Identify the angle AYZ. This is the angle between the tangent XYZ and the chord AY.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the questions based on Fig. 12.31 and the Alternate Segment Theorem, which states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. a) Name two angles equal to ∠ AYZ. Step 1: Identify the angle ∠ AYZ. This is the angle between the tangent XYZ and the chord AY. Step 2: Apply the Alternate Segment Theorem. The angles in the alternate segment subtended by chord AY are ∠ ABY and ∠ ACY. ∠ AYZ = ∠ ABY = ∠ ACY The two angles are ∠ ABY and ∠ ACY. b) Name two angles equal to ∠ CYX. Step 1: Identify the angle ∠ CYX. This is the angle between the tangent XYZ and the chord CY. Step 2: Apply the Alternate Segment Theorem. The angles in the alternate segment subtended by chord CY are ∠ CAY and ∠ CBY. ∠ CYX = ∠ CAY = ∠ CBY The two angles are ∠ CAY and ∠ CBY. c) Name an angle equal to ∠ BYZ. Step 1: Identify the angle ∠ BYZ. This is the angle between the tangent XYZ and the chord BY. Step 2: Apply the Alternate Segment Theorem. The angle in the alternate segment subtended by chord BY is ∠ BAY. ∠ BYZ = ∠ BAY The angle is ∠ BAY. d) Name an angle equal to ∠ BYX. Step 1: Identify the angle ∠ BYX. This is the angle between the tangent XYZ and the chord BY. Step 2: Apply the Alternate Segment Theorem. The angle in the alternate segment subtended by chord BY is ∠ BCY. ∠ BYX = ∠ BCY The angle is ∠ BCY. e) If ∠ AYZ = 58° what is ∠ ACY? Step 1: From part a), we know that ∠ AYZ = ∠ ACY. Step 2: Substitute the given value. ∠ ACY = 58° The angle is 58°. f) If ∠ BCY = 112° what is ∠ BYZ? Step 1: From part d), we know that ∠ BYX = ∠ BCY. ∠ BYX = 112° Step 2: Angles on a straight line sum to 180°. ∠ BYX and ∠ BYZ are adjacent angles on the straight line XYZ. ∠ BYX + ∠ BYZ = 180° Step 3: Substitute the value of ∠ BYX and solve for ∠ BYZ. 112° + ∠ BYZ = 180°; ∠ BYZ = 180° − 112°; ∠ BYZ = 68° The angle is 68°. g) If ∠ BCY = 125° what is ∠ BYX? Step 1: From part d), we know that ∠ BYX = ∠ BCY. Step 2: Substitute the given value. ∠ BYX = 125° The angle is 125°. h) If ∠ BYZ = 100° what is ∠ BAY? Step 1: From part c), we know that ∠ BYZ = ∠ BAY. Step 2: Substitute the given value. ∠ BAY = 100° The angle is 100°.

Identify the angle AYZ. This is the angle between the tangent XYZ and the chord AY.

Answers

Step-by-step solution

Part a· 2 steps

Identify the angle $\angle AYZ$ .

Apply the Alternate Segment Theorem.

Part b· 2 steps
$\angle CAY \text{ and } \angle CBY$

Identify the angle $\angle CYX$ .

Apply the Alternate Segment Theorem.

Part c· 2 steps
$\angle BAY$

Identify the angle $\angle BYZ$ .

Apply the Alternate Segment Theorem.

Part d· 2 steps
$\angle BCY$

Identify the angle $\angle BYX$ .

Apply the Alternate Segment Theorem.

Part e· 2 steps
$58^\circ$

From part a), we know that $\angle AYZ = \angle ACY$ .

Substitute the given value.

Part f· 3 steps
$68^\circ$

From part d), we know that $\angle BYX = \angle BCY$ .

Angles on a straight line sum to $180^\circ$ .

Substitute the value of $\angle BYX$ and solve for $\angle BYZ$ .

Part g· 2 steps
$125^\circ$

From part d), we know that $\angle BYX = \angle BCY$ .

Substitute the given value.

Part h· 2 steps
$100^\circ$

From part c), we know that $\angle BYZ = \angle BAY$ .

Substitute the given value.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Identify the angle AYZ. This is the angle between the tangent XYZ and the chord AY.

Answers

Step-by-step solution

Part a· 2 steps

Identify the angle ∠AYZ\angle AYZ∠AYZ.

Apply the Alternate Segment Theorem.

Part b· 2 steps∠CAY and ∠CBY\angle CAY \text{ and } \angle CBY∠CAY and ∠CBY

Identify the angle ∠CYX\angle CYX∠CYX.

Apply the Alternate Segment Theorem.

Part c· 2 steps∠BAY\angle BAY∠BAY

Identify the angle ∠BYZ\angle BYZ∠BYZ.

Apply the Alternate Segment Theorem.

Part d· 2 steps∠BCY\angle BCY∠BCY

Identify the angle ∠BYX\angle BYX∠BYX.

Apply the Alternate Segment Theorem.

Part e· 2 steps58∘58^\circ58∘

From part a), we know that ∠AYZ=∠ACY\angle AYZ = \angle ACY∠AYZ=∠ACY.

Substitute the given value.

Part f· 3 steps68∘68^\circ68∘

From part d), we know that ∠BYX=∠BCY\angle BYX = \angle BCY∠BYX=∠BCY.

Angles on a straight line sum to 180∘180^\circ180∘.

Substitute the value of ∠BYX\angle BYX∠BYX and solve for ∠BYZ\angle BYZ∠BYZ.

Part g· 2 steps125∘125^\circ125∘

From part d), we know that ∠BYX=∠BCY\angle BYX = \angle BCY∠BYX=∠BCY.

Substitute the given value.

Part h· 2 steps100∘100^\circ100∘

From part c), we know that ∠BYZ=∠BAY\angle BYZ = \angle BAY∠BYZ=∠BAY.

Substitute the given value.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Identify the angle $\angle AYZ$ .

Part b· 2 steps
$\angle CAY \text{ and } \angle CBY$

Identify the angle $\angle CYX$ .

Part c· 2 steps
$\angle BAY$

Identify the angle $\angle BYZ$ .

Part d· 2 steps
$\angle BCY$

Identify the angle $\angle BYX$ .

Part e· 2 steps
$58^\circ$

From part a), we know that $\angle AYZ = \angle ACY$ .

Part f· 3 steps
$68^\circ$

From part d), we know that $\angle BYX = \angle BCY$ .

Angles on a straight line sum to $180^\circ$ .

Substitute the value of $\angle BYX$ and solve for $\angle BYZ$ .

Part g· 2 steps
$125^\circ$

From part d), we know that $\angle BYX = \angle BCY$ .

Part h· 2 steps
$100^\circ$

From part c), we know that $\angle BYZ = \angle BAY$ .