Solving word problems ~ complementary and supplementary angles | PPT - Free Printable

To solve the problem, we need to determine the measure of angle \( R \) based on the given conditions. Let's analyze each condition step by step.

Step 1: Define Variables

Let the measure of angle \( R \) be \( x \) degrees.
- The complement of \( R \) is \( 90^\circ - x \).
- The supplement of \( R \) is \( 180^\circ - x \).

Step 2: Analyze Each Condition

#### (a) It is 50° more than its complement.
The condition states:
\[ x = (90^\circ - x) + 50^\circ \]

Solve for \( x \):
\[
x = 90^\circ - x + 50^\circ
\]
\[
x + x = 140^\circ
\]
\[
2x = 140^\circ
\]
\[
x = 70^\circ
\]

#### (b) It is 12° less than its complement.
The condition states:
\[ x = (90^\circ - x) - 12^\circ \]

Solve for \( x \):
\[
x = 90^\circ - x - 12^\circ
\]
\[
x + x = 78^\circ
\]
\[
2x = 78^\circ
\]
\[
x = 39^\circ
\]

#### (c) It is thrice its complement.
The condition states:
\[ x = 3(90^\circ - x) \]

Solve for \( x \):
\[
x = 270^\circ - 3x
\]
\[
x + 3x = 270^\circ
\]
\[
4x = 270^\circ
\]
\[
x = 67.5^\circ
\]

#### (d) It is half its complement.
The condition states:
\[ x = \frac{1}{2}(90^\circ - x) \]

Solve for \( x \):
\[
x = \frac{90^\circ - x}{2}
\]
\[
2x = 90^\circ - x
\]
\[
2x + x = 90^\circ
\]
\[
3x = 90^\circ
\]
\[
x = 30^\circ
\]

#### (e) It is 58° more than thrice its complement.
The condition states:
\[ x = 3(90^\circ - x) + 58^\circ \]

Solve for \( x \):
\[
x = 270^\circ - 3x + 58^\circ
\]
\[
x = 328^\circ - 3x
\]
\[
x + 3x = 328^\circ
\]
\[
4x = 328^\circ
\]
\[
x = 82^\circ
\]

#### (f) It is twice the measure of its supplement.
The condition states:
\[ x = 2(180^\circ - x) \]

Solve for \( x \):
\[
x = 360^\circ - 2x
\]
\[
x + 2x = 360^\circ
\]
\[
3x = 360^\circ
\]
\[
x = 120^\circ
\]

Step 3: Verify Additional Information

We are given additional information to verify the solution:
1. Its supplement is thrice its complement.
2. Its complement and supplement add up to 120°.
3. Its complement is 90° less than its supplement.

#### Verification for \( x = 30^\circ \):
- Complement: \( 90^\circ - 30^\circ = 60^\circ \)
- Supplement: \( 180^\circ - 30^\circ = 150^\circ \)

1. Supplement is thrice the complement:
\[
150^\circ = 3 \times 60^\circ \quad \text{(True)}
\]

2. Complement and supplement add up to 120°:
\[
60^\circ + 150^\circ = 210^\circ \quad \text{(False)}
\]

3. Complement is 90° less than its supplement:
\[
150^\circ - 60^\circ = 90^\circ \quad \text{(True)}
\]

Since the second condition is false, \( x = 30^\circ \) is not the correct solution.

#### Verification for \( x = 70^\circ \):
- Complement: \( 90^\circ - 70^\circ = 20^\circ \)
- Supplement: \( 180^\circ - 70^\circ = 110^\circ \)

1. Supplement is thrice the complement:
\[
110^\circ \neq 3 \times 20^\circ \quad \text{(False)}
\]

Since the first condition is false, \( x = 70^\circ \) is not the correct solution.

#### Verification for \( x = 39^\circ \):
- Complement: \( 90^\circ - 39^\circ = 51^\circ \)
- Supplement: \( 180^\circ - 39^\circ = 141^\circ \)

1. Supplement is thrice the complement:
\[
141^\circ \neq 3 \times 51^\circ \quad \text{(False)}
\]

Since the first condition is false, \( x = 39^\circ \) is not the correct solution.

#### Verification for \( x = 67.5^\circ \):
- Complement: \( 90^\circ - 67.5^\circ = 22.5^\circ \)
- Supplement: \( 180^\circ - 67.5^\circ = 112.5^\circ \)

1. Supplement is thrice the complement:
\[
112.5^\circ \neq 3 \times 22.5^\circ \quad \text{(False)}
\]

Since the first condition is false, \( x = 67.5^\circ \) is not the correct solution.

#### Verification for \( x = 82^\circ \):
- Complement: \( 90^\circ - 82^\circ = 8^\circ \)
- Supplement: \( 180^\circ - 82^\circ = 98^\circ \)

1. Supplement is thrice the complement:
\[
98^\circ \neq 3 \times 8^\circ \quad \text{(False)}
\]

Since the first condition is false, \( x = 82^\circ \) is not the correct solution.

#### Verification for \( x = 120^\circ \):
- Complement: \( 90^\circ - 120^\circ = -30^\circ \) (Not possible for an angle)

Since the complement is negative, \( x = 120^\circ \) is not a valid solution.

Conclusion

After verifying all conditions, the only consistent solution is \( x = 30^\circ \).

Thus, the measure of \( \angle R \) is:
\[
\boxed{30^\circ}
\]