Geometry worksheet featuring angle calculation problems with diagrams.

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

Problem 1

Goal: Find angles A, B, and C.
* Angle A: Angles on a straight line add up to $180^\circ$. Angle A and the $32^\circ$ angle form a straight line.
$$180^\circ - 32^\circ = 148^\circ$$
So, A = 148°.
* Angle B: Angle B and Angle A are on a straight line (or you can see that B and the $32^\circ$ angle are vertically opposite). Vertically opposite angles are equal.
So, B = 32°.
* Angle C: Angle C and Angle B are on a straight line.
$$180^\circ - 32^\circ = 148^\circ$$
So, C = 148°.

Problem 2

Goal: Find angles A, B, and C.
* Angle A: Angle A and the $29^\circ$ angle are vertically opposite (they face each other across the intersection). Therefore, they are equal.
So, A = 29°.
* Angle B: Angle B and Angle A form a straight line ($180^\circ$).
$$180^\circ - 29^\circ = 151^\circ$$
So, B = 151°.
* Angle C: Angle C and Angle B form a straight line. Alternatively, Angle C is vertically opposite to the $29^\circ$ angle's partner? No, looking at the diagram, Angle C is adjacent to the $29^\circ$ angle on the bottom line. Wait, let's look closer.
The line goes straight through. The angle labeled $29^\circ$ and Angle C are on a straight line.
$$180^\circ - 29^\circ = 151^\circ$$
So, C = 151°.
*(Self-Check: Angle A and Angle C are vertically opposite? No. Angle A is top-left. The $29^\circ$ is bottom-left. Angle C is bottom-right. Angle B is top-right.
Actually, usually in these diagrams:
Top-Left: A
Bottom-Left: $29^\circ$
Bottom-Right: C
Top-Right: B

If $29^\circ$ and A are on a straight vertical line? No, it's an X shape.
Let's assume standard positions:
Angle opposite $29^\circ$ is B. So B = 29°.
Angle adjacent to $29^\circ$ on the straight line is A. So $A + 29 = 180 \rightarrow$ A = 151°.
Angle adjacent to $29^\circ$ on the other straight line is C. So $C + 29 = 180 \rightarrow$ C = 151°.

Let's re-read the diagram labels carefully.
Label A is in the top obtuse angle.
Label B is in the right acute angle.
Label C is in the bottom obtuse angle.
The given angle is $29^\circ$ (left acute angle).

Therefore:
Angle B is vertically opposite to $29^\circ$. So B = 29°.
Angle A is supplementary to $29^\circ$. So A = 151°.
Angle C is vertically opposite to A. So C = 151°.

Problem 3

Goal: Find angles A, B, and C.
* Angle A: Angle A and the $130^\circ$ angle are on a straight line.
$$180^\circ - 130^\circ = 50^\circ$$
So, A = 50°.
* Angle B: Look at the triangle formed in the middle. The angles inside a triangle always add up to $180^\circ$.
The angles we know are:
1. Angle A ($50^\circ$)
2. The given angle ($32^\circ$)
3. The third angle is vertically opposite to Angle B? No, Angle B is outside. Let's look at the vertex where A, B, and C meet.

Let's use the exterior angle rule or just straight lines.
Angle A ($50^\circ$) and the angle next to it inside the triangle are on a straight line with the $130^\circ$? No, A *is* the angle inside the triangle at that vertex?
Looking at the arc for $130^\circ$, it covers the obtuse angle. Angle A is the acute angle next to it. So the interior angle of the triangle at that corner is indeed $50^\circ$.

Now, look at Angle B. Angle B and the interior angle at that top vertex are on a straight line? No, B is vertically opposite to the interior angle?
Let's trace the lines.
Line 1 goes from bottom-left to top-right.
Line 2 goes from top-left to bottom-right.
Line 3 is horizontal.

Let's find the third angle of the triangle first.
Sum of angles in triangle = $180^\circ$.
Angle 1 = $50^\circ$ (calculated above).
Angle 2 = $32^\circ$ (given).
Angle 3 (top vertex inside triangle) = $180^\circ - 50^\circ - 32^\circ = 98^\circ$.

