Question

5 find the value of x, y, and z.
6 as shown in the figure, a, b, and c are on ⊙o. if m∠oab = 46°, then m∠acb =

Explanation:

For problem 1:

Step1: Use inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. The arc with measure \(100^{\circ}\) has an inscribed - angle \(y\). So \(y=\frac{100^{\circ}}{2}=50^{\circ}\).

Step2: Consider triangle formed

In the triangle with angles \(y\), \(z\), and the angle subtended by the arc of \(50^{\circ}\). The angle subtended by the \(50^{\circ}\) arc at the circumference is \(x\). By the inscribed - angle theorem, \(x=\frac{50^{\circ}}{2}=25^{\circ}\).

Step3: Find \(z\)

In the triangle, since the sum of angles in a triangle is \(180^{\circ}\), and we know one angle is \(y = 50^{\circ}\) and another is \(x = 25^{\circ}\), and the third angle is \(z\). Also, using the property of angles in the circle - related triangle, \(z=x = 25^{\circ}\).

For problem 2:

Step1: Find the arc measure

The arc corresponding to the \(140^{\circ}\) central angle has measure \(140^{\circ}\). The remaining arc has measure \(360^{\circ}-140^{\circ}=220^{\circ}\).

Step2: Use inscribed - angle theorem for \(x\)

The inscribed angle \(x\) intercepts an arc. Let's consider the non - \(140^{\circ}\) arc. The angle \(x\) is half of the difference of the two arcs (using the property of angles formed by two chords in a circle). First, find the measure of the arc intercepted by \(x\). The arc intercepted by \(x\) has measure \(130^{\circ}\) (by angle - arc relationships in the circle). So \(x=\frac{130^{\circ}}{2}=65^{\circ}\).

Step3: Find \(y\)

The sum of angles in the triangle formed in the circle is \(180^{\circ}\). One angle is \(55^{\circ}\) and \(x = 65^{\circ}\), so \(y=180^{\circ}-(55^{\circ}+65^{\circ}) = 75^{\circ}\).

Step4: Find \(z\)

\(z = 55^{\circ}\) (vertically opposite angles or angle - arc relationships in the circle).

For problem 6:

Step1: In \(\triangle OAB\)

Since \(OA = OB\) (radii of the same circle), \(\triangle OAB\) is isosceles. So \(\angle OBA=\angle OAB = 46^{\circ}\).

Step2: Find \(\angle AOB\)

Using the angle - sum property of a triangle in \(\triangle OAB\), \(\angle AOB=180^{\circ}-2\times46^{\circ}=88^{\circ}\).

Step3: Use inscribed - central angle relationship

The measure of the inscribed angle \(\angle ACB\) is half of the measure of the central angle \(\angle AOB\) that subtends the same arc \(AB\). So \(m\angle ACB=\frac{88^{\circ}}{2}=44^{\circ}\).

Answer:

1. \(x = 25\), \(y = 50\), \(z = 25\)
2. \(x = 65\), \(y = 75\), \(z = 55\)
3. \(m\angle ACB=44^{\circ}\)