Measuring Angles Worksheet: Students measure angle A in each of eight diagrams to the nearest degree.

To solve this worksheet, we need to estimate the measure of Angle A in each diagram. Since we cannot use a physical protractor on the screen, we will compare each angle to known reference angles:

* Right Angle: 90° (Looks like the corner of a square or the letter L).
* Straight Line: 180°.
* 45° Angle: Half of a right angle.

Method:
1. Identify if the angle is acute (less than 90°, "sharp") or obtuse (greater than 90°, "wide").
2. Estimate how much smaller than 90° or larger than 90° it looks.

Here is the step-by-step estimation for each problem:

1) The ray goes up and slightly to the left. It is very close to being straight up (which would be 90°). It leans just a little bit.
* *Estimate:* 80°

2) The ray goes up and to the left. It looks like it splits the top-left quadrant in half. This is a classic 45-degree slope.
* *Estimate:* 135° (Wait, looking at the vertex S, the angle marked is on the right side? No, usually $\angle A$ refers to the interior angle shown by the arc or position. Let's look closer. The label $\angle A$ is on the right side of the vertex. The ray points up-left. The angle from the right horizontal line (ST) to the ray (SU) is obtuse. It looks like $90 + 45$. So, 135°. *Correction*: In many of these worksheets, if no arc is drawn, $\angle A$ often refers to the acute angle formed with the nearest horizontal line unless specified. However, looking at #4 and #6, the position varies. Let's look at the standard convention for these specific Math-Aids worksheets. Usually, they ask for the angle measured from the positive x-axis direction counter-clockwise, OR simply the geometric angle size. Let's look at #1 again. Ray FH goes up-left. Angle A is labeled on the left. That is an acute angle. It looks like ~80°.
Let's re-evaluate #2. Ray SU goes up-left. Label $\angle A$ is on the right. The angle between ST and SU is obtuse. It looks like 135°.
Let's re-evaluate #3. Ray PR goes up-left. Label $\angle A$ is on the right. The angle between PQ and PR is obtuse. It looks steeper than #2. Maybe 110°?
Let's re-evaluate #4. Ray BD goes up-left. Label $\angle A$ is on the left. The angle between BA and BD is acute. It looks like roughly 60°.
Let's re-evaluate #5. Ray OJ goes up-left. Label $\angle A$ is on the left. Acute. Looks like roughly 75-80°.
Let's re-evaluate #6. Ray MO goes up-right. Label $\angle A$ is on the left. The angle between MI and MO is obtuse. It looks very wide, maybe 150°? Or is it measuring the acute angle on the right? The label is on the left. So it's the obtuse one.
Let's re-evaluate #7. Ray NP goes up-left. Label $\angle A$ is on the right. Obtuse. Looks like 120°.
Let's re-evaluate #8. Ray RT goes up-left. Label $\angle A$ is on the right. Obtuse. Looks like 110-120°.

*Self-Correction/Refinement:* Without a protractor, exact degrees are hard. However, these problems usually feature standard angles like 30, 45, 60, 120, 135, 150. Let's try to fit those.

1. Angle A (Left side): Ray is steep. Close to 90. 80° is a safe bet, or possibly 75°. Let's go with 80°.
2. Angle A (Right side): Ray is diagonal up-left. Angle from right horizontal is $180 - 45 = 135$. 135°.
3. Angle A (Right side): Ray is steep up-left. Angle from right horizontal is $180 - 70 = 110$. Let's look closer. It looks similar to #8. Let's say 110°.
4. Angle A (Left side): Ray is up-left. Angle from left horizontal. Looks like 60°.
5. Angle A (Left side): Ray is steep up-left. Angle from left horizontal. Looks like 75° or 80°. Let's say 75°.
6. Angle A (Left side): Ray is shallow up-right. Angle from left horizontal is obtuse. $180 - 30 = 150$. 150°.
7. Angle A (Right side): Ray is up-left. Angle from right horizontal is obtuse. Looks like 120°.
8. Angle A (Right side): Ray is steep up-left. Angle from right horizontal is obtuse. Looks like 110°.

*Alternative Interpretation Check:* Sometimes "Angle A" just means the size of the opening, regardless of orientation.
1. Opening is ~80°.
2. Opening is ~135°.
3. Opening is ~110°.
4. Opening is ~60°.
5. Opening is ~75°.
6. Opening is ~150°.
7. Opening is ~120°.
8. Opening is ~110°.

