Circle Theorems worksheet with 12 problems requiring calculation of missing angles in circles.

Here are the step-by-step solutions for each problem on the worksheet.

Section A: Calculate the missing angles

1) Angle a
* Rule: The angle at the center is double the angle at the circumference subtended by the same arc.
* Calculation: The center angle is $118^\circ$. So, $a = 118^\circ \div 2$.
* Result: $a = 59^\circ$

2) Angles b and c
* Rule: Angles in the same segment (subtended by the same arc) are equal.
* Calculation for b: Angle $b$ and the $47^\circ$ angle both look at the same bottom-left arc. So, $b = 47^\circ$.
* Calculation for c: Angle $c$ and the $62^\circ$ angle both look at the same top-right arc. So, $c = 62^\circ$.

3) Angles d, e, and f
* Rule: Angles in the same segment are equal. Also, angles in a triangle add up to $180^\circ$.
* Calculation for d: Angle $d$ and the $12^\circ$ angle subtend the same small top arc. So, $d = 12^\circ$.
* Calculation for e: Angle $e$ and the $24^\circ$ angle subtend the same bottom-left arc. So, $e = 24^\circ$.
* Calculation for f: Look at the triangle containing angles $d$, $e$, and $f$. We know $d=12^\circ$ and $e=24^\circ$.
* $f = 180^\circ - 12^\circ - 24^\circ = 144^\circ$.

4) Angle g
* Rule: The angle at the center is double the angle at the circumference.
* Calculation: The angle at the circumference is $49^\circ$. So, $g = 49^\circ \times 2$.
* Result: $g = 98^\circ$

5) Angles h and i
* Rule: Angle at center is double angle at circumference. Angles in same segment are equal.
* Calculation for h: Angle $h$ is at the center, and $43^\circ$ is at the circumference on the same arc. So, $h = 43^\circ \times 2 = 86^\circ$.
* Calculation for i: Angle $i$ and the $43^\circ$ angle are in the same segment (both subtend the left-side chord). So, $i = 43^\circ$.

6) Angle j
* Rule: Angle at circumference is half the angle at the center.
* Calculation: The center angle is $116^\circ$. So, $j = 116^\circ \div 2$.
* Result: $j = 58^\circ$

7) Angles k, l, and m
* Rule: Vertically opposite angles are equal. Angles in same segment are equal. Angles in a triangle sum to $180^\circ$.
* Calculation for l: Angle $l$ and the $81^\circ$ angle are vertically opposite. So, $l = 81^\circ$.
* Calculation for k: Look at the top triangle with angles $67^\circ$, $k$, and the angle vertically opposite to $81^\circ$ (which is also $81^\circ$). Wait, let's use the "same segment" rule instead, it's simpler. Angle $k$ and the angle labeled $67^\circ$ are NOT in the same segment. Let's look at the triangle formed by the chord on the left. The angles are $67^\circ$, $l$ ($81^\circ$), and the third angle. Let's find the third angle of that left triangle first: $180 - 67 - 81 = 32^\circ$. This angle ($32^\circ$) and angle $m$ are in the same segment? No.
* Let's restart Problem 7 carefully.
* Angle l: Vertically opposite to $81^\circ$. So $l = 81^\circ$.
* Angle k: Consider the triangle with angles $67^\circ$, $k$, and the angle vertically opposite to the intersection. Actually, easier method: Angles subtended by the same arc are equal. The angle marked $67^\circ$ and angle $m$ subtend the same arc (the right-hand arc). So $m = 67^\circ$.
* Now look at the triangle containing $k$, $m$, and the intersection angle. The intersection angle is vertically opposite to $81^\circ$, so it is $81^\circ$.
* Sum of angles in that triangle: $k + m + 81^\circ = 180^\circ$.
* $k + 67^\circ + 81^\circ = 180^\circ$.
* $k + 148^\circ = 180^\circ$.
* $k = 32^\circ$.

