Question

In ΔABC (m∠C = 90°), the points D and E are the points where the angle bisectors of ∠A and ∠B intersect respectively sides BC and AC . Point G ∈ AB so that DG ⊥ AB and H ∈ AB so that EH ⊥ AB .Prove m∠HCG = 45° (with statement reason preferably).

Answer 1

This is a little long, but it gets you there.

1. ΔEBH ≅ ΔEBC . . . . HA theorem
2. EH ≅ EC . . . . . . . . . CPCTC
3. ∠ECH ≅ ∠EHC . . . base angles of isosceles ΔEHC
4. ΔAHE ~ ΔDGB ~ ΔACB . . . . AA similarity
5. ∠AEH ≅ ∠ABC . . . corresponding angles of similar triangle
6. ∠AEH = ∠ECH + ∠EHC = 2∠ECH . . . external angle is equal to the sum of opposite internal angles (of ΔECH)
7. ΔDAC ≅ ΔDAG . . . HA theorem
8. DC ≅ DG . . . . . . . . . CPCTC
9. ∠DCG ≅ ∠DGC . . . base angles of isosceles ΔDGC
10. ∠BDG ≅ ∠BAC . . . .corresponding angles of similar triangles
11. ∠BDG = ∠DCG + ∠DGC = 2∠DCG . . . external angle is equal to the sum of opposite internal angles (of ΔDCG)
12. ∠BAC + ∠ACB + ∠ABC = 180° . . . . sum of angles of a triangle
13. (∠BAC)/2 + (∠ACB)/2 + (∠ABC)/2 = 90° . . . . division property of equality (divide equation of 12 by 2)
14. ∠DCG + 45° + ∠ECH = 90° . . . . substitute (∠BAC)/2 = (∠BDG)/2 = ∠DCG (from 10 and 11); substitute (∠ABC)/2 = (∠AEH)/2 = ∠ECH (from 5 and 6)
15. This equation represents the sum of angles at point C: ∠DCG + ∠HCG + ∠ECH = 90°, ∴ ∠HCG = 45° . . . . subtraction property of equality, transitive property of equality. (Subtract ∠DCG+∠ECH from both equations (14 and 15).)