Question

when a transversal intersects parallel lines, special angle pairs are formed:
angle type\trelationship
consecutive (same side) interior angles\tsupplementary m∠3 + m∠6 = 180°/ m∠4 + m∠5 = 180°
consecutive (same side) interior angles\tsupplementary m∠1 + m∠8 = 180°/ m∠2 + m∠7 = 180°
alternative interior angles\tcongruent m∠4 = m∠6, m∠3 = m∠5
alternative exterior angles\tcongruent m∠2 = m∠9 / m∠1 = m∠7
corresponding angles\t∠1≅∠5, ∠2≅∠6, ∠4≅∠8, ∠3≅∠7
number 1: identify the angle pair as congruent or supplementary. give your reasoning.
a. ∠3 and ∠5 alternative interior / congruent
b. ∠1 and ∠8 same side exterior / supplementary
c. ∠2 and ∠6 corresponding / supplementary
d. ∠1 and ∠7 alternative exterior / congruent
e. ∠4 and ∠5 same side interior /
f. ∠3 and ∠7 corresponding / congruent
g. ∠5 and ∠6 none / supplementary
h. ∠2 and ∠4 none / congruent
number 2: if m∠6 = 142°, find each measure. give your reasoning.
a. m∠1 =
b. m∠2 =
c. m∠3 =
d. m∠4 =
e. m∠5 =
f. m∠7 =
g. m∠8 =

Explanation:

Step1: Recall angle - pair relationships

When a transversal intersects parallel lines, corresponding angles are congruent, alternate - interior angles are congruent, alternate - exterior angles are congruent, and same - side interior (consecutive interior) angles are supplementary, same - side exterior angles are supplementary.

Step2: Analyze each angle pair for congruence or supplementarity in Number 1

a. $\angle3$ and $\angle5$ are alternate interior angles. By the alternate - interior angles theorem, they are congruent.
b. $\angle1$ and $\angle8$ are same - side exterior angles. So they are supplementary.
c. $\angle2$ and $\angle6$ are corresponding angles. They are congruent, not supplementary.
d. $\angle1$ and $\angle7$ are alternate exterior angles. They are congruent.
e. $\angle4$ and $\angle5$ are same - side interior angles. They are supplementary.
f. $\angle3$ and $\angle7$ are corresponding angles. They are congruent.
g. $\angle5$ and $\angle6$ are adjacent angles on a straight - line. They are supplementary.
h. $\angle2$ and $\angle4$ are vertical angles. They are congruent.

Step3: Solve for angle measures in Number 2 given $m\angle6 = 142^{\circ}$

a. $\angle1$ and $\angle6$ are same - side exterior angles. So $m\angle1=180 - 142=38^{\circ}$ (same - side exterior angles are supplementary).
b. $\angle2$ and $\angle6$ are corresponding angles. So $m\angle2 = 142^{\circ}$ (corresponding angles are congruent).
c. $\angle3$ and $\angle2$ are supplementary (linear pair). So $m\angle3=180 - 142 = 38^{\circ}$.
d. $\angle4$ and $\angle6$ are alternate interior angles. So $m\angle4 = 142^{\circ}$ (alternate interior angles are congruent).
e. $\angle5$ and $\angle6$ are supplementary (linear pair). So $m\angle5=180 - 142 = 38^{\circ}$.
f. $\angle7$ and $\angle6$ are supplementary (same - side interior angles). So $m\angle7=180 - 142 = 38^{\circ}$.
g. $\angle8$ and $\angle6$ are vertical angles. So $m\angle8 = 142^{\circ}$ (vertical angles are congruent).

Answer:

1.
a. Congruent (alternate interior angles)
b. Supplementary (same - side exterior angles)
c. Congruent (corresponding angles)
d. Congruent (alternate exterior angles)
e. Supplementary (same - side interior angles)
f. Congruent (corresponding angles)
g. Supplementary (adjacent angles on a straight - line)
h. Congruent (vertical angles)
2.
a. $38^{\circ}$
b. $142^{\circ}$
c. $38^{\circ}$
d. $142^{\circ}$
e. $38^{\circ}$
f. $38^{\circ}$
g. $142^{\circ}$