Question

which diagrams show that the two triangles must be congruent?
i.
ii.
iii.
○ ii and iii only
○ i only
○ i and iii only
○ i and ii only

Explanation:

Step1: Analyze Diagram I

In Diagram I, we have two triangles with two sides marked as equal (the tick marks) and the included angle (the angle between the two sides) marked as equal (the angle symbol). By the Side - Angle - Side (SAS) 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 Diagram I shows congruent triangles.

Step2: Analyze Diagram II

In Diagram II, we have a quadrilateral with a diagonal, and we have two sides of each triangle (formed by the diagonal) marked as equal, and the included angle (at the vertex where the diagonal meets the base) marked as equal. By SAS, the two triangles ( \(\triangle PIT\) and \(\triangle PBT\)) are congruent. So Diagram II shows congruent triangles.

Step3: Analyze Diagram III

In Diagram III, we have two triangles with three sides? Wait, no. Let's look at the markings. The sides: we have some tick marks, but the way the triangles are formed, the congruence criteria (SSS, SAS, ASA, AAS) - let's check. The triangles here: we have two sides marked, but the included angle or the other sides - actually, in Diagram III, the triangles are part of a quadrilateral, but the markings don't satisfy a standard congruence criterion (like SAS, ASA, SSS). Wait, no, maybe I made a mistake. Wait, Diagram III: let's see, the quadrilateral has sides with tick marks, but the triangles formed by the diagonal - do we have enough information? Wait, no, in Diagram III, the markings are on the sides of the quadrilateral, but for the two triangles, we don't have the included angle or the third side to satisfy a congruence criterion. Wait, no, maybe my initial analysis was wrong. Wait, let's re - check.

Wait, Diagram I: SAS (two sides and included angle). Diagram II: SAS (two sides and included angle). Diagram III: Let's see, the triangles: do we have SSS? The sides: the tick marks - maybe not. Wait, maybe I misanalyzed Diagram III. Wait, the problem is asking which diagrams show that the two triangles must be congruent.

Wait, let's re - evaluate:

Diagram I: Two sides with tick marks (so equal) and the included angle (marked with the angle symbol) equal. So SAS - congruent.

Diagram II: Two sides (with two tick marks each) and the included angle (marked with the angle symbol) equal. So SAS - congruent.

Diagram III: Let's see, the triangles: the sides - the tick marks are on the sides of the quadrilateral, but the triangles formed by the diagonal: do we have three sides? No, the markings are on the sides of the quadrilateral, not the triangles. So Diagram III does not show congruent triangles.

So Diagrams I and II show congruent triangles.

Answer:

I and II only