Question

For each diagram, name a pair of overlapping triangles. Tell whether the triangles are congruent by the$\mathit{S}\mathit{S}\mathit{S}\mathbf{,}\mathbf{}\mathit{S}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathit{A}\mathit{S}\mathit{A}\mathbf{,}\mathbf{}\mathit{A}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathbf{\text{or}}\mathbf{}\mathit{H}\mathit{L}$ method.

Given: $\angle \mathbf{ABC}\cong \angle \mathbf{ACB}\mathbf{;}\mathbf{}\overline{\mathrm{AE}}\mathbf{\perp }\overline{\mathrm{EC}}\mathbf{;}\mathbf{}\overline{\mathrm{AD}}\mathbf{\perp }\overline{\mathrm{DB}}$

Short answer

The pair of overlapping triangles that are congruent are$\mathbf{△}\mathbf{ADB}$and $\mathbf{△}\mathbf{AEC}$.

The triangles$△ADB$ and$△AEC$ are congruent by the $\mathit{A}\mathit{A}\mathit{S}$method.

Step-by-step solution

- Observe the given diagram.

The given diagram is:

- Description of step.

It is being given that$\angle ABC\cong \angle ACB,\overline{AE}\perp \overline{EC}$ and $\overline{AD}\perp \overline{DB}$.

As, $\angle ABC\cong \angle ACB$, therefore by using the theorem $4-2$it can be said that $\overline{AB}\cong \overline{AC}$.

As, $\overline{AE}\perp \overline{EC}$, therefore it can be said that $m\angle AEC=90°$.

As, $\overline{AD}\perp \overline{DB}$, therefore it can be said that $m\angle ADB=90°$.

Therefore, $m\angle AEC=m\angle ADB=90°$.

Therefore, $\angle AEC\cong \angle ADB$.

In the triangles $△ADB$and $△AEC$, the angle that is common is $\angle A$.

Therefore, $\angle A\cong \angle A$by using the reflexive property.

In the triangles$△ADB$ and $△AEC$, it can be noticed that $\angle A\cong \angle A,\angle AEC\cong \angle ADB$ and$\overline{AB}\cong \overline{AC}$.

Therefore, the triangles $△ADB$and $△AEC$are the congruent triangles by using the $AAS$method.

- Write the conclusion.

The pair of overlapping triangles that are congruent are$△ADB$ and $△AEC$.

The triangles$△ADB$ and$△AEC$are congruent by the $AAS$method.