Question

Use the information given in each exercise to name the method $\mathbf{(}\mathit{S}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathit{A}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathit{S}\mathit{S}\mathit{S}\mathbf{,}\mathbf{}\mathit{A}\mathit{S}\mathit{A}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathit{H}\mathit{L}\mathbf{)}$ you could use to prove $\mathit{\Delta }\mathit{A}\mathit{O}\mathit{B}\mathbf{\cong }\mathit{\Delta }\mathit{A}\mathit{O}\mathit{C}$.You need not write the proofs.

Given $\overline{\mathbf{A}\mathbf{O}}\mathbf{\perp }\mathbf{\text{plane\hspace{0.17em}}}\mathit{M}\mathbf{;}\mathbf{}\overline{\mathbf{A}\mathbf{B}}\mathbf{\cong }\overline{\mathbf{A}\mathbf{C}}$

Short answer

$\Delta \text{AOB}\cong \Delta \text{AOC}$by $\mathit{H}\mathit{L}$congruency method.

Step-by-step solution

- Define methods SSS, SAS, ASA, AAS, and HL

$SSS$- Side – Side – Side method means if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

$SAS$ – Side – Angle – side method means if two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, then triangles are congruent.

$ASA$- Angle – side – angle method means if two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, then triangles are congruent.

$AAS$– Angle – angle – side method if two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, then triangles are congruent.

$HL$– Hypotenuse – leg method means if the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, then two triangles are congruent.

- Understand the figure

In the figure, $\overline{AO}\perp \text{plane\hspace{0.17em}}M$and also $\overline{AB}\cong \overline{AC}$.

Step 3- State the congruency method

In order to prove the congruency of triangles$\Delta \text{AOB}$and$\Delta \text{AOC}$of which sides, $BO$and $OC$lies on plane $M$ observe the following:

i. (Right - Angle) -Since,$\overrightarrow{\text{AO}}\perp$ plane M, so,$\text{AO}\perp \text{BO}$ and $\text{AO}\perp \text{OC}$, which implies that $\angle \text{AOB}\cong \angle \text{AOC}$.It is given that,$\text{BO}\cong \text{CO}$.

ii. (Hypotenuse) - It is given that $AB\cong AC$.

iii. (Leg) - Side $\text{AO}=\text{OA}$is common in both triangles.

Therefore, by $HL$congruency method $\Delta \text{AOB}\cong \Delta \text{AOC}$.