Question

what are the corresponding parts of the figures below? select all that apply. (figures of two triangles: one with vertices k, l, j and angles 26°, 43°, 111°; the other with vertices v, x, w and angles 43°, 111°, 26°. options: ∠l ≅ ∠x, ∠k ≅ ∠v, ∠j ≅ ∠x, ∠k ≅ ∠w)

Explanation:

Step1: Analyze Triangle Angles

First, recall that in congruent triangles (implied by equal angle measures and side markings), corresponding angles are equal. Let's list the angles:
- Triangle \( KJL \): \( \angle K = 26^\circ \), \( \angle J = 111^\circ \), \( \angle L = 43^\circ \)
- Triangle \( VWX \): \( \angle V = 43^\circ \), \( \angle X = 111^\circ \), \( \angle W = 26^\circ \)

Step2: Match Corresponding Angles

- \( \angle L = 43^\circ \) and \( \angle V = 43^\circ \)? Wait, no, wait: Wait, \( \angle L = 43^\circ \), \( \angle V = 43^\circ \)? Wait, no, let's re - check. Wait, \( \angle K = 26^\circ \), \( \angle W = 26^\circ \); \( \angle J = 111^\circ \), \( \angle X = 111^\circ \); \( \angle L = 43^\circ \), \( \angle V = 43^\circ \)? Wait, the options:
- \( \angle L \cong \angle X \): \( \angle L = 43^\circ \), \( \angle X = 111^\circ \) → No, that's wrong. Wait, maybe I misread the triangles. Wait, the first triangle (KJL): angles are \( \angle K = 26^\circ \), \( \angle J = 111^\circ \), \( \angle L = 43^\circ \). The second triangle (VWX): \( \angle V = 43^\circ \), \( \angle X = 111^\circ \), \( \angle W = 26^\circ \). So:
- \( \angle K = 26^\circ \cong \angle W = 26^\circ \)
- \( \angle J = 111^\circ \cong \angle X = 111^\circ \)
- \( \angle L = 43^\circ \cong \angle V = 43^\circ \)? But the options given:
- \( \angle L \cong \angle X \): No, \( \angle L = 43^\circ \), \( \angle X = 111^\circ \). Wait, maybe the side markings indicate congruence. Wait, the side markings: in \( KJL \), \( KJ \) has 1 mark, \( JL \) has 2 marks, \( KL \) has 3 marks? Wait, no, the original problem's options: let's re - evaluate the options.
Wait, the options are \( \angle L \cong \angle X \), \( \angle K \cong \angle V \), \( \angle J \cong \angle X \), \( \angle K \cong \angle W \).
Let's re - check angles:
- \( \angle K = 26^\circ \), \( \angle W = 26^\circ \) → \( \angle K \cong \angle W \) (correct)
- \( \angle J = 111^\circ \), \( \angle X = 111^\circ \) → \( \angle J \cong \angle X \) (correct)
- Wait, \( \angle L = 43^\circ \), \( \angle V = 43^\circ \), but the option \( \angle K \cong \angle V \): \( \angle K = 26^\circ \), \( \angle V = 43^\circ \) → incorrect. \( \angle L \cong \angle X \): \( \angle L = 43^\circ \), \( \angle X = 111^\circ \) → incorrect. Wait, maybe the initial analysis of the triangles' correspondence is wrong. Maybe the triangles are named differently. Let's assume the triangles are congruent by the order of angles. So if we match the angles with the same measure:
- \( 26^\circ \) angles: \( \angle K \) and \( \angle W \)
- \( 111^\circ \) angles: \( \angle J \) and \( \angle X \)
- \( 43^\circ \) angles: \( \angle L \) and \( \angle V \)

But the options given:
- \( \angle L \cong \angle X \): No
- \( \angle K \cong \angle V \): No
- \( \angle J \cong \angle X \): Yes (\( \angle J = 111^\circ \), \( \angle X = 111^\circ \))
- \( \angle K \cong \angle W \): Yes (\( \angle K = 26^\circ \), \( \angle W = 26^\circ \))

Wait, maybe the first option \( \angle L \cong \angle X \) is wrong, but let's check the original problem's checked options. Wait, the user's image shows some options checked, but we need to find the correct corresponding parts.