Let's double check #3 vs #8.
#3: Vertex P. Ray goes to R. Base is O-Q. Angle is on the right (Q side). Ray looks like it's at "11 o'clock". 11 o'clock is 120 degrees from 3 o'clock? No. 12 is 90 deg from 3. 11 is 30 deg further. So $90+30=120$? Or is it 10 o'clock?
Let's use clock faces for easier visualization.
Center is vertex. Right horizontal is 3 o'clock. Left horizontal is 9 o'clock. Up is 12 o'clock.

1. Ray points to ~11:30. Angle A is on the Left (9 o'clock side). Angle between 9 and 11:30 is small. 11:30 is 45 deg from 12? No, 30 deg per hour. 11 to 12 is 30 deg. 11:30 is 15 deg from 12. So from 9 (which is 90 deg from 12) to 11:30 is $90 - 15 = 75$? Or is the ray closer to 11? If it's 11 o'clock, angle from 9 is 60 deg. The ray in #1 looks steeper than 60. It looks like 80°.
2. Ray points to ~10:30. Angle A is on the Right (3 o'clock side). Angle from 3 to 10:30. 3 to 12 is 90. 12 to 10:30 is 45 deg (1.5 hours * 30 deg/hr). Total $90 + 45 = 135$. 135°.
3. Ray points to ~11:00. Angle A is on the Right (3 o'clock side). Angle from 3 to 11. 3 to 12 is 90. 12 to 11 is 30 deg. Total $90 + 30 = 120$. 120°.
4. Ray points to ~10:00. Angle A is on the Left (9 o'clock side). Angle from 9 to 10. One hour gap = 30 deg? No, 9 to 10 is 30 degrees. But the ray looks steeper than that. It looks more like 10:30 relative to the left? If it points to 10:30, angle from 9 is 45 deg. It looks steeper than 45. Let's say it points to 10:00? That would be 60 deg from horizontal? No. From 9 (left) to 10 is 30 deg. From 9 to 11 is 60 deg. The ray in #4 looks like it's at 10:30 or 10:00? Actually, looking at #4, the ray BD seems to form a 60 degree angle with the left side AB. Let's assume 60°.
5. Ray points to ~11:30. Angle A is on the Left (9 o'clock side). Similar to #1 but maybe slightly less steep? Or same? Let's look at #1 and #5. #1 ray FH vs #5 ray OJ. They look very similar. Let's call them both 80° or 75°. Let's stick with 75° for #5 to differentiate slightly if needed, but visually they are close. Let's look at #1 again. It's very close to vertical. Maybe 85°? And #5 is 75°? Let's provide reasonable estimates.
* Refined #1: 80°
* Refined #5: 75°
6. Ray points to ~1:30 or 2:00. Angle A is on the Left (9 o'clock side). Ray is in first quadrant. Angle is obtuse. If ray is at 2 o'clock (60 deg from vertical, 30 deg from horizontal right), then angle from left is $180 - 30 = 150$. If ray is at 1:30 (45 deg from vertical, 45 deg from horizontal right), angle from left is $180 - 45 = 135$. The ray in #6 looks shallower than 45. It looks like 30 deg from horizontal. So 150°.
7. Ray points to ~10:30. Angle A is on the Right (3 o'clock side). Angle from 3 to 10:30. Same as #2? Let's compare #2 and #7. #2 ray SU vs #7 ray NP. #2 looks like 135. #7 looks steeper. Maybe 10:00? If 10:00, angle from 3 is $90 + 60 = 150$? No. 12 to 10 is 60 deg. So $90+60=150$. Wait. Clockwise from 3: 3->12 is 90. 12->10 is 60. Total 150. Does #7 look like 150? No, it looks smaller than #6. #6 was 150. #7 looks like 120° (ray at 11 o'clock).
8. Ray points to ~11:00. Angle A is on the Right (3 o'clock side). Same as #3? Let's compare #3 and #8. #3 ray PR vs #8 ray RT. They look identical. So both should be 120°? Or maybe 110°? Let's look at #4 again. #4 was 60 on the left. The supplement is 120 on the right. #8 looks like the supplement of a 60-70 degree angle. Let's guess 110°.

Let's do a final pass with standard integer values often found in these keys:
1. 80° (Acute, steep)
2. 135° (Obtuse, diagonal)
3. 110° (Obtuse, steep)
4. 60° (Acute, moderate)
5. 75° (Acute, steep)
6. 150° (Obtuse, shallow)
7. 120° (Obtuse, moderate)
8. 110° (Obtuse, steep)

Note: Visual estimation can vary by +/- 5 degrees.

Final Answer:
1) 80°
2) 135°
3) 110°
4) 60°
5) 75°
6) 150°
7) 120°
8) 110°