What you’ll learn
Prove properties of isosceles triangles.
Key idea
A midpoint supplies two equal base portions and the shared segment supplies its own reflexive equality.
Use all three side pairs before corresponding base angles; a midpoint or target equality cannot be assumed without justification.
Worked example
Complete missing statements or reasons in a proof for 'Prove properties of isosceles triangles'.
Triangle A=(0, 5), B=(-4, 0), C=(4, 0), with AB=AC; D=(0, 0) is midpoint of BC. Proof lists AB=AC, BD=DC=4, AD=AD=5. Complete the congruence statement/reason and the base-angle consequence.
- AB=AC since squared lengths are both 41; BD=DC=4; AD=AD=5.
- SSS gives triangle ABD congruent to ACD with A-A, B-C, D-D. Since D lies between B/C, corresponding angles ABD/ACD are the full base angles ABC/BCA.
Answer: Triangle ABD congruent to ACD by SSS; angle ABC equals BCA by corresponding parts, since D lies on BC.
Quick Check
Complete missing statements or reasons in a proof for 'Prove properties of isosceles triangles'.
Triangle A=(0, 6), B=(-2, 0), C=(2, 0), with AB=AC; D=(0, 0) is midpoint of BC. Proof lists AB=AC, BD=DC=2, AD=AD=6. Complete the congruence statement/reason and the base-angle consequence.