Question

△ABC is an isosceles triangle with legs AB and AC. △AYX is also an isosceles triangle with legs AY and AX.

Triangle A Y X is shown. Line segment B C is drawn from side A Y to A X to form triangle A B C.

The proof that △ABC ~ △AYX is shown.

Statements Reasons
1. △ABC is isosceles with legs AB and AC;
△AYX is also isosceles with legs AY and AX. 1. given
2. AB ≅ AC and AY ≅ AX 2. definition of isosceles triangle
3. AB = AC and AY = AX 3. definition of congruency
4. AY • AC = AX • AC 4. multiplication property of equality
5. AY • AC = AX • AB 5. substitution property of equality
6. 6. division property of equality
7. 7. division property of equality
8. ? 8. ?
9. △ABC ~ △AYX 9. SAS similarity theorem
Which statement and reason are missing in the proof?

∠A ≅ ∠A; reflexive property
∠X ≅ ∠X; reflexive property
∠ABC ≅ ∠AYX; corresponding angles of similar triangles
∠ABC ≅ ∠AXY; corresponding angles of similar triangles

Answer 1

∠A ≅ ∠A; reflexive property

Reason: The included angle A is common to both triangles, so by reflexive property ∠A ≅ ∠A, which with the shown side ratios gives SAS similarity.