Question Video: Using Properties of Supplementary Angles and Intersecting Angles Between Parallel Lines and a Transversal

The given figure shows a pair of parallel lines and two transversals, one of which crosses at right angles. Write an expression for 𝑑 in terms of 𝑏. Using this expression for 𝑑, find a fully simplified expression for π‘Ž in terms of 𝑏.

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Video Transcript

The given figure shows a pair of parallel lines and two transversals, one of which crosses at right angles. Write an expression for 𝑑 in terms of 𝑏. Using this expression for 𝑑, find a fully simplified expression for π‘Ž in terms of 𝑏.

Let’s see what we have. We have two parallel lines. We could call them line one and line two, and then we have two transversals, which we could call line three and line four. The first thing we wanna do is write an expression for 𝑑 in terms of 𝑏. 𝑑 and 𝑏 occur at the intersection of line one and line three. They are adjacent angles on the same side of line one. This makes them supplementary angles. They must add up to 180 degrees. If we know that 𝑏 degrees plus 𝑑 degrees must equal 180 degrees and we want 𝑑 in terms of 𝑏, this means we want to try to get 𝑑 by itself and have 𝑏 on the other side of the equation. To do that, we can subtract 𝑏 degrees from both sides of the equation, which means 𝑑 degrees will be equal to 180 degrees minus 𝑏 degrees, and therefore 𝑑 will be equal to 180 minus 𝑏. This completes part one, our first expression.

Now, using this expression for 𝑑, we need to find a fully simplified expression for π‘Ž in terms of 𝑏, which means we first need to look at the relationship between π‘Ž degrees and 𝑑 degrees. When it comes to π‘Ž and 𝑑, they are in between the two parallel lines and on the same side of the transversal. They are consecutive interior angles, sometimes called cointerior angles. And when two parallel lines are crossed by a transversal, the consecutive interior angles are supplementary. They sum to 180. And that means π‘Ž degrees plus 𝑑 degrees must equal 180 degrees.

For this expression, we want to express π‘Ž in terms of 𝑏, and that means we’ll need to get π‘Ž by itself. First, we subtract 𝑑 from both sides, which gives us π‘Ž degrees equals 180 degrees minus 𝑑 degrees or π‘Ž equals 180 minus 𝑑. But remember, we want to substitute our expression we’ve already found for 𝑑. We know that 𝑑 equals 180 minus 𝑏. Remember though, we’re subtracting all of 𝑑, so we need to keep that in parentheses and then distribute that subtraction, which means subtract 180. But subtracting negative 𝑏 would be adding 𝑏, and that means π‘Ž equals 180 minus 180 plus 𝑏. 180 minus 180 equals zero, and zero plus 𝑏 equals 𝑏. A fully simplified expression for π‘Ž in terms of 𝑏 is that π‘Ž is equal to 𝑏.

In fact, if we take a closer look at π‘Ž and 𝑏, we see that the relationship between these two angles are alternate interior angles, and we know that alternate interior angles will be congruent. So 𝑑 in terms of 𝑏 would be 𝑑 equals 180 minus 𝑏, and π‘Ž in terms of 𝑏 would be π‘Ž equals 𝑏.

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