Problem
Count acute angles among intersecting lines
Four straight lines are drawn across the page. No two of them are parallel. No three of them meet at the same spot. Count every acute angle in the picture.
Reasoning · Grade 4-1 Triangles and Angles
Your answer
Watch out Four angles per crossing tempts you to count all four, but neighbours add to 180 — only two are acute.
How to solve
Strategy Draw a Diagram — Angles only exist where lines meet, so the first job is to mark the crossing points on the diagram and count them. Then I look closely at one single crossing and work out how many acute angles it holds. Because every crossing in this figure is built the same way, two lines cutting across each other, the answer for one crossing repeats at all of them, and the total is just a multiplication.
1STEP 1
Find and count the crossing points
Four lines cross at 6 points.
6 crossing points
Listing the pairs of lines in order is a sure way to catch every crossing without counting one of them twice.
4.G.A.1Make A Systematic List2STEP 2
Look at one crossing on its own
Each crossing makes 4 angles.
360°
An angle is what you get when two rays share an endpoint, so drawing the four rays shows straight away that there are four angles to judge at each crossing.
4.MD.C.5Draw A Diagram3STEP 3
Pair the four angles up
Neighbouring angles always add to 180 degrees.
a + b = 180°
The straight-line fact and the vertical-angle fact together mean you only ever have to decide about one angle at a crossing; the other three follow from it.
7.G.B.5Draw A DiagramAt a crossing the four angles pair up, so deciding about one of them settles the other three.
Why?
Two angles sitting side by side along one of the lines always fill 180 degrees between them.
🧱A straight angle is 180 degreesAngles sitting side by side along a straight line always fill up 180 degrees.
Why?
Because angle measures add, knowing one angle and the straight-line total hands over its neighbour by a single subtraction.
🧱Angle measure addsA ray drawn inside an angle cuts it into two smaller angles whose measures add back to the original, just as lengths do.
4STEP 4
Decide how many of the four are acute
So 2 of the four are acute.
a ＜ 90° → b = 180° - a ＞ 90°
Two numbers that add to 180 cannot both be under 90, so the sharp angle and the wide angle always come in a matched pair.
7.G.B.5Look For A Pattern5STEP 5
Multiply, because every crossing behaves the same way
Every crossing behaves alike: 6 × 2 = 12.
6 × 2 = 12
Six crossings each holding 2 acute angles is six groups of two, which is exactly what multiplication counts.
3.OA.A.1Look For A PatternAnswer
12 acute angles
6 × 2 = 12
The figure has 6 crossings and 4 angles at each one, so there are 24 angles in total. Exactly half of them are acute and half are obtuse, giving 12 and 12, which adds back to 24. The answer is a whole number, it is less than the total number of angles, and it is even, as it must be because acute angles at a crossing always come in vertical pairs. A spot check on the picture agrees: the big triangle formed by the horizontal line and two of the slanted lines has three sharp corners, and each of those corners is one of the 12.
Takeaway
Wherever two lines cross you always get two sharp corners and two wide ones, so just count the crossings and double.
- Find and count the crossing points
- Look at one crossing on its own
- Pair the four angles up
- Decide how many of the four are acute
- Multiply, because every crossing behaves the same way