Question

∠bac measures ____ °
∠bac and ∠dae are ____ angles.
∠dae measures ____ °

∠vzw and ∠vzy are ____ angles.
∠vzy measures ____ °
∠vzw and ∠yzx are ____ angles.
∠yzx measures ____ °
∠wzx and ∠yzx are ____ angles.
∠wzx measures ____ °

∠sqt and ∠rqs are ____ angles.
∠sqr measures ____ °
∠rqt measures ____ °

Explanation:

Step1: Identify angle type (∠VZW & ∠VZY)

These angles form a straight line, so they are supplementary (linear pair) angles.

Step2: Calculate ∠VZY

$\angle VZY = 180^\circ - \angle VZW = 180^\circ - 20^\circ = 160^\circ$

Step3: Identify angle type (∠VZW & ∠YZX)

These are opposite intersecting angles, so they are vertical angles.

Step4: Calculate ∠YZX

Vertical angles are equal: $\angle YZX = \angle VZW = 20^\circ$

Step5: Identify angle type (∠WZX & ∠YZX)

These angles form a straight line, so they are supplementary (linear pair) angles.

Step6: Calculate ∠WZX

$\angle WZX = 180^\circ - \angle YZX = 180^\circ - 20^\circ = 160^\circ$

Step7: Identify angle type (∠SQT & ∠RQS)

The diagram shows a right angle, so these are complementary angles (sum to 90°).

Step8: Calculate ∠SQR

$\angle SQR = 90^\circ - \angle SQT = 90^\circ - 30^\circ = 60^\circ$

Step9: Calculate ∠RQT

$\angle RQT = 90^\circ + \angle SQT = 90^\circ + 30^\circ = 120^\circ$

Answer:

∠VZW and ∠VZY are supplementary (linear pair) angles.
∠VZY measures $\boldsymbol{160}$°
∠VZW and ∠YZX are vertical angles.
∠YZX measures $\boldsymbol{20}$°
∠WZX and ∠YZX are supplementary (linear pair) angles.
∠WZX measures $\boldsymbol{160}$°
∠SQT and ∠RQS are complementary angles.
∠SQR measures $\boldsymbol{60}$°
∠RQT measures $\boldsymbol{120}$°