Question

state whether δabc and δaed are congruent. justify your answer.
options:
○ yes, by either sss or sas
○ yes, by sas only
○ yes, by sss only
○ no; there is not enough information to conclude that the triangles are congruent.
17 multiple choice 5 points
given the diagram below, which must be true?
options:
○ δfxs ≅ δgxt
○ δsxf ≅ δgxf
○ δxsf ≅ δxtg
○ δfxs ≅ δxgt

Explanation:

Question 1 (Triangles \( \triangle ABC \) and \( \triangle AED \))

Step 1: Analyze SSS (Side - Side - Side) Congruence

• From the diagram, we can see that \( AB = AE = 7\) (given by the length mark).
• \( AC = AD\) (marked with a single tick mark on each segment).
• \( BC = ED\) (marked with two and three tick marks respectively, indicating they are equal in length).
• By the SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. So \( \triangle ABC\cong\triangle AED\) by SSS.

Step 2: Analyze SAS (Side - Angle - Side) Congruence

• We know that \( AB = AE\), \( AC = AD\), and the included angle \( \angle BAC=\angle EAD\) (since the angles between the equal sides are equal, as indicated by the small angle marks).
• By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. So \( \triangle ABC\cong\triangle AED\) by SAS.

So the triangles are congruent by either SSS or SAS.

Step 1: Identify Equal Parts

• We have right angles at \( \angle S\) and \( \angle G\) (both are \( 90^{\circ}\)).
• \( SX = GX\) (marked with two tick marks on each segment).
• \( FX = TX\) (marked with one tick mark on each segment).
• \( FS = GT\) (marked with three tick marks on each segment).

Step 2: Analyze Triangle \( \triangle XSF \) and \( \triangle XTG \)

• In \( \triangle XSF\) and \( \triangle XTG\):
• \( SX = GX\) (given by tick marks).
• \( \angle XSF=\angle XTG = 90^{\circ}\) (right angles).
• \( FS = GT\) (given by tick marks).
• By the SAS (Side - Angle - Side) congruence criterion, \( \triangle XSF\cong\triangle XTG\).
• Let's check other options:
• For \( \triangle FXS\) and \( \triangle GXT\): The correspondence of vertices is not correct as per the equal sides and angles.
• For \( \triangle SXF\) and \( \triangle GXF\): The sides and angles do not match the congruence criteria.
• For \( \triangle FXS\) and \( \triangle XGT\): The correspondence of vertices and sides is incorrect.

Answer:

A. yes, by either SSS or SAS

Question 2 (Triangles in the second diagram)