Problem 12 · 2013 Math Kangaroo
Medium
Geometry & Measurement
spatial-reasoning
In the diagram, α = 55°, β = 40° and γ = 35°. How big is δ?
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Answer: E — 130°
Show hints
Hint 1 of 3
An exterior angle of a triangle equals the sum of the two interior angles it is not next to.
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Hint 2 of 3
Apply that idea twice, stepping up through the two stacked triangles toward δ.
Still stuck? Show hint 3 →
Hint 3 of 3
Watch how each step folds one more of the given angles into the running total.
Show solution
Approach: exterior-angle theorem, applied twice up the figure
- On the bottom line, the lower triangle has base angles α and β, so the angle at its top vertex (the exterior angle on the far side) collects α + β = 55° + 40° = 95°.
- That 95° angle and the angle γ = 35° are the two remote interior angles of the small triangle that has δ as its exterior angle.
- By the exterior-angle theorem, δ = 95° + 35° = 130°.
- So δ = 130°.
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