Question

1. which pair of triangles can be proven congruent by the aas theorem?

Explanation:

Step1: Recall AAS Theorem

AAS (Angle - Angle - Side) Theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step2: Analyze each option

• First option (Triangle TSR and QRT):
• In the first figure, we have a quadrilateral \(QTSR\) with \(QR\parallel TS\) (since \(Q\) and \(S\) are right angles, \(QR\perp TR\) and \(TS\perp TR\)). \(\angle TSR=\angle QRT = 90^{\circ}\), \(\angle STR=\angle QRT\) (alternate interior angles), and \(TR\) is common. But let's check the angles and sides. Wait, actually, for the first figure, \(\angle TSR=\angle RQT = 90^{\circ}\), \(\angle STR=\angle QRT\) (alternate interior angles), and \(TR = TR\) (common side). But let's check the other options.
• Second option (Triangle GKH and JKH):
• In the second figure, we have \(GK = JK\) (marked), \(KH\) is common, and \(\angle GKH=\angle JKH\) (marked as right angles? Wait, no, the markings show \(GK = JK\) and \(KH\) is common, and \(\angle GHK=\angle JHK\)? Wait, maybe this is SAS or SSS, not AAS.
• Third option (Triangle LKM and OKP):
• In the third figure, we have vertical angles \(\angle LKM=\angle OKP\), and two sides marked equal, but this looks like SAS (if we consider the vertical angles and the two sides), not AAS.
• Fourth option (Triangle ABC and DEF):
• Wait, the first option (the left - most figure) has two right angles, a common side, and another pair of equal angles. Wait, let's re - examine. The first figure: \(\angle TSR=\angle RQT = 90^{\circ}\), \(\angle STR=\angle QRT\) (alternate interior angles), and \(TR\) is the non - included side. So by AAS, \(\triangle TSR\cong\triangle RQT\). Wait, but the blue dot is on the fourth option? Wait, no, maybe I misread. Wait, the first option (the left - most triangle pair) has two right angles, a common side, and a pair of equal angles. Let's correct:

Wait, the first option (the left - most figure) is a rectangle - like figure with \(TR\) as a diagonal. \(\angle TSR=\angle RQT = 90^{\circ}\), \(\angle STR=\angle QRT\) (because \(QR\parallel TS\), alternate interior angles), and \(TR = TR\). So by AAS, \(\triangle TSR\cong\triangle RQT\).

Wait, maybe the first option (the left - most) is the correct one. But let's check the AAS conditions again. AAS requires two angles and a non - included side. In the first figure, \(\angle S=\angle Q = 90^{\circ}\), \(\angle STR=\angle QRT\), and \(TR\) is the side that is not included between the two angles (for \(\triangle TSR\), angles are \(\angle S\) and \(\angle STR\), side \(TR\); for \(\triangle RQT\), angles are \(\angle Q\) and \(\angle QRT\), side \(TR\)). So by AAS, these two triangles are congruent.

Answer:

The left - most pair of triangles (the first option)