Sure! Here are five problems involving congruent angles and complementary angles, along with their solutions.
Problem 1
Question: Angle A is congruent to Angle B. If Angle A measures \(x\) degrees, what is the measure of Angle B?
Solution:
Since Angle A is congruent to Angle B:
Angle A = Angle B
Therefore, if Angle A = \(x\), then Angle B = \(x\)
So, \(m\angle B = x\)
Problem 2
Question: Angle C and Angle D are complementary angles. If Angle C measures \(45\) degrees, what is the measure of Angle D?
Solution:
Complementary angles add up to \(90\) degrees:
\(m\angle C + m\angle D = 90\)
Substituting the value of Angle C:
\(45 + m\angle D = 90\)
Now, solve for Angle D:
\(m\angle D = 90 - 45 = 45\) degrees
Problem 3
Question: Angles E and F are congruent. If the measure of Angle E is \(2x\) degrees and Angle F is \(3x - 10\) degrees, find the value of \(x\) and the measures of angles E and F.
Solution:
Since Angle E is congruent to Angle F:
\(2x = 3x - 10\)
Now, solve for \(x\):
\(10 = 3x - 2x\)
\(10 = x\)
Now, substitute \(x\) back to find the measures:
Angle E = \(2x = 2(10) = 20\) degrees
Angle F = \(3x - 10 = 3(10) - 10 = 20\) degrees
Thus, both \(m\angle E = 20\) degrees and \(m\angle F = 20\) degrees.
Problem 4
Question: Angles G and H are complementary. If Angle G measures \(3x\) degrees and Angle H measures \(x + 30\) degrees, find \(x\) and the measures of angles G and H.
Solution:
Complementary angles add up to \(90\) degrees:
\(m\angle G + m\angle H = 90\)
Substituting their expressions:
\(3x + (x + 30) = 90\)
Now, combine like terms:
\(4x + 30 = 90\)
Now, isolate \(x\):
\(4x = 90 - 30\)
\(4x = 60\)
\(x = 15\)
Now substitute \(x\) back into the angle expressions:
Angle G = \(3(15) = 45\) degrees
Angle H = \(15 + 30 = 45\) degrees
Problem 5
Question: Angle I and Angle J are complementary angles. If Angle I measures \(x\) degrees and Angle J measures \(2x - 30\) degrees, find \(x\) and the measure of both angles.
Solution:
Since they are complementary:
\(m\angle I + m\angle J = 90\)
Substituting their expressions:
\(x + (2x - 30) = 90\)
Combine like terms:
\(3x - 30 = 90\)
Add \(30\) to both sides:
\(3x = 120\)
Now divide by \(3\):
\(x = 40\)
Now substitute \(x\) back into the angle expressions:
Angle I = \(40\) degrees
Angle J = \(2(40) - 30 = 80 - 30 = 50\) degrees
Summary of Results
- \(m\angle A = x\), \(m\angle B = x\)
- \(m\angle C = 45\) degrees, \(m\angle D = 45\) degrees
- \(m\angle E = 20\) degrees, \(m\angle F = 20\) degrees
- \(m\angle G = 45\) degrees, \(m\angle H = 45\) degrees
- \(m\angle I = 40\) degrees, \(m\angle J = 50\) degrees
Feel free to ask any further questions on geometry!
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