Question

which two triangles are congruent by the aas theorem? complete the congruence statement.
△□ ≅ △□

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 triangle

- For \(\triangle HIJ\) and \(\triangle RST\) and \(\triangle XYZ\):
- In \(\triangle HIJ\): We have one angle marked with one arc (\(\angle I\)), one angle marked with two arcs (\(\angle J\)) and a side (between \(H\) and \(J\)) marked with two tick marks.
- In \(\triangle RST\): We have one angle marked with one arc (\(\angle T\)), one angle marked with two arcs (\(\angle R\)) and a side (between \(S\) and \(R\)) marked with two tick marks. Wait, no, let's re - check. Wait, in \(\triangle HIJ\), angle \(I\) (1 arc), angle \(J\) (2 arcs), side \(HJ\) (2 ticks). In \(\triangle XYZ\), angle \(X\) (1 arc), angle \(Z\) (2 arcs), side \(YZ\) (1 tick). In \(\triangle RST\), angle \(T\) (1 arc), angle \(R\) (2 arcs), side \(SR\) (2 ticks). Wait, no, the correct pair: Let's look at the angles and non - included side.
- \(\triangle HIJ\): \(\angle I\) (1 arc), \(\angle J\) (2 arcs), side \(HJ\) (2 ticks). \(\triangle RST\): \(\angle T\) (1 arc), \(\angle R\) (2 arcs), side \(SR\) (2 ticks). Wait, no, the non - included side: For AAS, the side is not between the two angles. In \(\triangle HIJ\), the two angles \(\angle I\) and \(\angle J\), the non - included side is \(HJ\)? No, wait, the angles are \(\angle I\) and \(\angle J\), the side opposite to one of the angles? Wait, no, let's look at the markings again.
- Wait, \(\triangle HIJ\): angle at \(I\) (1 arc), angle at \(J\) (2 arcs), side \(HJ\) (2 ticks). \(\triangle XYZ\): angle at \(X\) (1 arc), angle at \(Z\) (2 arcs), side \(YZ\) (1 tick). \(\triangle RST\): angle at \(T\) (1 arc), angle at \(R\) (2 arcs), side \(SR\) (2 ticks). Wait, no, the correct pair is \(\triangle HIJ\) and \(\triangle RST\)? No, wait, no. Wait, \(\triangle HIJ\): \(\angle I\) (1 arc), \(\angle J\) (2 arcs), side \(HJ\) (2 ticks). \(\triangle XYZ\): \(\angle X\) (1 arc), \(\angle Z\) (2 arcs), side \(YZ\) (1 tick). \(\triangle RST\): \(\angle T\) (1 arc), \(\angle R\) (2 arcs), side \(SR\) (2 ticks). Wait, no, I think I made a mistake. Wait, the correct pair is \(\triangle HIJ\) and \(\triangle RST\)? No, wait, let's check the AAS conditions. The two angles and a non - included side. So, in \(\triangle HIJ\), \(\angle I\) (1 arc), \(\angle J\) (2 arcs), and side \(HJ\) (the side not between \(\angle I\) and \(\angle J\)? Wait, no, the side between \(\angle H\) and \(\angle J\) is \(HJ\). Wait, the angles \(\angle I\) and \(\angle J\), the non - included side would be \(HI\)? No, maybe I got the triangles wrong. Wait, the correct pair is \(\triangle HIJ\cong\triangle RST\) by AAS? No, wait, no. Wait, looking at the markings: \(\triangle HIJ\) has \(\angle I\) (1 arc), \(\angle J\) (2 arcs), side \(HJ\) (2 ticks). \(\triangle RST\) has \(\angle T\) (1 arc), \(\angle R\) (2 arcs), side \(SR\) (2 ticks). And \(\triangle XYZ\) has \(\angle X\) (1 arc), \(\angle Z\) (2 arcs), side \(YZ\) (1 tick). So the two triangles with two angles (1 arc and 2 arcs) and a non - included side (with 2 ticks) are \(\triangle HIJ\) and \(\triangle RST\)? Wait, no, wait, the correct pair is \(\triangle HIJ\) and \(\triangle RST\)? Wait, no, let's re - examine. The AAS requires two angles and a non - included side. So in \(\triangle HIJ\), angle \(I\) (1 arc), angle \(J\) (2 arcs), and side \(HJ\) (the side opposite to angle \(I\)? No, the side between \(H\) and \(J\) is \(HJ\). The angles at \(I\) and \(J\), the non - included side is \(HI\)? No, maybe the correct pair is \(\triangle HIJ\) and \(\triangle RST\). Wait, actually, the correct congruence is \(\triangle HIJ\cong\triangle RST\) by AAS. Wait, no, let's check the angle - angle - side. \(\angle I\cong\angle T\) (1 arc), \(\angle J\cong\angle R\) (2 arcs), and \(HJ\cong SR\) (2 ticks, non - included side). So yes, \(\triangle HIJ\cong\triangle RST\) by AAS. Wait, but maybe I mixed up the triangles. Wait, the other option: \(\triangle HIJ\) and \(\triangle XYZ\)? No, because the side markings are different. So the congruence statement is \(\triangle HIJ\cong\triangle RST\).

Wait, no, maybe the correct answer is \(\triangle HIJ\cong\triangle RST\). Wait, let's do it properly. AAS: two angles and a non - included side. So in \(\triangle HIJ\): \(\angle I\) (1 arc), \(\angle J\) (2 arcs), side \(HJ\) (non - included between \(\angle I\) and \(\angle J\)? Wait, no, the side between \(\angle H\) and \(\angle J\) is \(HJ\). The angles \(\angle I\) and \(\angle J\), the non - included side is \(HI\)? No, I think I made a mistake. Let's start over.

The AAS theorem: If in two triangles, two angles and a non - included side of one triangle are equal to the corresponding two angles and non - included side of the other triangle, then the triangles are congruent.

Looking at the triangles:

- Triangle \(HIJ\): Angles at \(I\) (1 arc), \(J\) (2 arcs), side \(HJ\) (2 ticks).
- Triangle \(RST\): Angles at \(T\) (1 arc), \(R\) (2 arcs), side \(SR\) (2 ticks).
- Triangle \(XYZ\): Angles at \(X\) (1 arc), \(Z\) (2 arcs), side \(YZ\) (1 tick).

So, \(\angle I\cong\angle T\) (1 arc), \(\angle J\cong\angle R\) (2 arcs), and \(HJ\cong SR\) (2 ticks, non - included side). So by AAS, \(\triangle HIJ\cong\triangle RST\).

Answer:

\(\triangle HIJ\cong\triangle RST\)