Question

Write proof in two-column form.

Given: $\overline{\text{AD}}\perp \overline{\text{BC}}$; $\overline{\text{BA}}\perp \overline{\text{AC}}.$

Prove: $\angle \mathbf{1}\cong \angle \mathbf{2}\mathbf{.}$

Short answer

The two-column proof is:

Statements	Reasons
1. $\overline{\text{AD}}\perp \overline{\text{BC}}$; $\overline{\text{BA}}\perp \overline{\text{AC}}$	1. Given
2. $\angle \text{BAC}$and $\angle \text{BDA}$are right angles.	2. Definition of$\perp$lines.
3. $\Delta \text{BAC}$and $\Delta \text{BDA}$are right triangles.	3. Definition of right triangles.
4. $\angle \mathbf{1}$and $\angle \text{B}$are complementary angles; $\angle \mathbf{2}$and $\angle \text{B}$are complementary angles.	4. The acute angles of a right triangle are complementary.
5.$\angle \mathbf{1}\cong \angle \mathbf{2}$	5. If two angles are complementary of the same angle, then the two angles are congruent.

Step-by-step solution

Step 1. Consider the diagram.

Here,$\overline{AD}\perp \overline{BC}$;$\overline{BA}\perp \overline{AC}$

Step 2. State the proof.

The two triangles are said to be congruent if they are copies of each other and if their vertices are superposed, then say that the corresponding angles and the sides of the triangles are congruent.

The two-column proof is:

Statements	Reasons
1.$\overline{AD}\perp \overline{BC}$;$\overline{BA}\perp \overline{AC}$	1. Given
2. $\angle BAC$and$\angle BDA$ are right angles.	2. Definition of$\perp$lines.
3. $\Delta BAC$and $\Delta BDA$are right triangles.	3. Definition of right triangles.
4. $\angle 1$and $\angle B$are complementary angles; $\angle 2$and $\angle B$are complementary angles.	4. The acute angles of a right triangle are complementary.
5.$\angle 1\cong \angle 2$	5. If two angles are complementary of the same angle, then the two angles are congruent.

Step 3. State the conclusion.

Therefore, $\angle 1\cong \angle 2$( proved).