a Two inscribed angles intersecting the same circle at the same point have the same measure, as long as their vertices also lie on the same circle.
B
b By the Inscribed angle Theorem the measure of an inscribed angle is half the measure of its intercepted arc.
C
c Two inscribed angles intersecting the same circle at the same point have the same measure, as long as their vertices also lie on the same circle.
D
d The measure of an arc intercepted by a diameter is equal to the measure of a straight angle, that is, equal to 180^(∘).
E
e The measure of an arc is defined to be equal to the measure of its corresponding central angle.
F
f The Angle Addition Postulate states that if two angles have the same vertex and are side by side, the new angle formed measures the sum of the two side by side angles.
A
a m ∠ D = 64^(∘)
B
b m BF = 128^(∘)
C
c m ∠ E = 64^(∘)
D
d m CBF = 180^(∘)
E
e m ∠ BAF = 128^(∘)
F
f m∠ BAC = 52^(∘)
Practice makes perfect
a We want to find the value of m ∠ D given that m ∠ C = 64^(∘) and CF is a diameter. This time, we only need to focus on angles ∠ C and ∠ D.
Notice that ∠ C and ∠ D are two inscribed angles with the same intercepted arc. For this reason their measures must be the same.
m ∠ C = m ∠ D
Since we are given that m ∠ C = 64^(∘), we can conclude that m ∠ D = 64^(∘).
b We want to find the value of m BF given that m ∠ C = 64^(∘) and CF is a diameter. This time, we only need to focus on ∠ C and BF.
The measure of an inscribed angle is half the measure of its intercepted arc.
Notice that ∠ C is an inscribed angle with BF as its intercepted arc. By the mentioned theorem we have the following relation between their measures.
m ∠ C = 1/2 m BF
We can substitute the known value of m ∠ C into equation above and solve for m BF. Let's do it!
c We want to find the value of m ∠ E given that m ∠ C = 64^(∘) and CF is a diameter. Like in Part A, we only need to focus on angles ∠ C and ∠ E.
Notice that ∠ C and ∠ E are two inscribed angles with the same intercepted arc. For this reason their measures must be the same.
m ∠ C = m ∠ E
Since we are given that m ∠ C = 64^(∘), we can conclude that m ∠ E = 64^(∘).
d We want to find the value of m CBF given that m ∠ C = 64^(∘) and CF is a diameter. This time we should focus on diameter CF and CBF.
Since CF is a diameter, the measure of its intercepted arc CBF must be the same as the measure of a straight angle, that is, m CBF = 180^(∘).
e We want to find the value of m ∠ BAF given that m∠ C = 64^(∘) and A is the center of the given circle.
In Part B we found that m BF = 128^(∘). Since the measure of an arc is defined to be equal to the measure of its corresponding central angle, we have m ∠ BAF = 128^(∘).
f This time we want to find the value of m ∠ BAC. In Part E we found that m ∠ BAF = 128^(∘) and we are given that CF is a diameter. Let's recall the Angle Addition Postulate.
Angle Addition Postulate
If two angles have the same vertex and are side by side, the new angle formed measures the sum of the two side by side angles.
We can use this postulate to relate the sum of measures of ∠ BAF and ∠ BAC to the measure of the straight angle ∠ CAF.
m ∠ BAF + m∠ BAC = m ∠ CAF
Since we know that m ∠ BAF = 128^(∘) and, since ∠ CAF is a straight angle, m ∠ CAF = 180^(∘), we can substitute this values into equation above and solve it for m ∠ CAF.
