Problem
Find the hidden isosceles triangle
A square ABCD is drawn. An equilateral triangle EBC stands inside it on the bottom side BC. A straight line runs from A through E. Find the marked angle.
Reasoning · Grade 4-2 Triangles and Angles
Your answer
Watch out The hidden triangle looks ordinary, but matching sides make it isosceles.
How to solve
Strategy Identify Subproblems — Nothing in the picture touches the marked angle directly, so I cannot get at it in one move. Instead I hunt for a smaller triangle whose angles I can work out completely, and then pass what I learn along to E. The square and the equilateral triangle share the side BC, which means several segments secretly have the same length; marking those equal lengths on the drawing turns triangle ABE into an isosceles triangle that nobody has pointed out. Once that hidden triangle is solved, the straight line through A, E and F carries the answer the rest of the way.
1STEP 1
Mark every segment that has the same length
Matching sides hide an isosceles triangle.
AB = BC = EB → △ ABE is isosceles with AB = EB
A square has four equal sides and an equilateral triangle has three equal sides; when the two shapes share a side, all of those segments are the same length, and putting little tick marks on the drawing makes the hidden isosceles triangle jump out.
2.G.A.1Draw A Diagram2STEP 2
Find the apex angle of the hidden triangle
Its apex angle is 90 − 60 = 30 degrees.
∠ ABE = ∠ ABC - ∠ EBC = 90° - 60° = 30°
Angle measure adds up just like length: when one ray sits inside a bigger angle it splits that angle into two pieces, so the piece you want comes from a single subtraction.
4.MD.C.7Identify Subproblems3STEP 3
Get the base angles of triangle ABE
Its base angles are 75 degrees each.
∠ BAE = ∠ AEB = (180° - 30°) ÷ 2 = 150° ÷ 2 = 75°
Equal sides face equal angles, so an isosceles triangle needs only one number: subtract the apex angle from 180° and split what is left into two fair shares.
8.G.A.5Identify SubproblemsBecause triangle ABE has two equal sides, its two base angles are equal and the apex fixes both at once.
Why?
Equal sides always face equal angles, so the two base angles cannot differ however the triangle is drawn.
🧱Equal sides face equal anglesFold an isosceles triangle down its middle: the two equal sides land on each other and carry the two base angles together, so those angles are equal — and the reverse holds too.
Why?
The three angles of the triangle add to one straight angle, so the apex angle leaves a fixed amount to be split evenly.
🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.
4STEP 4
Add up the three angles that sit along the straight line at E
At E the three angles lie along a straight line.
∠ AEB + ∠ BEC + ∠ FEC = 180°
Re-reading the drawing as "one straight angle cut into three neighbouring pieces" turns a geometry puzzle into a simple addition sentence with one number missing.
7.G.B.5Organize Information In More Ways5STEP 5
Solve for the marked angle
Subtracting gives 45 degrees.
∠ FEC = 180° - 75° - 60° = 45°
Once a straight angle has been split into named pieces, finding the missing piece is ordinary subtraction — exactly the protractor skill practised in Grade 4.
4.MD.C.7Identify SubproblemsAnswer
45 degrees
180 − 75 − 60 = 45
The answer is an angle, so degrees are the right unit, and 45° is a sensible size: in the figure ① is clearly acute and clearly smaller than the 60° corner of the equilateral triangle right beside it, which is exactly what 45 ＜ 60 says. The three pieces at E also close up perfectly, 75° + 60° + 45° = 180°, so no part of the straight angle has been lost or counted twice. As one more check, ∠ BAE = 75° makes ∠ EAD = 90° - 75° = 15°, a small angle — and that is why the line from A meets the right side at F only a little below D, exactly as drawn.
Takeaway
When a square and an equilateral triangle share a side, all of those sides are the same length — mark them, and a hidden isosceles triangle appears that hands you the angle you need.
- Mark every segment that has the same length
- Find the apex angle of the hidden triangle
- Get the base angles of triangle ABE
- Add up the three angles that sit along the straight line at E
- Solve for the marked angle