Question

match each angle in column i with its reference angle in column ii
31° 40° 45° 30° 89° 60°
drag each reference angle above to the corresponding angle below. answers may be used once, more than once, or not at all.
i\tii
91°\t
220°\t
-135°\t
-30°\t
780°\t
510°\t

Explanation:

To solve for the reference angle of each given angle, we use the rules for reference angles based on the quadrant of the angle:

Recall:

• For an angle \(\theta\) in standard position:
• If \(0^\circ < \theta < 90^\circ\) (Quadrant I), reference angle \(= \theta\).
• If \(90^\circ < \theta < 180^\circ\) (Quadrant II), reference angle \(= 180^\circ - \theta\).
• If \(180^\circ < \theta < 270^\circ\) (Quadrant III), reference angle \(= \theta - 180^\circ\).
• If \(270^\circ < \theta < 360^\circ\) (Quadrant IV), reference angle \(= 360^\circ - \theta\).
• For negative angles or angles greater than \(360^\circ\), first find a coterminal angle (add or subtract \(360^\circ\) until the angle is between \(0^\circ\) and \(360^\circ\)), then find the reference angle.

Step 1: Find reference angle for \(91^\circ\)

• \(91^\circ\) is in Quadrant II (\(90^\circ < 91^\circ < 180^\circ\)).
• Reference angle \(= 180^\circ - 91^\circ = 89^\circ\).

Step 2: Find reference angle for \(220^\circ\)

• \(220^\circ\) is in Quadrant III (\(180^\circ < 220^\circ < 270^\circ\)).
• Reference angle \(= 220^\circ - 180^\circ = 40^\circ\).

Step 3: Find reference angle for \(-135^\circ\)

• First, find a coterminal angle: \(-135^\circ + 360^\circ = 225^\circ\) (Quadrant III).
• Reference angle \(= 225^\circ - 180^\circ = 45^\circ\).

Step 4: Find reference angle for \(-30^\circ\)

• Find a coterminal angle: \(-30^\circ + 360^\circ = 330^\circ\) (Quadrant IV).
• Reference angle \(= 360^\circ - 330^\circ = 30^\circ\).

Step 5: Find reference angle for \(780^\circ\)

• Find a coterminal angle: \(780^\circ - 2 \times 360^\circ = 780^\circ - 720^\circ = 60^\circ\) (Quadrant I).
• Reference angle \(= 60^\circ\) (since it's in Quadrant I).

Step 6: Find reference angle for \(510^\circ\)

• Find a coterminal angle: \(510^\circ - 360^\circ = 150^\circ\) (Quadrant II).
• Reference angle \(= 180^\circ - 150^\circ = 30^\circ\).

Matching:

• \(91^\circ\) → \(89^\circ\)
• \(220^\circ\) → \(40^\circ\)
• \(-135^\circ\) → \(45^\circ\)
• \(-30^\circ\) → \(30^\circ\)
• \(780^\circ\) → \(60^\circ\)
• \(510^\circ\) → \(30^\circ\)

Final Matches:

Column I	Column II (Reference Angle)
\(220^\circ\)	\(40^\circ\)
\(-135^\circ\)	\(45^\circ\)
\(-30^\circ\)	\(30^\circ\)
\(780^\circ\)	\(60^\circ\)
\(510^\circ\)	\(30^\circ\)

Answer:

To solve for the reference angle of each given angle, we use the rules for reference angles based on the quadrant of the angle:

Recall:

• For an angle \(\theta\) in standard position:
• If \(0^\circ < \theta < 90^\circ\) (Quadrant I), reference angle \(= \theta\).
• If \(90^\circ < \theta < 180^\circ\) (Quadrant II), reference angle \(= 180^\circ - \theta\).
• If \(180^\circ < \theta < 270^\circ\) (Quadrant III), reference angle \(= \theta - 180^\circ\).
• If \(270^\circ < \theta < 360^\circ\) (Quadrant IV), reference angle \(= 360^\circ - \theta\).
• For negative angles or angles greater than \(360^\circ\), first find a coterminal angle (add or subtract \(360^\circ\) until the angle is between \(0^\circ\) and \(360^\circ\)), then find the reference angle.

Step 1: Find reference angle for \(91^\circ\)

• \(91^\circ\) is in Quadrant II (\(90^\circ < 91^\circ < 180^\circ\)).
• Reference angle \(= 180^\circ - 91^\circ = 89^\circ\).

Step 2: Find reference angle for \(220^\circ\)

• \(220^\circ\) is in Quadrant III (\(180^\circ < 220^\circ < 270^\circ\)).
• Reference angle \(= 220^\circ - 180^\circ = 40^\circ\).

Step 3: Find reference angle for \(-135^\circ\)

• First, find a coterminal angle: \(-135^\circ + 360^\circ = 225^\circ\) (Quadrant III).
• Reference angle \(= 225^\circ - 180^\circ = 45^\circ\).

Step 4: Find reference angle for \(-30^\circ\)

• Find a coterminal angle: \(-30^\circ + 360^\circ = 330^\circ\) (Quadrant IV).
• Reference angle \(= 360^\circ - 330^\circ = 30^\circ\).

Step 5: Find reference angle for \(780^\circ\)

• Find a coterminal angle: \(780^\circ - 2 \times 360^\circ = 780^\circ - 720^\circ = 60^\circ\) (Quadrant I).
• Reference angle \(= 60^\circ\) (since it's in Quadrant I).

Step 6: Find reference angle for \(510^\circ\)

• Find a coterminal angle: \(510^\circ - 360^\circ = 150^\circ\) (Quadrant II).
• Reference angle \(= 180^\circ - 150^\circ = 30^\circ\).

Matching:

• \(91^\circ\) → \(89^\circ\)
• \(220^\circ\) → \(40^\circ\)
• \(-135^\circ\) → \(45^\circ\)
• \(-30^\circ\) → \(30^\circ\)
• \(780^\circ\) → \(60^\circ\)
• \(510^\circ\) → \(30^\circ\)

Final Matches:

Column I	Column II (Reference Angle)
\(220^\circ\)	\(40^\circ\)
\(-135^\circ\)	\(45^\circ\)
\(-30^\circ\)	\(30^\circ\)
\(780^\circ\)	\(60^\circ\)
\(510^\circ\)	\(30^\circ\)