Question

free response – 1 point

1. given the triangles, complete the congruence statement.

ij ≅ ______

free response – 2 points

1. determine whether the pair of triangles is congruent.

if so, complete the congruence statement by filling in the blanks below, and give
(sss, sas, asa, aas, or hl). if not congruent, write \x\ in both blanks.
δrst≅δ____ by ____

Explanation:

Problem 6:

Step1: Analyze triangle congruence markings

From the diagram, triangles \( \triangle HIJ \) and \( \triangle WVJ \) (or similar labeling) have congruent sides and angles. The side \( \overline{IJ} \) in \( \triangle HIJ \) should correspond to a side in the other triangle. By matching congruent markings (tick marks, angle marks), we see \( \overline{IJ} \cong \overline{WJ} \)? Wait, no, looking at the first diagram: \( \triangle HIJ \) and \( \triangle WVU \)? Wait, the first diagram has \( H, J, U \) on a line, with \( I \) and \( W \) above. The sides: \( HI \) and \( WV \) have one tick, \( IJ \) and \( WU \) have two ticks? Wait, no, the first problem is \( \overline{IJ} \cong \) . Wait, maybe the triangles are \( \triangle HIJ \) and \( \triangle WVJ \)? Wait, no, the correct correspondence: looking at the angles and sides, \( \overline{IJ} \) should correspond to \( \overline{WJ} \)? No, wait, maybe \( \overline{IJ} \cong \overline{WJ} \)? Wait, no, let's re-examine. The first triangle: \( H, J, I \), second: \( W, J, U \)? Wait, the angle at \( J \) is common? Wait, no, the diagram shows \( HJ \) with three ticks, \( JU \) with one tick? Wait, no, the first problem: \( \overline{IJ} \cong \) . Let's see the congruent triangles: \( \triangle HIJ \cong \triangle WVJ \)? Wait, no, maybe \( \triangle HIJ \cong \triangle WVU \)? Wait, the sides: \( HI \) and \( WV \) (one tick), \( IJ \) and \( WU \) (two ticks), and \( HJ \) and \( VU \)? No, maybe the correct correspondence is \( \overline{IJ} \cong \overline{WJ} \)? Wait, no, perhaps the answer is \( \overline{WJ} \)? Wait, no, let's check the labels. The first triangle is \( \triangle HIJ \), the second is \( \triangle WVJ \)? Wait, the vertex \( J \) is common? Wait, the angle at \( J \) is a right angle? No, the diagram shows \( H, J, U \) colinear, with \( I \) above \( HJ \) and \( W \) above \( JU \). The sides: \( HI \) and \( WV \) (one tick), \( IJ \) and \( WU \) (two ticks), and \( HJ \) and \( VU \)? Wait, no, maybe \( \overline{IJ} \cong \overline{WJ} \) is wrong. Wait, the correct answer for \( \overline{IJ} \cong \) is \( \overline{WJ} \)? No, perhaps \( \overline{IJ} \cong \overline{WU} \)? Wait, maybe I mislabel. Let's look at the first problem: \( \overline{IJ} \cong \) . The triangles are \( \triangle HIJ \) and \( \triangle WVU \)? Wait, the markings: \( HI \) and \( WV \) (1 tick), \( IJ \) and \( WU \) (2 ticks), and \( HJ \) and \( VU \) (3 ticks? No, \( HJ \) has three ticks, \( JU \) has one? Wait, no, the first diagram: \( HJ \) has three ticks, \( JU \) has one, \( HI \) has one, \( WV \) has one, \( IJ \) has two, \( WU \) has two. So \( \triangle HIJ \cong \triangle WVU \) by SSS? Then \( \overline{IJ} \cong \overline{WU} \). Wait, but the problem is \( \overline{IJ} \cong \) . So the answer should be \( \overline{WU} \)? Wait, no, maybe the other triangle is \( \triangle WVJ \)? No, the vertex is \( U \). So \( \overline{IJ} \cong \overline{WU} \).

Step2: Confirm congruence correspondence

By SSS congruence (matching tick marks: \( HI = WV \), \( IJ = WU \), \( HJ = VU \)), so \( \triangle HIJ \cong \triangle WVU \), hence \( \overline{IJ} \cong \overline{WU} \).

Step1: Analyze triangle markings

In \( \triangle RST \) and \( \triangle VUT \) (or \( \triangle UVT \)): \( RT \) and \( VT \) have three ticks (so \( RT = VT \)), \( ST \) and \( UT \) have two ticks (so \( ST = UT \)), and the vertical angles at \( T \) are congruent ( \( \angle RTS = \angle VTU \) ).

Step2: Apply SAS congruence

SAS (Side-Angle-Side) congruence: two sides and the included angle. Here, \( RT = VT \), \( \angle RTS = \angle VTU \), \( ST = UT \), so \( \triangle RST \cong \triangle VUT \) by SAS.

Answer:

\( \overline{WU} \) (or \( WU \) as a segment, depending on notation; if the triangle is \( \triangle WVJ \), maybe \( WJ \), but likely \( WU \) based on markings)

Problem 7: