properties of angles
find the measure of each angle marked with α.
(there are multiple angle - related diagrams with angle measures and the angle α to be found, such as diagrams with right angles, straight angles, and other angle - related configurations. for example, some diagrams have a right angle at point a, some have a straight line forming a straight angle, and there are also angle - addition and subtraction - related diagrams to find the measure of angle α.)

Question

Accepted Answer

Let's solve the first problem (Problem 1) as an example:

# Explanation:
## Step1: Identify the angle type
We have a right angle at $ A $ (since $ \angle BAC $ is a right angle, $ 90^\circ $) and another angle of $ 35^\circ $ ( $ \angle CAD = 35^\circ $ ). We need to find $ \alpha=\angle DAB $.

## Step2: Use angle subtraction
Since $ \angle BAC = 90^\circ=\angle CAD+\angle DAB $, we can write the equation:
$ \alpha = 90^\circ - 35^\circ $

## Step3: Calculate the value
$ \alpha= 55^\circ $

# Answer:
$ 55^\circ $

For Problem 2:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $, with one angle $ 69^\circ $ ( $ \angle BAD = 69^\circ $ ). We need to find $ \alpha=\angle DAC $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle BAD+\angle DAC = 180^\circ $. Thus, $ \alpha=180^\circ - 69^\circ $

## Step3: Calculate the value
$ \alpha = 111^\circ $

# Answer:
$ 111^\circ $

For Problem 3:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 216^\circ $. We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
The sum of angles around a point is $ 360^\circ $, so $ \alpha + 216^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 216^\circ $

## Step3: Calculate the value
$ \alpha = 144^\circ $

# Answer:
$ 144^\circ $

For Problem 4:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 197^\circ $. We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
Sum of angles around a point is $ 360^\circ $, so $ \alpha + 197^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 197^\circ $

## Step3: Calculate the value
$ \alpha = 163^\circ $

# Answer:
$ 163^\circ $

For Problem 5:

# Explanation:
## Step1: Identify the angle relationship
We have a right angle ( $ 90^\circ $ ) at $ A $, with one angle $ 60^\circ $ ( $ \angle DAB = 60^\circ $ ). We need to find $ \alpha=\angle CAD $.

## Step2: Use angle subtraction
Since $ \angle BAC = 90^\circ=\angle CAD+\angle DAB $, we write:
$ \alpha = 90^\circ - 60^\circ $

## Step3: Calculate the value
$ \alpha= 30^\circ $

# Answer:
$ 30^\circ $

For Problem 6:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $, with one angle $ 147^\circ $ ( $ \angle DAB = 147^\circ $ ). We need to find $ \alpha=\angle CAD $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle DAB+\angle CAD = 180^\circ $. Thus, $ \alpha=180^\circ - 147^\circ $

## Step3: Calculate the value
$ \alpha = 33^\circ $

# Answer:
$ 33^\circ $

For Problem 7:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 36^\circ $ ( $ \angle CAD = 36^\circ $ ). We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
Sum of angles around a point is $ 360^\circ $, so $ \alpha + 36^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 36^\circ $

## Step3: Calculate the value
$ \alpha = 324^\circ $

# Answer:
$ 324^\circ $

For Problem 8:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $, with one angle $ 71^\circ $ ( $ \angle CAD = 71^\circ $ ). We need to find $ \alpha=\angle DAB $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle CAD+\angle DAB = 180^\circ $. Thus, $ \alpha=180^\circ - 71^\circ $

## Step3: Calculate the value
$ \alpha = 109^\circ $

# Answer:
$ 109^\circ $

For Problem 9:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $, with one angle $ 77^\circ $ ( $ \angle BAD = 77^\circ $ ). We need to find $ \alpha=\angle DAC $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle BAD+\angle DAC = 180^\circ $. Thus, $ \alpha=180^\circ - 77^\circ $

## Step3: Calculate the value
$ \alpha = 103^\circ $

# Answer:
$ 103^\circ $

For Problem 10:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 191^\circ $. We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
Sum of angles around a point is $ 360^\circ $, so $ \alpha + 191^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 191^\circ $

## Step3: Calculate the value
$ \alpha = 169^\circ $

# Answer:
$ 169^\circ $

For Problem 11:

# Explanation:
## Step1: Identify the angle relationship
We have a right angle ( $ 90^\circ $ ) at $ A $, with one angle $ 64^\circ $ ( $ \angle CAD = 64^\circ $ ). We need to find $ \alpha=\angle DAB $.