8) Angle n
* Rule: Reflex angle at center is double the obtuse angle at circumference? Or simpler: Use the triangle formed by radii.
* Let's use the property that the angle at the center ($142^\circ$) and the angle at the circumference on the *major* arc are related. But $n$ is on the minor arc side? No, $n$ is part of a triangle with the center.
* Let's look at the triangle formed by the two radii and the chord. It is an isosceles triangle.
* The angle at the center inside this triangle is $142^\circ$.
* The other two angles are equal (base angles of isosceles triangle). Let them be $x$.
* $2x + 142^\circ = 180^\circ \rightarrow 2x = 38^\circ \rightarrow x = 19^\circ$.
* Angle $n$ is one of these base angles.
* Result: $n = 19^\circ$

9) Angle p
* Rule: Angles around a point sum to $360^\circ$. Angle at center is double angle at circumference.
* First, find the interior center angle: $360^\circ - 232^\circ = 128^\circ$.
* Angle $p$ is at the circumference subtended by this $128^\circ$ arc.
* $p = 128^\circ \div 2$.
* Result: $p = 64^\circ$

10) Angle q
* Rule: Angle at circumference is half angle at center.
* The reflex angle at the center corresponding to arc $q$ is $360^\circ - 76^\circ = 284^\circ$.
* Wait, angle $q$ subtends the major arc. The angle at the center for the major arc is $360 - 76 = 284$.
* So $q = 284 \div 2 = 142^\circ$.
* *Alternative Check:* The quadrilateral formed by the two radii and the two chords to $q$ has angles: Center ($76^\circ$), two $90^\circ$? No, not tangents.
* Let's stick to the basic theorem. Angle $q$ faces the major arc. The central angle of the major arc is $360^\circ - 76^\circ = 284^\circ$.
* Therefore, $q = 284^\circ / 2 = 142^\circ$.

11) Angle r
* Rule: Opposite angles in a cyclic quadrilateral sum to $180^\circ$.
* The quadrilateral is formed by the two radii and the two chords? No, the shape shown is a quadrilateral inscribed in the circle? No, one vertex is the center.
* Let's look at the shape. It's a quadrilateral with vertices: Center, Point on left, Point on right, Point top-right ($r$). This is not a standard cyclic quad because one vertex is the center.
* Let's use the "Arrowhead" or reflex angle rule.
* Reflex angle at center = $360^\circ - 122^\circ = 238^\circ$.
* Angle $r$ is at the circumference facing this reflex angle.
* $r = 238^\circ \div 2 = 119^\circ$.

12) Angles s and t
* Rule: Angles in same segment are equal. Exterior angle of a triangle equals sum of interior opposite angles.
* Angle s: Angle $s$ and the $44^\circ$ angle are in the same segment (subtending the top-left arc). So, $s = 44^\circ$.
* Angle t: Look at the triangle formed by the intersecting chords on the right side. The angles are $25^\circ$, $s$ ($44^\circ$), and the third angle.
* Actually, $t$ is an exterior angle to the triangle containing the $25^\circ$ angle and the angle vertically opposite to the one adjacent to $44^\circ$.
* Let's use the simplest triangle: The triangle with vertices at the circle edge (left), circle edge (bottom), and the intersection point.
* The angles inside this small triangle are $25^\circ$ and $s$ ($44^\circ$).
* Angle $t$ is the exterior angle to this triangle? No, $t$ is inside the top triangle.
* Let's look at the triangle containing $t$. Its vertices are the intersection point, top point, and right point.
* We need the angle at the right point. That angle is in the same segment as the $25^\circ$ angle. So the angle at the right point is $25^\circ$.
* We need the angle at the top point. That angle is in the same segment as... we don't have it directly.
* Let's try another way. Angle $t$ is vertically opposite to the angle in the bottom triangle.
* Bottom triangle angles: $25^\circ$ and $s=44^\circ$.
* Third angle in bottom triangle = $180 - 25 - 44 = 111^\circ$.
* Angle $t$ and this $111^\circ$ angle are on a straight line? No, they are vertically opposite to the top/bottom triangles?
* Let's look at the diagram again. $t$ is the angle between the chords at the top intersection.
* The angle vertically opposite to $t$ is inside the bottom triangle.
* The sum of angles in the bottom triangle is $180^\circ$. The known angles are $25^\circ$ and $s$ ($44^\circ$).
* So the third angle (vertically opposite to $t$) is $180 - 25 - 44 = 111^\circ$.
* Therefore, $t = 111^\circ$.

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Final Answer:
a) 59°
b) 47°
c) 62°
d) 12°
e) 24°
f) 144°
g) 98°
h) 86°
i) 43°
j) 58°
k) 32°
l) 81°
m) 67°
n) 19°
p) 64°
q) 142°
r) 119°
s) 44°
t) 111°