Question

1. name the pair of congruent sides, if such a pair exists, in the figure below.

\\(\overline{ca} \cong \overline{cb}\\)
\\(\overline{ba} \cong \overline{bc}\\)
no congruent sides
\\(\overline{ab} \cong \overline{ac}\\)

Explanation:

🆕 New Concept Discovered: Converse of the Isosceles Triangle Theorem
Sides opposite to equal angles are equal in length.

Step 1: Find the missing angle measure

The sum of the interior angles of any triangle is always \( 180^\circ \).

$$ m\angle A + m\angle B + m\angle C = 180^\circ $$

Substitute the given values:

$$ 56^\circ + m\angle B + 62^\circ = 180^\circ $$

$$ 118^\circ + m\angle B = 180^\circ $$

$$ m\angle B = 180^\circ - 118^\circ = 62^\circ $$

Step 2: Identify equal angles

We now have the measures of all three angles:

• \( m\angle A = 56^\circ \)
• \( m\angle B = 62^\circ \)
• \( m\angle C = 62^\circ \)

Since \( m\angle B = m\angle C \), the triangle has two congruent angles.

Step 3: Determine the congruent sides

According to the Converse of the Isosceles Triangle Theorem, if two angles of a triangle are congruent, then the sides opposite those angles are also congruent.

• The side opposite to \( \angle B \) is \( \overline{AC} \) (or \( \overline{CA} \)).
• The side opposite to \( \angle C \) is \( \overline{AB} \) (or \( \overline{BA} \)).

Therefore, the sides opposite these equal angles must be congruent:

$$ \overline{AB} \cong \overline{AC} $$

Answer:

\( \overline{AB} \cong \overline{AC} \) (the fourth option)