## Step2: Use angle subtraction
Since $ \angle BAC = 90^\circ=\angle CAD+\angle DAB $, we can write the equation:
$ \alpha = 90^\circ - 64^\circ $

## Step3: Calculate the value
$ \alpha= 26^\circ $

# Answer:
$ 26^\circ $

For Problem 12:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $, with one angle $ 117^\circ $ ( $ \angle CAD = 117^\circ $ ). We need to find $ \alpha=\angle DAB $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle CAD+\angle DAB = 180^\circ $. Thus, $ \alpha=180^\circ - 117^\circ $

## Step3: Calculate the value
$ \alpha = 63^\circ $

# Answer:
$ 63^\circ $

For Problem 13:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $ (since $ B $, $ A $, $ C $ are colinear), with one angle $ 53^\circ $ ( $ \angle BAD = 53^\circ $ ). We need to find $ \alpha=\angle DAC $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle BAD+\angle DAC = 180^\circ $. Thus, $ \alpha=180^\circ - 53^\circ $

## Step3: Calculate the value
$ \alpha = 127^\circ $

# Answer:
$ 127^\circ $

For Problem 14:

# Explanation:
## Step1: Identify the angle relationship
We have a right angle ( $ 90^\circ $ ) at $ A $, with one angle $ 22^\circ $ ( $ \angle BAD = 22^\circ $ ). We need to find $ \alpha=\angle DAC $.

## Step2: Use angle subtraction
Since $ \angle BAC = 90^\circ=\angle BAD+\angle DAC $, we can write the equation:
$ \alpha = 90^\circ - 22^\circ $

## Step3: Calculate the value
$ \alpha= 68^\circ $

# Answer:
$ 68^\circ $

For Problem 15:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 131^\circ $. We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
Sum of angles around a point is $ 360^\circ $, so $ \alpha + 131^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 131^\circ $

## Step3: Calculate the value
$ \alpha = 229^\circ $

# Answer:
$ 229^\circ $

For Problem 16:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 301^\circ $. We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
Sum of angles around a point is $ 360^\circ $, so $ \alpha + 301^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 301^\circ $

## Step3: Calculate the value
$ \alpha = 59^\circ $

# Answer:
$ 59^\circ $

For Problem 17:

# Explanation:
## Step1: Identify the angle relationship
We have a right angle ( $ 90^\circ $ ) at $ A $, with one angle $ 32^\circ $ ( $ \angle CAD = 32^\circ $ ). We need to find $ \alpha=\angle DAB $.

## Step2: Use angle subtraction
Since $ \angle BAC = 90^\circ=\angle CAD+\angle DAB $, we can write the equation:
$ \alpha = 90^\circ - 32^\circ $

## Step3: Calculate the value
$ \alpha= 58^\circ $

# Answer:
$ 58^\circ $

For Problem 18:

# Explanation:
## Step1: Identify the angle relationship
We have a full circle ( $ 360^\circ $ ) around point $ A $, with one angle $ 178^\circ $. We need to find $ \alpha $ (the remaining angle).

## Step2: Use full circle property
Sum of angles around a point is $ 360^\circ $, so $ \alpha + 178^\circ= 360^\circ $. Thus, $ \alpha=360^\circ - 178^\circ $

## Step3: Calculate the value
$ \alpha = 182^\circ $

# Answer:
$ 182^\circ $

For Problem 19:

# Explanation:
## Step1: Identify the angle relationship
We have a right angle ( $ 90^\circ $ ) at $ A $, with one angle $ 74^\circ $ ( $ \angle CAD = 74^\circ $ ). We need to find $ x=\angle DAB $ (note: the problem uses $ x $ here, but it's similar to $ \alpha $ in other problems).

## Step2: Use angle subtraction
Since $ \angle BAC = 90^\circ=\angle CAD+\angle DAB $, we can write the equation:
$ x = 90^\circ - 74^\circ $

## Step3: Calculate the value
$ x= 16^\circ $

# Answer:
$ 16^\circ $

For Problem 20:

# Explanation:
## Step1: Identify the angle relationship
We have a straight angle ( $ 180^\circ $ ) at $ A $, with one angle $ 159^\circ $ ( $ \angle CAD = 159^\circ $ ). We need to find $ \alpha=\angle DAB $.

## Step2: Use straight angle property
A straight angle is $ 180^\circ $, so $ \angle CAD+\angle DAB = 180^\circ $. Thus, $ \alpha=180^\circ - 159^\circ $

## Step3: Calculate the value
$ \alpha = 21^\circ $

# Answer:
$ 21^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Identify the angle type

Step2: Use angle subtraction

Step3: Calculate the value

Step1: Identify the angle relationship

Step2: Use straight angle property

Step3: Calculate the value

Step1: Identify the angle relationship

Step2: Use full circle property

Step3: Calculate the value

Answer: