Street Intersection Three streets form an intersection. ∠C and ∠D are complementary angles. If the measure of ∠C is x° and the measure of ∠D is 66° greater than ∠C. What is the value of x? Use pencil and paper. Find the measures of ∠C and ∠D. The figure is not drawn to scale.
Please give and explanation for how you found the answer. Thank you!
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Let's solve this step by step.
Step 1: Let's set up an equation based on the given information. We know that ∠C and ∠D are complementary angles, which means that their measures add up to 90°. So we have:
x + ∠D = 90° (1)
Step 2: We also know that the measure of ∠D is 66° greater than ∠C. So we can express ∠D in terms of x as:
∠D = ∠C + 66° (2)
Step 3: Now we can substitute equation (2) into equation (1) to solve for x:
x + (∠C + 66°) = 90°
Simplifying the equation, we have:
x + ∠C + 66° = 90°
Step 4: Next, let's combine like terms:
x + ∠C = 90° - 66°
x + ∠C = 24°
Step 5: Rearranging the equation to isolate x, we get:
x = 24° - ∠C (3)
Step 6: Now we can substitute the expression for x from equation (3) into equation (2) to solve for ∠D:
∠D = ∠C + 66°
∠D = 24° - ∠C + 66°
Simplifying the equation, we have:
∠D = 90° - ∠C
So, the measure of ∠D is 90° - ∠C.
Step 7: Since ∠C and ∠D are complementary angles, their measures add up to 90°. Therefore, we can conclude that:
x + (90° - x) = 90°
Simplifying the equation, we have:
90° = 90°
Step 8: This equation is always true, which means that x can be any value.
Thus, the value of x is not uniquely determined. The measure of ∠C and ∠D will depend on the chosen value for x.
From the given information, we know that the measure of ∠D is 66° greater than ∠C. So, if we choose x = 12°, then we have:
∠C = 12°
∠D = 12° + 66° = 78°
Therefore, for x = 12°, the measures of ∠C and ∠D are 12° and 78°, respectively.
To summarize, the value of x is not uniquely determined, but one possible solution is x = 12°, with ∠C = 12° and ∠D = 78°.
To find the value of x and the measures of angles ∠C and ∠D, we need to set up an equation based on the given information and solve for x.
Given:
∠C and ∠D are complementary angles, which means they add up to 90 degrees.
∠D is 66 degrees greater than ∠C.
Let's set up the equation:
∠D = ∠C + 66° (Since ∠D is 66 degrees greater than ∠C)
∠C + ∠D = 90° (Since ∠C and ∠D are complementary angles and they add up to 90 degrees)
Now we have a system of equations to solve:
∠D = ∠C + 66°
∠C + ∠D = 90°
We can substitute the value from the first equation into the second equation:
∠C + (∠C + 66°) = 90°
Combining like terms:
2∠C + 66° = 90°
Subtracting 66° from both sides to isolate 2∠C:
2∠C = 90° - 66°
2∠C = 24°
Divide both sides by 2 to solve for ∠C:
∠C = 12°
Therefore, the value of x (which represents the measure of ∠C) is 12 degrees.
To find the measure of ∠D, substitute the value of ∠C into the first equation:
∠D = ∠C + 66°
∠D = 12° + 66°
∠D = 78°
So, the measures of angles ∠C and ∠D are 12 degrees and 78 degrees, respectively.
complementary angles sum to 90º
C + D = 90
substituting ... x + x + 66 = 90 ... x = 12