Question

1. complete the following table.

angle calculations

angle	complement	supplement	resulting angle measure after the angle is bisected
	12°
15°
		132°
90°
			34°
			68°
	49°
		100°
			127°

Explanation:

To solve the table, we use the following concepts:
- **Complement of an angle**: If two angles are complementary, their sum is \(90^\circ\). So, complement of an angle \(\theta\) is \(90^\circ - \theta\).
- **Supplement of an angle**: If two angles are supplementary, their sum is \(180^\circ\). So, supplement of an angle \(\theta\) is \(180^\circ - \theta\).
- **Resulting angle after bisecting**: When an angle \(\theta\) is bisected, the resulting angle is \(\frac{\theta}{2}\).

Let's solve each row one by one:

Row 1: Angle = \(73^\circ\)

Step 1: Find Complement

Complement = \(90^\circ - 73^\circ = 17^\circ\)

Step 2: Find Supplement

Supplement = \(180^\circ - 73^\circ = 107^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{73^\circ}{2} = 36.5^\circ\)

Row 2: Complement = \(12^\circ\)

Step 1: Find Angle

Angle = \(90^\circ - 12^\circ = 78^\circ\)

Step 2: Find Supplement

Supplement = \(180^\circ - 78^\circ = 102^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{78^\circ}{2} = 39^\circ\)

Row 3: Angle = \(15^\circ\)

Step 1: Find Complement

Complement = \(90^\circ - 15^\circ = 75^\circ\)

Step 2: Find Supplement

Supplement = \(180^\circ - 15^\circ = 165^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{15^\circ}{2} = 7.5^\circ\)

Row 4: Supplement = \(132^\circ\)

Step 1: Find Angle

Angle = \(180^\circ - 132^\circ = 48^\circ\)

Step 2: Find Complement

Complement = \(90^\circ - 48^\circ = 42^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{48^\circ}{2} = 24^\circ\)

Row 5: Angle = \(90^\circ\)

Step 1: Find Complement

Complement = \(90^\circ - 90^\circ = 0^\circ\)

Step 2: Find Supplement

Supplement = \(180^\circ - 90^\circ = 90^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{90^\circ}{2} = 45^\circ\)

Row 6: Resulting angle (bisected) = \(34^\circ\)

Step 1: Find Angle

Since bisected angle is \(\frac{\theta}{2} = 34^\circ\), so Angle = \(34^\circ \times 2 = 68^\circ\)

Step 2: Find Complement

Complement = \(90^\circ - 68^\circ = 22^\circ\)

Step 3: Find Supplement

Supplement = \(180^\circ - 68^\circ = 112^\circ\)

Row 7: Complement = \(49^\circ\)

Step 1: Find Angle

Angle = \(90^\circ - 49^\circ = 41^\circ\)

Step 2: Find Supplement

Supplement = \(180^\circ - 41^\circ = 139^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{41^\circ}{2} = 20.5^\circ\)

Row 8: Resulting angle (bisected) = \(68^\circ\)

Step 1: Find Angle

Since bisected angle is \(\frac{\theta}{2} = 68^\circ\), so Angle = \(68^\circ \times 2 = 136^\circ\)

Step 2: Find Complement

Complement = \(90^\circ - 136^\circ\) (But \(136^\circ > 90^\circ\), so no complement)

Step 3: Find Supplement

Supplement = \(180^\circ - 136^\circ = 44^\circ\)

Row 9: Supplement = \(100^\circ\)

Step 1: Find Angle

Angle = \(180^\circ - 100^\circ = 80^\circ\)

Step 2: Find Complement

Complement = \(90^\circ - 80^\circ = 10^\circ\)

Step 3: Find Resulting angle (bisected)

Resulting angle = \(\frac{80^\circ}{2} = 40^\circ\)

Row 10: Resulting angle (bisected) = \(127^\circ\)

Step 1: Find Angle

Since bisected angle is \(\frac{\theta}{2} = 127^\circ\), so Angle = \(127^\circ \times 2 = 254^\circ\) (But angles in standard position are less than \(180^\circ\) for complement and supplement, but mathematically it's \(254^\circ\))

Step 2: Find Complement

Complement = \(90^\circ - 254^\circ\) (Negative, so no complement)

Step 3: Find Supplement

Supplement = \(180^\circ - 254^\circ = -74^\circ\) (Negative, which is not a standard angle, but mathematically it's \(-74^\circ\))

Now, let's tabulate the results:

| Angle | Complement | Supplement | Resulting angle (bisected) |
|-------|------------|------------|----------------------------|
| \(73^\circ\) | \(17^\circ\) | \(107^\circ\) | \(36.5^\circ\) |
| \(78^\circ\) | \(12^\circ\) | \(102^\circ\) | \(39^\circ\) |
| \(15^\circ\) | \(75^\circ\) | \(165^\circ\) | \(7.5^\circ\) |
| \(48^\circ\) | \(42^\circ\) | \(132^\circ\) | \(24^\circ\) |
| \(90^\circ\) | \(0^\circ\) | \(90^\circ\) | \(45^\circ\) |
| \(68^\circ\) | \(22^\circ\) | \(112^\circ\) | \(34^\circ\) |
| \(41^\circ\) | \(49^\circ\) | \(139^\circ\) | \(20.5^\circ\) |
| \(136^\circ\) | - | \(44^\circ\) | \(68^\circ\) |
| \(80^\circ\) | \(10^\circ\) | \(100^\circ\) | \(40^\circ\) |
| \(254^\circ\) | - | \(-74^\circ\) | \(127^\circ\) |

(Note: For angles greater than \(90^\circ\), the complement is not defined in the context of positive acute angles, and for angles greater than \(180^\circ\), the supplement may be negative, but we have calculated them mathematically.)

Answer:

The completed table is as follows:

Angle	Complement	Supplement	Resulting angle (bisected)
\(78^\circ\)	\(12^\circ\)	\(102^\circ\)	\(39^\circ\)
\(15^\circ\)	\(75^\circ\)	\(165^\circ\)	\(7.5^\circ\)
\(48^\circ\)	\(42^\circ\)	\(132^\circ\)	\(24^\circ\)
\(90^\circ\)	\(0^\circ\)	\(90^\circ\)	\(45^\circ\)
\(68^\circ\)	\(22^\circ\)	\(112^\circ\)	\(34^\circ\)
\(41^\circ\)	\(49^\circ\)	\(139^\circ\)	\(20.5^\circ\)
\(136^\circ\)	-	\(44^\circ\)	\(68^\circ\)
\(80^\circ\)	\(10^\circ\)	\(100^\circ\)	\(40^\circ\)
\(254^\circ\)	-	\(-74^\circ\)	\(127^\circ\)