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The questions you provided are about constructing angles based on given angles or combinations of them. Here’s a breakdown and how each question can be approached:

### 27. Construct an angle:
   - a. Congruent to ∠1
   - b. Congruent to ∠2
   - c. Congruent to ∠3
   
   **Approach**: Use a compass and straightedge method to construct angles congruent to the given angles by copying the angles exactly. For each part:
   - To construct an angle congruent to ∠1, ∠2, or ∠3, use the following steps:
     1. Place the compass at the vertex of the given angle and draw an arc intersecting both sides of the angle.
     2. Without adjusting the compass, place the compass point on a new vertex and draw an arc that will serve as the base of the new angle.
     3. Measure the distance between the intersections of the arc and the original angle’s sides, then transfer that distance to the new arc.
     4. Draw the angle using a straightedge to connect the vertex to the point where the arc meets the transferred length.

### 28. Construct an angle which is twice the measure of ∠2.
   **Approach**: 
   - Use an angle bisector technique to divide ∠2 into two equal parts and then extend the second half to get an angle that is twice the size of ∠2.

### 29. Construct an angle which is three times the measure of ∠1.
   **Approach**: 
   - First, copy ∠1 three times in succession, either by using the compass technique or by drawing ∠1 and then bisecting each new angle twice to achieve a total measure that is three times ∠1.

### 30. Construct an angle which is equal to the sum of the measures of ∠1 and ∠2. Name your angle ∠LED.
   **Approach**:
   - Copy ∠1 and ∠2 adjacent to each other, sharing a common vertex, and the resulting angle will be the sum of both. You can name it ∠LED as instructed.

### 31. Construct an angle which is equal to the sum of the measure of ∠1 and half the measure of ∠3.
   **Approach**:
   - First, bisect ∠3 using a compass and straightedge to get half of its measure. Then, construct an angle equal to the sum of ∠1 and the half of ∠3 by placing them adjacent to each other.

These problems are primarily geometric constructions involving using a compass and straightedge to recreate and combine angles. Let me know if you need further clarifications on any particular construction!

### Expansion Questions:
1. What is the geometric procedure to bisect an angle using only a compass and straightedge?
2. How would you verify that an angle you constructed is congruent to a given angle?
3. Can you describe the steps to add two angles together using only geometric methods?
4. How do you construct an angle that is exactly one-third of a given angle?
5. How does the construction of an angle that is double another angle differ from constructing an angle congruent to a given one?

**Tip**: When constructing angles, it's essential to keep your compass at a consistent length between steps to ensure accuracy.

The user provided an image with angle construction tasks such as constructing congruent angles, multiplying angles, and adding angle measures. The tasks involve compass and straightedge constructions.

Learn how to construct angles congruent to given angles, multiply angle measures, and add angle measures using a compass and straightedge. This task involves the Angle Bisector Theorem and the Angle Addition Postulate, suitable for students in Grades 8-10.

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