Question

in the figure shown, ( overline{gm}perpoverline{mo} ) and ( overline{eo}perpoverline{mo} ). which theorem can be used to prove ( \triangle gmocong\triangle eom )?

Explanation:

Step1: Identify right angles

Since \(GM\perp MO\) and \(EO\perp MO\), \(\angle GMO=\angle EOM = 90^{\circ}\). So, \(\triangle GMO\) and \(\triangle EOM\) are right - angled triangles.

Step2: Identify hypotenuse and leg

The hypotenuse for both triangles is \(GO\) (common side) and one leg: \(GM = EO\) (given by the equal - length marks).
By the HL (Hypotenuse - Leg) congruence criterion for right - angled triangles, if the hypotenuse and one leg of a right - angled triangle are equal to the hypotenuse and one leg of another right - angled triangle, then the two right - angled triangles are congruent. So, \(\triangle GMO\cong\triangle EOM\) by the HL theorem.

Answer:

HL (Hypotenuse - Leg) Theorem