Problem
Find angles by grouping equal pairs
Triangle ABC has a 40 degree angle at A. Point D inside joins to B and C, forming triangle BDC. BD halves the angle at B and CD halves the angle at C. Find the size of angle BDC.
Reasoning · Grade 4-1 Triangles and Angles
Your answer
Watch out Trying to pin each angle down stalls — only their sum is fixed.
How to solve
Strategy Organize Information in More Ways — There is not enough information to pin down ∠B or ∠C separately, so chasing them one at a time is a dead end. Instead I re-group the information: the two bottom halves ∠DBC and ∠DCB never need to be known apart — only their sum matters for triangle BDC. So I treat the pair as a single chunk, find that chunk from the big triangle, and carry it straight into the small triangle.
1STEP 1
See the two triangles that share the base
The two triangles share the base BC.
Naming the two triangles and noticing they share the base BC is just reading the picture carefully — the shared base is what lets an answer travel from one triangle to the other.
4.G.A.1Draw A Diagram2STEP 2
Get the bottom pair of the big triangle as one chunk
The big triangle's bottom two angles add to 140 degrees.
∠ ABC + ∠ ACB = 180° - 40° = 140°
The triangle angle sum gives a total, not the separate parts, so the honest thing to record is the total — and a total is all the next step will ask for.
8.G.A.5Organize Information In More Ways3STEP 3
Halve the whole chunk at once
Both are halved, so the halves add to 70 degrees.
∠ DBC + ∠ DCB = 1/2(∠ ABC + ∠ ACB) = 140° ÷ 2 = 70°
Angle measure adds up like length, so taking half of each piece and adding is the same as adding first and then halving — the same reason half of 8 apples plus half of 6 apples is half of 14 apples.
4.MD.C.7Organize Information In More WaysHalving the whole chunk at once gives the same answer as halving each angle and adding, so the pieces never have to be split apart.
Why?
Angle measures add up just like lengths, so a total of two angles behaves exactly like a single amount.
🧱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.
Why?
Half of a sum is the sum of the halves, which is why a shared factor can be applied to the whole chunk in one go.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
4STEP 4
Finish inside the small triangle
In the small triangle, BDC is 180 − 70 = 110 degrees.
∠ BDC = 180° - 70° = 110°
Once a triangle's other two angles are known as a single total, the third angle is a one-step subtraction — no need to ever split that total apart.
8.G.A.5Identify SubproblemsAnswer
110 degrees
180 − 70 = 110
110° is obtuse, which matches the wide arc drawn at D in the figure, and it is less than 180° as any triangle angle must be. It is also bigger than the 40° at A, which makes sense: D sits low inside the triangle, so BC subtends a much wider angle from D than from the far-away vertex A. A quick formula check: the answer came out as 90° + 40° ÷ 2 = 110°, so if the angle at A were 0° the answer would slide to 90° and if it were 180° (a flattened triangle) it would slide to 180° — both sensible extremes.
Takeaway
When you cannot find two angles separately, find their sum as one chunk — often the chunk is all the next triangle ever asked for.
- See the two triangles that share the base
- Get the bottom pair of the big triangle as one chunk
- Halve the whole chunk at once
- Finish inside the small triangle