P is a point on line AB and PQ is a ray such that ∠ QPA = 7x and ∠ QPB = 5x. Then, the value of (8x – 10°) is :

Q: P is a point on line AB and PQ is a ray such that ∠ QPA = 7x and ∠ QPB = 5x. Then, the value of (8x – 10°) is :

To solve the problem, we need to determine the value of $8x - 10^\circ$ given the angles $\angle QPA = 7x$ and $\angle QPB = 5x$. ### Explanation: 1. **Understanding the Geometry:** - Since $P$ is a point on line $AB$ and $PQ$ is a ray, the angles $\angle QPA$ and $\angle QPB$ are formed on a straight line. - The sum of angles on a straight line is $180^\circ$. 2. **Setting Up the Equation:** - According to the problem, $\angle QPA = 7x$ and $\angle QPB = 5x$. - Therefore, the equation based on the straight line is: \[ \angle QPA + \angle QPB = 180^\circ \] \[ 7x + 5x = 180^\circ \] \[ 12x = 180^\circ \] 3. **Solving for $x$:** - Divide both sides by 12 to find $x$: \[ x = \frac{180^\circ}{12} = 15^\circ \] 4. **Calculating $8x - 10^\circ$:** - Substitute $x = 15^\circ$ into the expression $8x - 10^\circ$: \[ 8x - 10^\circ = 8(15^\circ) - 10^\circ \] \[ = 120^\circ - 10^\circ \] \[ = 110^\circ \] ### Conclusion: - The value of $8x - 10^\circ$ is $110^\circ$, which corresponds to Option 2. ### Justification for Other Options: - **Option 1 (118°):** This value is incorrect because substituting $x = 15^\circ$ into $8x - 10^\circ$ yields $110^\circ$, not $118^\circ$. - **Option 3 (140°):** This value is incorrect for the same reason; the calculation clearly shows $110^\circ$. - **Option 4 (102°):** This value is also incorrect as the correct calculation results in $110^\circ$. Therefore, Option 2 is the correct answer, as it accurately reflects the calculated value of $8x - 10^\circ$.

Question

Accepted Answer

To solve the problem, we need to determine the value of $8x - 10^\circ$ given the angles $\angle QPA = 7x$ and $\angle QPB = 5x$.

### Explanation:

1. **Understanding the Geometry:**
   - Since $P$ is a point on line $AB$ and $PQ$ is a ray, the angles $\angle QPA$ and $\angle QPB$ are formed on a straight line.
   - The sum of angles on a straight line is $180^\circ$.

2. **Setting Up the Equation:**
   - According to the problem, $\angle QPA = 7x$ and $\angle QPB = 5x$.
   - Therefore, the equation based on the straight line is:
     $$
     \angle QPA + \angle QPB = 180^\circ
     $$
     $$
     7x + 5x = 180^\circ
     $$
     $$
     12x = 180^\circ
     $$

3. **Solving for $x$:**
   - Divide both sides by 12 to find $x$:
     $$
     x = \frac{180^\circ}{12} = 15^\circ
     $$

4. **Calculating $8x - 10^\circ$:**
   - Substitute $x = 15^\circ$ into the expression $8x - 10^\circ$:
     $$
     8x - 10^\circ = 8(15^\circ) - 10^\circ
     $$
     $$
     = 120^\circ - 10^\circ
     $$
     $$
     = 110^\circ
     $$

### Conclusion:

- The value of $8x - 10^\circ$ is $110^\circ$, which corresponds to Option 2.

### Justification for Other Options:

- **Option 1 (118°):** This value is incorrect because substituting $x = 15^\circ$ into $8x - 10^\circ$ yields $110^\circ$, not $118^\circ$.
- **Option 3 (140°):** This value is incorrect for the same reason; the calculation clearly shows $110^\circ$.
- **Option 4 (102°):** This value is also incorrect as the correct calculation results in $110^\circ$.

Therefore, Option 2 is the correct answer, as it accurately reflects the calculated value of $8x - 10^\circ$.