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Isosceles Triangle Example

Question

In the figure above, ABC is an isosceles triangle, AB = AC, draw a line from vertex A and perpendicular to the opposite side BC at point D.

Prove (a). BD = CD (b). angle BAD = angle CAD

Proof (a).

since AB = AC (Given) (then triangle ABC is an isosceles triangle)

so angle B = angle C (two base angles of an isosles triangle are equal)

since AD is perpendicular to BC (Given)

triangle ABD and triangle ACD are right triangles

in right triangles ABD and ACD,

since AB = AC, AD = AD (common side)

so triangle ABD is congruent to triangle ACD (hypotenuse, leg)

so BD = CD (In two congruent triangles, corresonding sides are congruent.)

Note: To prove line segments BD = CD, we need to proof that triangles ABD is congruent to ACD. In order to prove these two triangles are congruent, since they are right triangles, we need to find the hypotenuse and a leg of one right triangle are congruent to corresponding parts of another right triangle. We also use the property of an isosceles triangle, that is, its two base angles are congruent. AD is leg of both right triangles ABD and ACD.

Proof (b).

Since right triangles ABD and ACD are congruent (AB = AC, AD = AD hypotenuse, leg), then angle BAD = angle CAD (Corresponding angles are congruent in two congruent triangles).

The altitude drawn to the base of an isosceles triangle bisects the vertex angle and the base. This is a property of isosceles triangle.