Solving Complementary Angles Algebraically
Two angles are complementary if their sum is equal to 90 degrees. Let the measures of the two angles be '$x$' and '$y$'.
From the problem statement, we have the following conditions:
- The angles are complementary: $x + y = 90^\circ$.
- One angle's measure is 10 degrees more than three times the other: Let $x = 3y + 10^\circ$.
Step-by-Step Solution
- Substitute the expression for '$x$' from the second condition into the first equation:
$(3y + 10^\circ) + y = 90^\circ$
- Combine like terms:
$4y + 10^\circ = 90^\circ$
- Isolate the term with '$y$' by subtracting $10^\circ$ from both sides:
$4y = 90^\circ - 10^\circ$
$4y = 80^\circ$
- Solve for '$y$' by dividing both sides by 4:
$y = \frac{80^\circ}{4}$
$y = 20^\circ$
- Find the measure of the other angle, '$x$':
$x = 3y + 10^\circ = 3(20^\circ) + 10^\circ = 60^\circ + 10^\circ = 70^\circ$
- Identify the smaller angle. The two angles are $20^\circ$ and $70^\circ$. The smaller angle is $20^\circ$.
Therefore, the measure of the smaller angle is $20^\circ$.
Similar Questions
- In a right triangle, one of the acute angles is $30^\circ$. What is the measure of the exterior angle at the right-angled vertex?
- If the angle subtended by arc AB at the center is $140^\circ$, what is the reflex angle made by this arc at the center?
- When a transversal intersects two parallel lines, one of the alternate interior angles is 115°. What is the measure of the angle that lies opposite to it on the alternate interior side?
Two parallel lines are intersected by a transversal. If a pair of corresponding angles is formed, and one of the angles measures 130°, what is the measure of the other angle?
- At an intersection of two lines, two angles are vertically opposite. If one of the angles measures $45^\circ$, what is the measure of the other angle?
Important Questions from Lines and Angles
Two parallel lines are intersected by a transversal, the corresponding angles are:
If l, m, n are lines such that, l is parallel to n and m is parallel to n, then, ______
There are three points P, Q and R on a straight line such that PQ : QR = 3 : 5. If n is the number of possible values of PQ : PR, then what is n equal to ?
Two cars start from a point at the same time and travel on two different roads at right angles to each other. Their speed is 18 km/h and 72 km/h respectively. What will be the distance between them after 6 seconds?
If angles of a triangle are in the ration of 2 : 3 : 4, then the measure of the smallest angle is:
