We want to find m ∠ QSO given that QS is a diameter of the given circle. Notice that ∠ QSO is an inscribed angle. Let's recall the Inscribed Angle Theorem.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
By this theorem we know that the measure of ∠ QSO is half that of the arc QO.
m ∠ QSO = 1/2 m QO
Next, notice that by the Arc Addition Postulate measures of QO and OS must sum to the measure of QSO.
mQO + mOS = m QSO
Finally, since QS is a diameter of our circle, the measure of QSO is 180^(∘). This, combined with the fact that based on the given diagram, m OS is 80^(∘), allows us to solve the equations above for mQO.
We want to find the measure of ∠ QPO from the diagram above. Notice that this angle intercepts the same arc as ∠ QSO, whose measure we found in Part A. By the Inscribed Angles of a Circle Theorem those measures must be equal.
m ∠ QPO = m ∠ QSO
In Part A we found that m ∠ QSO is 50^(∘). Let's substitute this value into equation above to find the sought measure.
m ∠ QPO = 50^(∘)
c Consider the given diagram.
In Part A we found that ∠ QSO is 50^(∘) and we are given that m ∠ NOS = 63^(∘). Therefore, ∠ ONS is the last interior angle of △ SNO that we do not know. Let's recall the Triangle Angle-Sum Theorem.
Triangle Angle-Sum Theorem
The sum of the interior angles of any triangle is 180^(∘).
using this theorem we can write an equation for m∠ ONS.
m∠ ONS + 63 + 50 = 180
Let's solve it!
Notice that we are given the measure of ∠ POS and PS, arc, whose measure we want to find, is the intercepted arc of this angle. By the Inscribed Angle Theorem the measure of the inscribed angle is half that of the intercepted arc.
m∠ POS = 1/2 mPS
From the diagram we know that m∠ POS = 63^(∘). Let's substitute this value into equation above and solve for mPS.
Notice that by the Arc Addition Postulate measures of PQ and PS must sum to the measure of QPS.
mPQ + mPS = m QPS
Finally, since QS is a diameter of our circle, the measure of QPS is 180^(∘). This, combined with the fact that in Part D we found that m PS is 126^(∘), allows us to solve the equations above for mPQ.
We want to find the measure of ∠ PQN from the diagram above. Notice that this angle intercepts the same arc as ∠ POS, whose measure we are given on the diagram. By the Inscribed Angles of a Circle Theorem those measures must be equal.
m ∠ PQN = m ∠ POS
We know from the diagram that m ∠ PON = 63^(∘). Let's substitute this value into equation above to find the sought measure.
m ∠ QPO = 63^(∘)
