Question

1. consider the diagram to the right. use 3 points to name

(a) an angle adjacent to <hra:
(b) an angle supplementary to <trx:
(c) an angle vertical to <crt
(d) an angle complementary to <art:

1. in each polygon that has been drawn, identify all side pairs that are parallel.

(a)
parallel:
(b)
parallel:

1. consider the diagram to the right. identify one example of each of the following types of angle pairs.

(a) alternate interior angles: <rsw and
(b) same side interior angles: <uws and
(c) alternate exterior angles: <vwt and
(d) same side exterior angles: <vwt and
(e) corresponding angles: <psq and

Explanation:

1. (a)

Step1: Recall adjacent angles definition

Adjacent angles share a common side and vertex.
$\angle HRA$ shares side $RH$ and vertex $R$ with $\angle ARC$.

1. (b)

Step1: Recall supplementary angles definition

Supplementary angles sum to $180^{\circ}$.
$\angle TRX$ and $\angle XRA$ form a linear - pair (sum to $180^{\circ}$).

1. (c)

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by two intersecting lines.
$\angle CRT$ and $\angle XRA$ are vertical angles.

1. (d)

Step1: Recall complementary angles definition

Complementary angles sum to $90^{\circ}$.
Since $\angle ART+\angle XRT = 90^{\circ}$ (assuming $\angle ARX = 90^{\circ}$).

2. (a)

Step1: Recall parallelogram property

In a parallelogram, opposite sides are parallel.
In parallelogram $PARE$, $PA\parallel ER$ and $PE\parallel AR$.

2. (b)

Step1: Recall trapezoid and right - angle properties

In the given figure, $TR\parallel PA$ (because of the right - angles and the shape of the trapezoid - like figure).

3. (a)

Step1: Recall alternate interior angles definition

Alternate interior angles are between two parallel lines and on opposite sides of the transversal.
If we assume $RQ\parallel UT$ and transversal $PV$, $\angle RSW$ and $\angle SWU$ are alternate interior angles.

3. (b)

Step1: Recall same - side interior angles definition

Same - side interior angles are between two parallel lines and on the same side of the transversal.
If $RQ\parallel UT$ and transversal $PV$, $\angle UWS$ and $\angle RSW$ are same - side interior angles.

3. (c)

Step1: Recall alternate exterior angles definition

Alternate exterior angles are outside two parallel lines and on opposite sides of the transversal.
If $RQ\parallel UT$ and transversal $PV$, $\angle VWT$ and $\angle VSR$ are alternate exterior angles.

3. (d)

Step1: Recall same - side exterior angles definition

Same - side exterior angles are outside two parallel lines and on the same side of the transversal.
If $RQ\parallel UT$ and transversal $PV$, $\angle VWT$ and $\angle VSP$ are same - side exterior angles.

3. (e)

Step1: Recall corresponding angles definition

Corresponding angles are in the same relative position with respect to parallel lines and a transversal.
If $RQ\parallel UT$ and transversal $PV$, $\angle PSQ$ and $\angle PWT$ are corresponding angles.

Answer:

1. (a) $\angle ARC$; (b) $\angle XRA$; (c) $\angle XRA$; (d) $\angle XRT$
2. (a) $PA\parallel ER$, $PE\parallel AR$; (b) $TR\parallel PA$
3. (a) $\angle SWU$; (b) $\angle RSW$; (c) $\angle VSR$; (d) $\angle VSP$; (e) $\angle PWT$