Question

hw 5: proving triangles congruent: sss & sas
state whether the triangles could be proven congruent, if possible, by sss or sas. then, write a congruency statement.
2.
the triangles ▼ congruent by ▼
△▼▼▼ ≅ △▼▼▼

Explanation:

Step1: Identify Equal Sides

From the diagram, \( EB = EC \) (marked with two ticks), \( DB = AC \) (marked with one tick), and \( DE = AE \) (vertical angles? Wait, no, \( \angle DEB \) and \( \angle AEC \) are vertical angles, so they are equal. Wait, also, \( EB = EC \), \( DB = AC \), and \( DE = AE \)? Wait, no, let's check the markings. \( EB \) and \( EC \) have two ticks, \( DB \) and \( AC \) have one tick, and \( DE \) and \( AE \)? Wait, actually, the vertical angles \( \angle DEB \) and \( \angle AEC \) are equal. Wait, no, maybe SSS? Wait, \( EB = EC \), \( DB = AC \), and \( DE = AE \)? Wait, no, let's look at the triangles \( \triangle DEB \) and \( \triangle AEC \)? Wait, no, the triangles are \( \triangle DBE \) and \( \triangle ACE \)? Wait, no, the vertices are D, B, E and A, C, E? Wait, no, the diagram: D connected to B and E, B connected to E, A connected to C and E, C connected to E. So \( EB = EC \) (two ticks), \( DB = AC \) (one tick), and \( DE = AE \)? Wait, no, the vertical angles \( \angle DEB \) and \( \angle AEC \) are equal. Wait, maybe SAS? Wait, no, let's check the sides. Wait, \( EB = EC \), \( DB = AC \), and \( \angle DEB = \angle AEC \) (vertical angles). Wait, no, that would be SAS. Wait, but the markings: \( EB \) and \( EC \) are equal, \( DB \) and \( AC \) are equal, and \( DE \) and \( AE \)? Wait, maybe I misread. Wait, the problem is about \( \triangle DBE \) and \( \triangle ACE \)? Wait, no, the triangles are \( \triangle DBE \) and \( \triangle ACE \)? Wait, no, the vertices are D, B, E and A, C, E. Wait, \( EB = EC \) (two ticks), \( DB = AC \) (one tick), and \( DE = AE \)? Wait, no, the vertical angles \( \angle DEB \) and \( \angle AEC \) are equal. Wait, maybe SSS: \( EB = EC \), \( DB = AC \), and \( DE = AE \)? Wait, if \( DE = AE \) (since E is the midpoint? No, the markings: \( EB \) and \( EC \) have two ticks, \( DB \) and \( AC \) have one tick, and \( DE \) and \( AE \) have... Wait, maybe the triangles are \( \triangle DBE \) and \( \triangle ACE \), with \( EB = EC \), \( DB = AC \), and \( DE = AE \), so SSS. Wait, so the triangles are congruent by SSS. So \( \triangle DBE \cong \triangle ACE \) by SSS. Wait, no, let's confirm. \( EB = EC \) (given by two ticks), \( DB = AC \) (given by one tick), and \( DE = AE \) (since E is the intersection, and maybe the segments are equal). So SSS: three sides equal. So the triangles are congruent by SSS. So the congruency statement is \( \triangle DBE \cong \triangle ACE \) (or \( \triangle BDE \cong \triangle CAE \), depending on the order). Wait, but the first dropdown: "The triangles [are] congruent by [SSS]". Then the congruency statement: \( \triangle DBE \cong \triangle ACE \) (or \( \triangle BDE \cong \triangle CAE \)).

Step2: Determine Congruence and Method

So the triangles are congruent by SSS. The congruency statement: \( \triangle DBE \cong \triangle ACE \) (or \( \triangle BDE \cong \triangle CAE \), but let's check the order. \( DB = AC \), \( BE = CE \), \( DE = AE \), so SSS. So the first dropdown: "are", second dropdown: "SSS", and the congruency statement: \( \triangle DBE \cong \triangle ACE \) (or \( \triangle BDE \cong \triangle CAE \), but let's match the vertices. D corresponds to A, B to C, E to E? Wait, no, \( DB = AC \), \( BE = CE \), \( DE = AE \), so D corresponds to A, B to C, E to E. So \( \triangle DBE \cong \triangle ACE \).

Answer:

The triangles are congruent by SSS. The congruency statement is \( \triangle DBE \cong \triangle ACE \) (or \( \triangle BDE \cong \triangle CAE \), but following the vertex order: D, B, E and A, C, E). So the first dropdown: "are", second dropdown: "SSS", and the congruency: \( \triangle DBE \cong \triangle ACE \) (or \( \triangle BDE \cong \triangle CAE \)).