if angle CDE is a straight angle, ray DE bisects angle GDH, measure angle GDE =(8x-1), measure angle EDH =(6x+15), and measure angle CDF =43, find each measure
The correct answer and explanation is :
We are given the following information about angles and their measures:
- Angle $CDE$ is a straight angle, meaning its measure is $180^\circ$.
- Ray $DE$ bisects angle $GDH$.
- Measure of angle $GDE$ is $8x – 1$.
- Measure of angle $EDH$ is $6x + 15$.
- Measure of angle $CDF$ is $43^\circ$.
Step 1: Understand the relationship between angles.
Since ray $DE$ bisects angle $GDH$, angle $GDE$ and angle $EDH$ must be congruent, meaning their measures are equal.
Thus, we can set up the equation:
$$
8x – 1 = 6x + 15
$$
This equation reflects the fact that the measures of angles $GDE$ and $EDH$ are equal.
Step 2: Solve for $x$.
To solve for $x$, follow these steps:
- Subtract $6x$ from both sides:
$$
8x – 6x – 1 = 15
$$
$$
2x – 1 = 15
$$
- Add 1 to both sides:
$$
2x = 16
$$
- Divide both sides by 2:
$$
x = 8
$$
Step 3: Find the measures of angles $GDE$ and $EDH$.
Now that we know $x = 8$, we can substitute this value back into the expressions for the measures of angles $GDE$ and $EDH$:
- Measure of angle $GDE$:
$$
GDE = 8x – 1 = 8(8) – 1 = 64 – 1 = 63^\circ
$$
- Measure of angle $EDH$:
$$
EDH = 6x + 15 = 6(8) + 15 = 48 + 15 = 63^\circ
$$
Thus, both angle $GDE$ and angle $EDH$ measure $63^\circ$.
Step 4: Calculate measure of angle $GDH$.
Since ray $DE$ bisects angle $GDH$, angle $GDH$ is the sum of angles $GDE$ and $EDH$:
$$
GDH = GDE + EDH = 63^\circ + 63^\circ = 126^\circ
$$
Step 5: Find the measure of angle $CDE$.
We know that angle $CDE$ is a straight angle, meaning its measure is $180^\circ$. Therefore:
$$
CDE = 180^\circ
$$
Step 6: Find the measure of angle $CDF$.
Finally, angle $CDF$ is given as $43^\circ$.
Summary of the measures:
- $\text{Angle } GDE = 63^\circ$
- $\text{Angle } EDH = 63^\circ$
- $\text{Angle } GDH = 126^\circ$
- $\text{Angle } CDE = 180^\circ$
- $\text{Angle } CDF = 43^\circ$
Explanation:
The problem involves several key concepts in geometry, including angle bisectors, straight angles, and solving linear equations. By applying the property of the angle bisector (i.e., that two adjacent angles formed by a bisector are congruent), we set up an equation to find $x$. Substituting this value into the given expressions allowed us to calculate the measures of the unknown angles. This method highlights the importance of understanding geometric relationships, as well as using algebra to solve for unknown values in angle-related problems.