12) x = 100°, y = 80° and z = 160°
13) x = 65°
Solution:
Question 12:
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half of the intercepted arc.
[tex]$\Rightarrow x =\frac{1}{2}\times 200^\circ[/tex]
⇒ x = 100°
Sum of the adjacent angles in a straight line is 180°.
x + y = 180°
100° + y = 180°
Subtract 100° from both sides, we get
y = 80°
By the above mentioned theorem,
[tex]$\Rightarrow y =\frac{1}{2}\times z[/tex]
[tex]$\Rightarrow 80^\circ =\frac{1}{2}\times z[/tex]
Multiply by 2 on both sides, we get
⇒ 160° = z
Therefore, x = 100°, y = 80° and z = 160°.
Question 13:
If two chords intersects in the interior of the circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
[tex]$\Rightarrow \angle x =\frac{1}{2} (100^\circ + 30^\circ )[/tex]
[tex]$\Rightarrow \angle x =\frac{1}{2} \times 130^\circ[/tex]
⇒ ∠x = 65°
Therefore x = 65°.
