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Problem 581

# To construct an angle whose measure is equal to the sum of the measures of two given angles.

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To construct an angle whose measure is equal to the sum of the measures of two given angles, follow these steps: 1) Construct the two given angles separately using a compass and straightedge. 2) Calculate the sum of their measures. 3) Use the compass and straightedge to construct the final angle, using the sum of the measures as the angle's measure. This involves drawing a straight line segment as the initial side, placing the compass on the vertex and drawing an arc, measuring the sum using a protractor, and marking the final angle's measure on the arc. Finally, draw a straight line segment from the vertex to the point marked on the arc as the terminal side of the angle.
See the step by step solution

## Step 1: Construct the first given angle

Using a compass and a straightedge, we can construct the first given angle as follows: 1. Draw a straight line segment (the initial side of the angle). 2. Place the compass on the vertex of the angle and draw an arc that intersects the initial side. 3. Measure the given angle using a protractor. 4. Use compass to mark the angle's measure on the arc (from initial side). 5. Draw another straight line segment (the terminal side of the angle) from the vertex to the point marked on the arc.

## Step 2: Construct the second given angle

Following the same instructions from Step 1, construct the second given angle: 1. Draw another straight line segment (the initial side of the second angle). 2. Place the compass on the vertex of the second angle and draw an arc that intersects the initial side. 3. Measure the second given angle using a protractor. 4. Use compass to mark the second angle's measure on the arc (from initial side). 5. Draw another straight line segment (the terminal side of the angle) from the vertex to the point marked on the arc.

## Step 3: Add the measures of the two angles

Calculate the sum of the two given angles. This will be the final measure of the angle that we need to construct.

## Step 4: Construct the angle with the sum of the measures

Using a compass and a straightedge, we can construct the angle with the sum of the measures as follows: 1. Draw a straight line segment (the initial side of the final angle). 2. Place the compass on the vertex of the final angle and draw an arc that intersects the initial side. 3. Measure the sum of the given angles using a protractor. 4. Use compass to mark the final angle's measure on the arc (from initial side). 5. Draw another straight line segment (the terminal side of the angle) from the vertex to the point marked on the arc. Now we have successfully constructed an angle whose measure is equal to the sum of the measures of the two given angles.

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