Pearson Geometry Common Core, 2011
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Pearson Geometry Common Core, 2011 View details
3. Inscribed Angles
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Exercise 10 Page 784

The Inscribed Quadrilateral Theorem says that opposite angles of a quadrilateral inscribed in a circle are supplementary.

a = 112, b = 120, and c = 38

Practice makes perfect

Consider the given diagram.

Let's find the values of a, b, and c one at a time.

Finding a

In the given diagram, the vertices of the quadrilateral are all on the circle. This means that we have an inscribed quadrilateral. The Inscribed Quadrilateral Theorem says that opposite angles of a quadrilateral inscribed in a circle are supplementary.

Using this theorem, we can find the value of a. a+68=180 ⇔ a=112

Finding b

By the Arc Addition Postulate, we know that the measure of the intercepted arc of the inscribed angle whose measure is 112^(∘) is the sum of 104^(∘) and b^(∘).

The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Therefore, 112^(∘) is half of the sum of 104^(∘) and b^(∘). We can write and solve an equation to find the value of b.
112=1/2(104+b)
Solve for b
224=104+b
120=b
b=120

Finding c

Once again, by the Arc Addition Postulate we know that the measure of the intercepted arc of the inscribed angle whose measure is 71^(∘) is the sum of 104^(∘) and c^(∘).

By the Inscribed Angle Theorem we know that 71^(∘) is half of (104+c)^(∘).
71=1/2(104+c)
Solve for c
142 = 104 + c
38 = c
c = 38