What you’ll learn
Prove triangle angle sum.
Key idea
The parallel construction justifies two alternate-interior substitutions before the target conclusion.
The proof must justify copied angles and the actual straight turn; it cannot cite its own target theorem.
Worked example
Complete missing statements or reasons in a proof for 'Prove triangle angle sum'.
Triangle ABC has B=30, C=90 degrees. A line through A is constructed parallel to BC; flank angles x and y beside A correspond to B and C by alternate interior angles. Complete the final statement and reason from mx+mA+my=180.
- Parallelism gives mx=30 and my=90 degrees without assuming triangle angle sum.
- x,A,y are adjacent along a straight line: mx+mA+my=180; substitution gives A+B+C=180 and A=60.
Answer: A+B+C=180 degrees and A=60 — substitute the two alternate-interior equalities into the straight-angle sum.
Quick Check
Complete missing statements or reasons in a proof for 'Prove triangle angle sum'.
Triangle ABC has B=70, C=20 degrees. A line through A is constructed parallel to BC; flank angles x and y beside A correspond to B and C by alternate interior angles. Complete the final statement and reason from mx+mA+my=180.