Wait, let's start over. The key is that corresponding angles in congruent triangles (with equal angle measures and side congruence markings) are equal. So:

- \( \angle K = 26^\circ \), \( \angle W = 26^\circ \) → \( \angle K \cong \angle W \)
- \( \angle J = 111^\circ \), \( \angle X = 111^\circ \) → \( \angle J \cong \angle X \)
- \( \angle L = 43^\circ \), \( \angle V = 43^\circ \) → \( \angle L \cong \angle V \), but that's not an option. Wait, the options are \( \angle L \cong \angle X \), \( \angle K \cong \angle V \), \( \angle J \cong \angle X \), \( \angle K \cong \angle W \).

So the correct corresponding parts (angles) are:

- \( \angle J \cong \angle X \) (since both are \( 111^\circ \))
- \( \angle K \cong \angle W \) (since both are \( 26^\circ \))

Wait, the option \( \angle L \cong \angle X \): \( \angle L = 43^\circ \), \( \angle X = 111^\circ \) → incorrect. \( \angle K \cong \angle V \): \( \angle K = 26^\circ \), \( \angle V = 43^\circ \) → incorrect.

So the correct options are \( \angle J \cong \angle X \) and \( \angle K \cong \angle W \), and also, wait, maybe I made a mistake in angle - side correspondence. Let's check the side markings. The first triangle (KJL) has: \( KJ \) with 1 mark, \( JL \) with 2 marks, \( KL \) with 3 marks? Wait, no, the first triangle (KJL): side \( KJ \) (between \( K \) and \( J \)) has 1 mark, \( JL \) (between \( J \) and \( L \)) has 2 marks, \( KL \) (between \( K \) and \( L \)) has 3 marks? The second triangle (VWX): side \( VW \) (between \( V \) and \( W \)) has 1 mark, \( WX \) (between \( W \) and \( X \)) has 2 marks, \( XV \) (between \( X \) and \( V \)) has 3 marks? So the correspondence of sides: \( KJ \cong VW \), \( JL \cong WX \), \( KL \cong XV \). Then the corresponding angles: \( \angle K \cong \angle V \)? No, \( \angle K \) is between \( KJ \) and \( KL \), \( \angle V \) is between \( XV \) and \( VW \). Wait, maybe the angle correspondence is based on the order of the letters. If \( \triangle KJL \cong \triangle VWX \) (by side - angle - side or angle - side - angle), then \( \angle K \cong \angle V \), \( \angle J \cong \angle W \), \( \angle L \cong \angle X \)? No, this is getting confusing. Wait, the angle measures:

In \( \triangle KJL \): \( \angle K = 26^\circ \), \( \angle J = 111^\circ \), \( \angle L = 43^\circ \)

In \( \triangle VWX \): \( \angle V = 43^\circ \), \( \angle X = 111^\circ \), \( \angle W = 26^\circ \)

So:

- \( \angle K = 26^\circ \cong \angle W = 26^\circ \)
- \( \angle J = 111^\circ \cong \angle X = 111^\circ \)
- \( \angle L = 43^\circ \cong \angle V = 43^\circ \)

Now, looking at the options:

- \( \angle L \cong \angle X \): \( \angle L = 43^\circ \), \( \angle X = 111^\circ \) → No
- \( \angle K \cong \angle V \): \( \angle K = 26^\circ \), \( \angle V = 43^\circ \) → No
- \( \angle J \cong \angle X \): \( \angle J = 111^\circ \), \( \angle X = 111^\circ \) → Yes
- \( \angle K \cong \angle W \): \( \angle K = 26^\circ \), \( \angle W = 26^\circ \) → Yes

Answer:

The correct corresponding parts (angles) are \( \angle J \cong \angle X \) and \( \angle K \cong \angle W \) (and also, if we consider the other correct one, but based on the options, the correct options among the given are \( \angle J \cong \angle X \), \( \angle K \cong \angle W \). Wait, but in the image, some options are checked incorrectly. But based on angle measures:

• \( \angle J \cong \angle X \) (both \( 111^\circ \))
• \( \angle K \cong \angle W \) (both \( 26^\circ \))

So the correct options are:

\( \angle J \cong \angle X \), \( \angle K \cong \angle W \) (among the given options, these are the correct ones. If we have to choose from the given options in the image - related question, the correct corresponding angle pairs are \( \angle J \cong \angle X \) and \( \angle K \cong \angle W \))