We are given isosceles `Delta ABC` with vertex A, altitudes `bar(CE),bar(BD)` . We are asked to prove that `Delta HBC` is isosceles, where H is the intersection of the altitudes.

** Depending on the text and/or instructor, you may have to include more/less detail. **

Statement: Reason:

(1) `bar(AB) cong bar(AC)` (1) Given

(2) `/_ABC cong /_ACB` (2) Isosceles triangle theorem *

(3) `/_CEB,/_BDC` are rt angles (3) Definition of altitude

(4) `/<CEB cong /_BDC` (4) Rt angles are congruent

(5) `bar(BC) cong bar(BC)` (5) Reflexive property of `cong`

(6) `Delta BEC cong Delta CDB` (6) AAS

(7) `/_DBC cong /_ECB` (7) CPCTC **

(8)`` `bar(HB) cong bar(HC)` (8) converse of isos. triangle thm*

(9) `Delta HBC` is isosceles (9) Definition of isosceles

QED.

* Isosceles triangle theorem -- If two sides of a triangle are congruent then the angles opposite those sides are congruent.

** CPCTC -- corresponding parts of congruent triangles are congruent

* Converse of isosceles triangle theorem -- If two angles of a triangle are congruent, then the sides opposite those angles are congruent