d Angles ∠ QPT and ∠ SPR are vertical angles and therefore they are congruent.
A
a 138^(∘)
B
b 180^(∘)
C
c 222^(∘)
D
d 138^(∘)
Practice makes perfect
a We are asked to determine the measure of arc RS. Since the arc is described only with the endpoints R and S, it is a minor arc. Let's highlight RS on the given diagram.
To find the measure of RS, we will use the Arc Addition Postulate. Note that QS is a diameter of the circle, so QRS is a semicircle and mQRS= 180^(∘). With the Arc Addition Postulate and knowing that mQRS= 180^(∘) and mQR= 42^(∘), we can find mRS.
b We are asked to determine the measure of arc QRS. Since the arc is described with the endpoints Q and S and a point on the arc R, it is a major arc. Let's highlight QRS on the given diagram.
Note that the endpoints Q and S of the arc are the endpoints of a diameter of the circle. Therefore, QRS is a semicircle and its measure is 180^(∘).
mQRS=180^(∘)
c We are asked to determine the measure of arc QST. Since the arc is described with the endpoints Q and T and a point on the arc S, it is a major arc. Let's highlight QST on the given diagram.
To find the measure of QST, we will first find the measure of ST. Note that ∠ QPR and ∠ SPT are vertical angles and therefore they are congruent. This means that m∠ SPT=mST= 42^(∘).
From Part B we have that mQRS= 180^(∘). Arc QST consists of two adjacent arcs QRS and ST, so its measure is equal to the sum of measures of QRS and ST by the Arc Addition Postulate.
d We are asked to determine the measure of arc QT. Since the arc is described only with the endpoints Q and T, it is a minor arc. Let's highlight QT on the given diagram.
Note that ∠ QPT and ∠ SPR are vertical angles and therefore they are congruent. This means that the measures of QT and RS are equal.
mQT=mRS
Recall that from Part A we have that mRS= 138^(∘).
mQT=mRS ⇒ mQT= 138^(∘)
