Finding Variables from Angles on a Straight Line
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11. (a) In the diagram, AOB is a straight line, $\angle AOC = 3(x + y)^{\circ}$, $\angle COB = 45^{\circ}$, $\angle AOD = (5x + y)^{\circ}$ and $\angle DOB = y^{\circ}$. Find the
This question includes visual content: A geometric diagram showing a straight line AB with a point O in the middle. Two rays, OC and OD, originate from point O. Ray OC is above the line and ray OD is below the line. The angle AOC is labeled $3(x+y)^{\circ}$, the angle COB is labeled $45^{\circ}$. Below the line, the angle AOD is labeled $(5x+y)^{\circ}$ and the angle DOB is labeled $y^{\circ}$.
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Step by Step Written Solution
Let's solve for the variables x and y based on the geometry shown in this diagram. We are given that AOB is a straight line, which is a key piece of information.
Geometry Problem
Given:
- $AOB$ is a straight line
- $\angle AOC = 3(x+y)^{\circ}$
- $\angle COB = 45^{\circ}$
- $\angle AOD = (5x+y)^{\circ}$
- $\angle DOB = y^{\circ}$
Since AOB is a straight line, the sum of the angles on either side must equal 180 degrees. Let's look at the angles above the line first.
From the upper side of the line, we can see that angle AOC plus angle COB equals 180 degrees. Substituting the given values, we get three times the quantity x plus y, plus forty-five, equals 180.
First, we subtract 45 from both sides of the equation.
This simplifies to 3 times x plus y equals 135.
Now, divide both sides by 3 to simplify further. 135 divided by 3 is 45.
We can rearrange this to express x in terms of y, giving us our first equation: x equals 45 minus y.
Now let's look at the angles below the straight line. Angle AOD plus angle DOB must also sum to 180 degrees.
Step 2: Angles Below the Line
Combining the y terms gives us 5x plus 2y equals 180. This is our second equation.
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