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
number1: 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 for part a

$\angle1$ and $\angle6$ are non - related in the standard parallel - line and transversal angle pairs. But $\angle1$ and $\angle5$ are corresponding angles and $\angle5$ and $\angle6$ are supplementary ($\angle5+\angle6 = 180^{\circ}$). Since $\angle6=142^{\circ}$, then $\angle5 = 180 - 142=38^{\circ}$. And $\angle1=\angle5$, so $m\angle1 = 38^{\circ}$.

Step3: Analyze for part b

$\angle2$ and $\angle6$ are corresponding angles. So $m\angle2=m\angle6 = 142^{\circ}$.

Step4: Analyze for part c

$\angle3$ and $\angle5$ are supplementary. Since $\angle5 = 38^{\circ}$, then $m\angle3=180 - 38 = 142^{\circ}$.

Step5: Analyze for part d

$\angle4$ and $\angle6$ are alternate interior angles. So $m\angle4=m\angle6 = 142^{\circ}$.

Step6: Analyze for part e

$\angle5$ and $\angle6$ are supplementary. $m\angle5=180 - m\angle6=180 - 142 = 38^{\circ}$.

Step7: Analyze for part f

$\angle7$ and $\angle5$ are corresponding angles. So $m\angle7=m\angle5 = 38^{\circ}$.

Step8: Analyze for part g

$\angle8$ and $\angle6$ are vertical angles. So $m\angle8=m\angle6 = 142^{\circ}$.

Answer:

a. $m\angle1 = 38^{\circ}$
b. $m\angle2 = 142^{\circ}$
c. $m\angle3 = 142^{\circ}$
d. $m\angle4 = 142^{\circ}$
e. $m\angle5 = 38^{\circ}$
f. $m\angle7 = 38^{\circ}$
g. $m\angle8 = 142^{\circ}$