Exercise 36 Page 568

We are given that PC and PB are tangent to the circle in the following diagram.

We will find the congruent angles in the diagram. To do so focus on the arcs CB, BA, and AC. Let's examine whether the angles that form these arcs are congruent. To be able to talk about the angles easily, we will label one point from PC and one point from PB.

Let's begin by examining whether the angles that form CB are congruent.

Arc	Angle Measure	Theorem	Arc Measure
CB	m∠ BPC	Angles Outside the Circle Theorem	1/2(mBAC-mCB)
	m∠ BAC	Inscribed Angle Theorem	2mCB ✓
	m∠ PBC	Tangent and Intersected Chord Theorem	2mCB ✓
	m∠ BCP	Tangent and Intersected Chord Theorem	2mCB ✓

Since the measures of ∠ BAC, ∠ PBC, and ∠ BCP are equal to 2mCB, these angles are congruent. ∠ BAC≅∠ PBC≅∠ BCP From here, we will focus on BA.

Arc	Angle Measure	Theorem	Arc Measure
BA	m∠ ACB	Inscribed Angle Theorem	2mBA ✓
BA	m∠ ABE	Tangent and Intersected Chord Theorem	2mBA ✓

Because the measure of both ∠ ACB and ∠ CBE are equal to 2mAC, these angles are congruent. ∠ ACB≅∠ ABE Last, we will check whether the angles that form AC are congruent.

Arc	Angle Measure	Theorem	Arc Measure
AC	m∠ CBA	Inscribed Angle Theorem	2mAC ✓
AC	m∠ DCA	Tangent and Intersected Chord Theorem	2mAC ✓

The measures of both ∠ CBA and ∠ DCA are equal to 2mAC, so these angles congruent. To sum up, all of the congruent angles are as follows. ∠ BAC≅∠ PBC≅∠ BCP ∠ ACB≅∠ ABE ∠ CBA≅∠ DCA

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