Now find A, B, C based on their positions relative to this triangle.
* Angle A: It is adjacent to the $130^\circ$ angle on a straight line. We already calculated this as 50°.
* Angle B: Angle B is vertically opposite to the interior angle we just found ($98^\circ$)? Or is B adjacent?
Looking at the diagram, B is the angle "above" the triangle vertex. The interior angle ($98^\circ$) and Angle B are on a straight line (the diagonal one).
So, $B + 98^\circ = 180^\circ$.
$$B = 180^\circ - 98^\circ = 82^\circ$$
* Angle C: Angle C is adjacent to Angle A on the horizontal straight line? No, C is below the horizontal line.
Let's look at the vertex on the left.
We have the $130^\circ$ angle.
Angle A is $50^\circ$.
Angle C is vertically opposite to the interior angle of the triangle? No.
Let's look at the intersection point on the left.
There is a straight diagonal line and a straight horizontal line.
The angle $130^\circ$ and Angle C are vertically opposite? No, they are adjacent on the diagonal line?
Actually, Angle C and the $130^\circ$ angle form a linear pair on the *diagonal* line? No, they share the horizontal ray.
So Angle C + $130^\circ$ = $180^\circ$ (angles on a straight horizontal line).
$$C = 180^\circ - 130^\circ = 50^\circ$$

Let's double check.
If C = 50, then C and A are vertically opposite. Yes, they look vertically opposite in the diagram.
If A = 50 and C = 50, that matches.

So:
A = 50°
C = 50°

Now for B again.
The angle inside the triangle at the top is $98^\circ$.
Angle B is adjacent to this $98^\circ$ angle on the straight diagonal line.
So $B = 180 - 98 = 82^\circ$.

Wait, let's look at label B again. Is B the whole angle or part of it?
Usually, B represents the angle marked by the arc.
Is there another way to find B?
Angle B and the angle inside the triangle ($98^\circ$) are supplementary.
So B = 82°.

Problem 4

Goal: Find angles A, B, and C.
This problem involves parallel lines (indicated by the arrows on the lines).
* Angle A: Angle A and the $72^\circ$ angle are corresponding angles? No, they are on the same side of the transversal but one is interior and one is exterior?
Let's look at the position.
The $72^\circ$ angle is bottom-right, between the parallel lines? No, it's below the bottom parallel line.
Angle A is top-left, above the top parallel line.

Let's use vertically opposite and alternate interior/exterior rules.
1. The angle vertically opposite to the $72^\circ$ angle is inside the parallel lines, top-left of that intersection. Let's call it $x$. So $x = 72^\circ$.
2. Angle A and angle $x$ are consecutive interior angles? No.

Let's try a simpler path:
The angle corresponding to the $72^\circ$ angle is at the top intersection, in the same position (bottom-right relative to the cross). Let's call this angle $y$. So $y = 72^\circ$.
Angle A and Angle $y$ are vertically opposite.
Therefore, A = 72°.

* Angle B: Angle B and Angle A are on a straight line.
$$180^\circ - 72^\circ = 108^\circ$$
So, B = 108°.

* Angle C: Angle C and Angle B are vertically opposite? No, they are adjacent on the straight transversal line?
Looking at the top intersection:
Top-Left: A ($72^\circ$)
Top-Right: B ($108^\circ$)
Bottom-Right: Corresponding to $72^\circ$ ($72^\circ$)
Bottom-Left: C

Angle C and Angle B are on a straight line (the horizontal parallel line).
$$180^\circ - 108^\circ = 72^\circ$$
Alternatively, Angle C and Angle A are on a straight line (the transversal).
$$180^\circ - 72^\circ = 108^\circ$$

Let's look closely at the placement of C.
C is in the bottom-left position of the top intersection.
A is in the top-left position.
A and C form a linear pair along the transversal line.
So $A + C = 180^\circ$.
$$72^\circ + C = 180^\circ$$
$$C = 108^\circ$$

Let's verify with alternate interior angles.
The interior angle on the bottom right is $72^\circ$ (vertically opposite to the exterior 72? No, the 72 is exterior bottom right. The interior bottom right is vertically opposite to the interior top left? No.)

Let's restart Problem 4 carefully.
Given: Parallel lines cut by a transversal.
Given Angle: $72^\circ$ at the bottom intersection, in the bottom-right position (exterior).

Find A:
Angle A is at the top intersection, top-left position (exterior).
Angle A and the given $72^\circ$ angle are alternate exterior angles.
Alternate exterior angles are equal when lines are parallel.
Therefore, A = 72°.

Find B:
Angle B is at the top intersection, top-right position.
Angle A and Angle B form a linear pair on the horizontal line.
$$A + B = 180^\circ$$
$$72^\circ + B = 180^\circ$$
$$B = 108^\circ$$

Find C:
Angle C is at the top intersection, bottom-left position.
Angle C and Angle B form a linear pair on the transversal line? No, they are vertically opposite to... wait.
Angle C and Angle A form a linear pair on the transversal line.
$$A + C = 180^\circ$$
$$72^\circ + C = 180^\circ$$
$$C = 108^\circ$$

Check: Angle C and Angle B are vertically opposite?
Top-Right is B. Bottom-Left is C. Yes, they are vertically opposite.
So $C = B = 108^\circ$. This is consistent.

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Final Answer:

1.
A = 148°
B = 32°
C = 148°

2.
A = 151°
B = 29°
C = 151°

3.
A = 50°
B = 82°
C = 50°

4.
A = 72°
B = 108°
C = 108°