Basic triangle proofs congruency

Question

Given: \overline{AD} \cong \overline{AB},AD≅AB, \overline{AB} \perp \overline{BC}AB⊥BC and \overline{AD} \perp \overline{DC}.AD⊥DC.

Prove: \triangle ABC \cong \triangle ADC△ABC≅△ADC.

Answer 1

The very first step should be to draw a diagram of what is given.

You should notice that side AC is present in both triangles. As it is congruent with itself, we have SSA which is not a congruency theorem. You should have also noticed that this is a right triangle, so the hypotenuse-leg (right angel hypotenuse-side) theorem might apply. Which states that if we a congruent hypotenuse and congruent leg between two triangles, the triangle is congruent.

1. Angle B is a right angle because AB⊥BC (the definition of perpendicular lines)
2. Angle D is a right angle because AD⊥DC (the definition of perpendicular lines)
3. AC is a hypotenuse in both triangles because it is opposite a right angle. (defn of hypotenuse)
4. AD and AB are congruent legs (given)
5. AC is congruent with itself (identity)
6. By the hypotenuse-leg theorem △ABC≅△ADC