given that segment AD is congruent to segment CD and that segment DB is perpendicular to segment AC, prove that angle A is congruent to angle C.

Answer 1

Draw a figure.

In right triangles ABD and CBD BD is common and AD=CD.

These 2 triangles are congruent by SAS (or by right triangles with 2 equal legs).

Therefore, ∠A = ∠C because they are corresponding parts of congruent triangles.

Please note that I have used older (and perhaps less precise which is more compact and almost never ambiguous) terminology: technically as is current teaching the length of AD = length of CD and the measure of ∠A = measure of ∠C.