IXL - Proofs involving parallel lines I (Geometry practice) - Free Printable

We are told that lines QS and TV are parallel, and a transversal line (the diagonal one going through P-R-U-W) cuts across them.

We’re given:
m∠QRU = 135°

We need to find:
m∠VUR

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Let’s label the points clearly:

- Line QS is the top horizontal line. Points Q, R, S lie on it (left to right).
- Line TV is the bottom horizontal line. Points T, U, V lie on it (left to right).
- The transversal goes from top-left (P) down through R (on QS), then through U (on TV), then to W (bottom-right).

So angle ∠QRU is at point R, between points Q, R, U.

That means:
→ From R, we go left along QS to Q, and down along the transversal to U. So ∠QRU is the angle above the transversal and to the left of it — actually, wait — let's think carefully.

Actually, since Q is to the left of R on the top line, and U is below and to the right of R (since the transversal goes down to the right), then ∠QRU is the angle formed by going from Q → R → U.

So that’s the angle inside the two parallel lines? Let’s sketch mentally:

At point R:
- Ray RQ goes left along the top line.
- Ray RU goes down-right along the transversal.

So ∠QRU is the angle between the leftward ray and the downward-right ray — which would be an obtuse angle, 135°, as given.

Now, we want ∠VUR.

Point U is on the bottom line TV. V is to the right of U. R is above and to the left of U.

So ∠VUR is the angle at U between points V, U, R.

That is:
→ From U, go right to V (along the bottom line), and up-left to R (along the transversal).

So this angle is between the rightward ray UV and the upward-left ray UR.

Notice: ∠QRU and ∠VUR are on opposite sides of the transversal, and both are between the two parallel lines? Wait — no.

Actually, let’s use standard angle relationships with parallel lines.

When a transversal crosses two parallel lines, several pairs of angles are related:

- Corresponding angles
- Alternate interior angles
- Consecutive interior angles (same-side interior)
- Vertical angles
- Linear pairs

Let’s identify what kind of pair ∠QRU and ∠VUR form.

First, note that ∠QRU is at point R, and ∠VUR is at point U.

They are on opposite sides of the transversal.

Also, ∠QRU is above the bottom line but below the top line? Actually, since it’s formed by RQ (left along top line) and RU (down to U), it’s actually the angle outside the space between the two parallel lines? Hmm.

Wait — perhaps better to consider adjacent angles or supplementary angles.

Note: At point R, the straight line QS means that angles on one side add to 180°.

So if ∠QRU = 135°, then the angle next to it on the same side — say, ∠SRU — should be 180° - 135° = 45°.

Because Q-R-S is a straight line, so ∠QRU + ∠SRU = 180°.

Yes! That makes sense.

So ∠SRU = 45°.

Now, look at ∠SRU and ∠VUR.

∠SRU is at R, between S (right on top line), R, and U (down on transversal).

∠VUR is at U, between V (right on bottom line), U, and R (up on transversal).

These two angles — ∠SRU and ∠VUR — are in the same relative position: both are on the right side of the transversal, and both are below their respective horizontal lines? Wait — actually:

- ∠SRU: at R, between rightward ray RS and downward ray RU → so it’s the angle below the top line and to the right of the transversal.
- ∠VUR: at U, between rightward ray UV and upward ray UR → so it’s the angle above the bottom line and to the right of the transversal.

Since the lines are parallel, these are corresponding angles!

Corresponding angles are equal when lines are parallel.

Therefore, ∠SRU = ∠VUR.

We found ∠SRU = 45°, so ∠VUR = 45°.

Alternatively, you can think of ∠QRU and ∠VUR as consecutive interior angles? No, because they are not both inside the parallel lines on the same side.

Wait — another way: ∠QRU and ∠TUR might be consecutive interior? But we don’t have ∠TUR.

But our first method is solid:

1. ∠QRU = 135°
2. Since QRS is straight, ∠SRU = 180° - 135° = 45°
3. ∠SRU and ∠VUR are corresponding angles (both on the right side of transversal, one above bottom line, one below top line — but since lines are parallel, corresponding angles match)
- Actually, more precisely: ∠SRU is at R, between the top line and transversal, on the right side.
- ∠VUR is at U, between the bottom line and transversal, on the right side.
- And since the lines are parallel, yes — corresponding angles are equal.

Thus, m∠VUR = 45°

Double-check: Could they be alternate interior? No, because alternate interior would be on opposite sides of the transversal. Here both are on the right side.

Another check: What about vertical angles or linear pairs at U?

At point U, the straight line TV means that ∠TUV = 180°.

If we knew ∠TUR, we could find ∠VUR as 180° - ∠TUR.

What is ∠TUR?

∠TUR is at U, between T (left on bottom line), U, R (up on transversal).

Compare to ∠QRU: both are on the left side of the transversal.

∠QRU is at R, between Q (left on top), R, U.

∠TUR is at U, between T (left on bottom), U, R.

These are alternate exterior angles? Or maybe consecutive?

Actually, ∠QRU and ∠TUR are on the same side of the transversal (left side), and one is above the bottom line, one is below the top line — but since the lines are parallel, they are consecutive interior angles? Wait, no — consecutive interior are both between the lines.

Actually, ∠QRU is NOT between the lines — because from R, going to Q is along the top line, and to U is down — so the angle opens outward.

Perhaps easier to stick with our first method.

We have:

- ∠QRU = 135°
- Adjacent angle on straight line: ∠SRU = 45°
- ∠SRU and ∠VUR are corresponding angles → equal → 45°

Yes.

Final Answer: 45