ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that∠ B = ∠ C

Step 1: Draw an isosceles triangle ABC with AB = AC.

Step 2: Draw a line AP perpendicular to BC. This line AP is known as the altitude of the triangle from vertex A.

Step 3: Now, we have two right triangles ABP and ACP.

Step 4: In right triangle ABP and ACP, we have AB = AC (given) and AP = AP (common).

Step 5: By the Hypotenuse-Leg (HL) theorem of congruence (which states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent), we can say that triangle ABP is congruent to triangle ACP.

Step 6: Therefore, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, we have ∠ B = ∠ C.

So, in an isosceles triangle, the base angles are always equal.