Problem
Draw an auxiliary parallel through the bend
Two parallel lines a and b run one above the other. A path crosses line a and bends once between the two lines. After the bend it runs down and crosses line b. Find the angle at the bend.
Reasoning · Grade 4-2 Parallel Lines and Angles
Your answer
Watch out The two angles seem comparable, but they connect only once you draw the helper.
How to solve
Strategy Draw a Diagram — Neither given angle is anywhere near the bend, so as the picture stands there is no chain of angle facts joining them to ①. The fix is to add a line to the drawing: a helper line through the bend point, parallel to a and to b. That single extra line does two jobs at once — it cuts ① into an upper part and a lower part, and it gives each part a parallel-line partner among the given angles. Then ① is just the sum of two subproblems that can each be solved on its own.
1STEP 1
Draw a helper line through the bend
Draw a helper parallel to both lines at the bend.
a ∥ c ∥ b
You are allowed to draw extra lines whenever you like; a line drawn parallel to two lines you already have is the natural bridge between them.
4.G.A.1Draw A DiagramDrawing a helper line through the bend parallel to the two given lines is the move that connects them.
Why?
A line parallel to both given lines keeps a constant gap from each, so it can stand between them without ever meeting either.
🧱Parallel lines keep a constant gapTwo parallel lines never meet and the distance between them is the same everywhere you measure it.
Why?
Each of the two given lines now crosses a parallel of the other, so the angles at the bend repeat the angles already known.
🧱A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.
2STEP 2
Split ① into an upper piece and a lower piece
The helper splits the angle into two pieces.
① = (upper piece) + (lower piece)
An angle chopped into two by a ray inside it is exactly the two smaller angles added — angle measure behaves just like length that way.
4.MD.C.7Identify Subproblems3STEP 3
The upper piece is 45°
The upper piece is 45 degrees.
upper piece = 45°
When one straight line cuts two parallel lines, the picture at the second crossing is a copy of the picture at the first crossing turned halfway round — so the matching inside angles have to be the same size.
8.G.A.5Draw A Diagram4STEP 4
The lower piece is 60°
The lower piece is 60 degrees.
180° - 120° = 60°
Two angles that sit side by side and together fill one side of a straight line always add to 180°, so one subtraction turns an awkwardly placed angle into the one you actually want.
7.G.B.5Draw A Diagram5STEP 5
Add the two pieces
Adding gives 105 degrees.
① = 45° + 60° = 105°
Once the two parts are known, finishing the problem is one addition — the hard work was all in choosing where to draw the helper line.
4.MD.C.7Identify SubproblemsAnswer
105 degrees
45 + 60 = 105
105° is an obtuse angle, a little more than a square corner, which matches the drawing: the two arms of the bend clearly open wider than 90° but nothing like a straight line. The two pieces are also believable on their own — the upper arm slants at 45°, exactly halfway between flat and upright, and the lower arm is noticeably steeper, which fits a 60° piece. As a check on the size, 45° + 60° = 105° is less than 180°, so the path really does bend rather than run straight, as the figure shows.
Takeaway
When a bend sits between two parallel lines, draw one more parallel line through the bend — it splits the mystery angle into two pieces you already know.
- Draw a helper line through the bend
- Split ① into an upper piece and a lower piece
- The upper piece is 45°
- The lower piece is 60°
- Add the two pieces