Question

ghlj and gstu are both parallelograms. why is ∠l ≅ ∠t?

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, opposite angles are congruent, and also, corresponding angles (or angles related by parallel sides) can be equal. For a parallelogram, a key property is that opposite angles are equal, and also, if two parallelograms have their sides parallel in the same orientation, their corresponding angles will be equal.

Step2: Analyze angles in each parallelogram

• In parallelogram \( GHLJ \), \( \angle L \) is equal to \( \angle G \) (opposite angles of a parallelogram are congruent: \( \angle L \cong \angle G \)).
• In parallelogram \( GSTU \), \( \angle T \) is equal to \( \angle G \) (opposite angles of a parallelogram are congruent: \( \angle T \cong \angle G \)).

Step3: Transitive property of congruence

Since \( \angle L \cong \angle G \) and \( \angle T \cong \angle G \), by the transitive property of congruence (if \( a \cong b \) and \( c \cong b \), then \( a \cong c \)), we can conclude that \( \angle L \cong \angle T \). Alternatively, another way: In a parallelogram, opposite angles are equal, and also, if two parallelograms are such that their sides are parallel (since \( GHLJ \) and \( GSTU \) share the side \( GS \) and have parallel sides \( HL \parallel GJ \) and \( ST \parallel GU \), and \( H L \) and \( ST \) are also parallel as they are both parallel to the base), so the corresponding angles (angles at \( L \) and \( T \)) which are both opposite to \( \angle G \) (or corresponding angles with respect to the parallel sides) will be equal. The core reason is that in a parallelogram, opposite angles are congruent, and both \( \angle L \) and \( \angle T \) are opposite to \( \angle G \) (or their respective parallelograms' angles opposite to the same angle \( \angle G \)), so they must be congruent to each other.

Answer:

\(\angle L \cong \angle T\) because in a parallelogram, opposite angles are congruent. For parallelogram \(GHLJ\), \(\angle L \cong \angle G\); for parallelogram \(GSTU\), \(\angle T \cong \angle G\). By the transitive property of congruence, \(\angle L \cong \angle T\). (Or, more simply: In a parallelogram, opposite angles are equal, and both \(\angle L\) (in \(GHLJ\)) and \(\angle T\) (in \(GSTU\)) are opposite to \(\angle G\), so they are congruent to each other.)