 121° and m∠6 = 75°, find the measure of each missing angle. a. m∠1 = b. m∠2 = c. m∠3 = d. m∠4 = e. m∠5 = f. m∠7 = g. m∠8 = h. m∠9 = i. m∠10 = j. m∠11 = k. m∠13 = l. m∠14 = example 5 given m∠7 = 38° and m∠10 = 102°, find the measure of each missing angle.

Explanation:

Step1: Identify vertical - angle relationships

Vertical angles are equal. $\angle1$ and $\angle5$ are vertical angles, $\angle2$ and $\angle6$ are vertical angles, $\angle3$ and $\angle7$ are vertical angles, $\angle4$ and $\angle8$ are vertical angles, $\angle9$ and $\angle13$ are vertical angles, $\angle10$ and $\angle14$ are vertical angles, $\angle11$ and $\angle15$ (not labeled but for concept - building) are vertical angles, $\angle12$ and $\angle16$ (not labeled but for concept - building) are vertical angles.

Step2: Identify corresponding - angle relationships

Corresponding angles are equal when lines are parallel. For example, if we assume the relevant lines are parallel, $\angle1$ and $\angle9$ are corresponding angles, $\angle2$ and $\angle10$ are corresponding angles etc.

Step3: Identify alternate - interior and alternate - exterior angle relationships

Alternate - interior angles are equal when lines are parallel. For example, $\angle3$ and $\angle5$ are alternate - interior angles. Alternate - exterior angles are equal when lines are parallel. For example, $\angle2$ and $\angle14$ are alternate - exterior angles.

For Example 4:

Given $m\angle12 = 121^{\circ}$ and $m\angle6=75^{\circ}$
- a. $\angle1$ and $\angle12$ are vertical angles, so $m\angle1 = 121^{\circ}$
- b. $\angle2$ and $\angle6$ are vertical angles, so $m\angle2 = 75^{\circ}$
- c. $\angle3$ and $\angle1$ are supplementary (linear - pair), so $m\angle3=180 - 121=59^{\circ}$
- d. $\angle4$ and $\angle2$ are supplementary (linear - pair), so $m\angle4 = 180 - 75 = 105^{\circ}$
- e. $\angle5$ and $\angle1$ are vertical angles, so $m\angle5=121^{\circ}$
- f. $\angle7$ and $\angle3$ are vertical angles, so $m\angle7 = 59^{\circ}$
- g. $\angle8$ and $\angle4$ are vertical angles, so $m\angle8 = 105^{\circ}$
- h. $\angle9$ and $\angle1$ are corresponding angles (assuming parallel lines), so $m\angle9 = 121^{\circ}$
- i. $\angle10$ and $\angle2$ are corresponding angles (assuming parallel lines), so $m\angle10 = 75^{\circ}$
- j. $\angle11$ and $\angle3$ are corresponding angles (assuming parallel lines), so $m\angle11 = 59^{\circ}$
- k. $\angle13$ and $\angle5$ are corresponding angles (assuming parallel lines), so $m\angle13 = 121^{\circ}$
- l. $\angle14$ and $\angle6$ are corresponding angles (assuming parallel lines), so $m\angle14 = 75^{\circ}$

For Example 5:

Given $m\angle7 = 38^{\circ}$ and $m\angle10=102^{\circ}$
- a. $\angle1$ and $\angle5$ are vertical angles. $\angle5$ and $\angle13$ are corresponding angles, $\angle7$ and $\angle13$ are alternate - exterior angles. Since $m\angle7 = 38^{\circ}$, $m\angle1=m\angle5 = 38^{\circ}$
- b. $\angle2$ and $\angle6$ are vertical angles. $\angle6$ and $\angle14$ are corresponding angles. Since $m\angle10 = 102^{\circ}$, and $\angle10$ and $\angle14$ are vertical angles, $m\angle2=m\angle6 = 102^{\circ}$
- c. $\angle3$ and $\angle7$ are vertical angles, so $m\angle3 = 38^{\circ}$
- d. $\angle4$ and $\angle8$ are vertical angles. $\angle8$ and $\angle12$ are vertical angles. $\angle7$ and $\angle12$ are supplementary (linear - pair if we consider the relevant lines). Since $m\angle7 = 38^{\circ}$, $m\angle4=142^{\circ}$
- e. $m\angle5 = 38^{\circ}$ (from above)
- f. $m\angle7 = 38^{\circ}$ (given)
- g. $\angle8$ and $\angle4$ are vertical angles, so $m\angle8 = 142^{\circ}$
- h. $\angle9$ and $\angle1$ are corresponding angles, so $m\angle9 = 38^{\circ}$
- i. $\angle10$ and $\angle2$ are corresponding angles, so $m\angle10 = 102^{\circ}$ (given)
- j. $\angle11$ and $\angle3$ are corresponding angles, so $m\angle11 = 38^{\circ}$
- k. $\angle13$ and $\angle5$ are corresponding angles, so $m\angle13 = 38^{\circ}$
- l. $\angle14$ and $\angle6$ are corresponding angles, so $m\angle14 = 102^{\circ}$

Answer:

Example 4:
a. $121^{\circ}$
b. $75^{\circ}$
c. $59^{\circ}$
d. $105^{\circ}$
e. $121^{\circ}$
f. $59^{\circ}$
g. $105^{\circ}$
h. $121^{\circ}$
i. $75^{\circ}$
j. $59^{\circ}$
k. $121^{\circ}$
l. $75^{\circ}$

Example 5:
a. $38^{\circ}$
b. $102^{\circ}$
c. $38^{\circ}$
d. $142^{\circ}$
e. $38^{\circ}$
f. $38^{\circ}$
g. $142^{\circ}$
h. $38^{\circ}$
i. $102^{\circ}$
j. $38^{\circ}$
k. $38^{\circ}$
l. $102^{\circ}$