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McGraw Hill Integrated II, 2012

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McGraw Hill Integrated II, 2012 View details

Practice Test

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Exercise 12 Page 467

What are the properties of the incenter of a triangle?

32

Let's modify the given figure a bit, keeping in mind that the question asks for the measure of an angle.

We focus on the angle measures given on the figure and ignore the information about the lengths of certain segments.
The Incenter Theorem tells us that the incenter of a triangle is the point of concurrency of the angle bisectors. Let's add angle markers indicating congruent angles.

A question mark indicates angle ∠ HAC, the measure of which we are asked to find. Let's look for the angles of triangle △ ABC first.

Finding m∠ ABC and m∠ BCA.

Since HB bisects angle ∠ ABC, the given m∠ ABH=30 is half of the measure of angle ∠ ABC.

m∠ ABC=2m∠ ABH

m∠ ABH= 30

m∠ ABC=2( 30)

Multiply

m∠ ABC=60

Similarly, we can get the angle at C.

m∠ BCA=2m∠ BCH

m∠ BCH= 28

m∠ BCA=2( 28)

Multiply

m∠ BCA=56

Finding m∠ BAC.

Now that we know the measure of two angles of triangle △ ABC, we can use the Triangle Angle-Sum Theorem to find the third angle.

m∠ ABC+m∠ BCA+m∠ BAC=180

m∠ ABC= 60, m∠ BCA= 56

60+ 56+m∠ BAC=180

▼

Solve for m∠ BAC

Add terms

116+m∠ BAC=180

LHS-116=RHS-116

m∠ BAC=64

Answering the Question

Since HA bisects angle ∠ BAC, the measure of angle ∠ HAC is half of m∠ BAC=64.

m∠ HAC=1/2m∠ BAC

m∠ BAC= 64

m∠ HAC=1/2( 64)

Multiply

m∠ HAC=32

The measure of angle ∠ HAC is m∠ HAC=32.