Question

write a statement that indicates that the triangles in each pair are congruent.
13)
14)
15)
16)

Explanation:

Step1: Identify corresponding vertices

For triangle congruence statements, we match the vertices of the congruent triangles. In problem 14, by observing the markings (equal sides and angles), we can see that vertex \(Y\) corresponds to vertex \(I\), vertex \(H\) corresponds to vertex \(H\) (common vertex in the congruence relation here based on side - angle - side or other congruence criteria from markings), and vertex \(J\) corresponds to vertex \(J\).

Step2: Write the congruence statement

Using the order of corresponding vertices, the congruence statement is \(\triangle YHI\cong\triangle IHJ\). But wait, no, looking again at the figure (assuming standard congruence notation where we list vertices in order of correspondence). Wait, actually, looking at the side - side - side or angle - side - angle etc. from the markings. Wait, no, for problem 14:
Looking at the two triangles formed by the diagonal \(HI\) (assuming the figure is a quadrilateral split into two triangles). Wait, no, actually, looking at the side markings:
If we consider the left - hand triangle as \(\triangle YHI\) and the right - hand triangle as \(\triangle IHJ\) (no, wait, no. Wait, actually, looking at the figure, the two triangles: one triangle has vertices \(Y\), \(H\), \(I\) (assuming \(I\) is a vertex) and the other has \(I\), \(H\), \(J\). But no, wait, the correct correspondence:
Looking at the side markings: \(YH = IH\) (two - tick marks), \(HI=HI\) (common side), \(IJ = YJ\) (three - tick marks). Also, angles: \(\angle YHI=\angle IHJ\) (from the angle markings). So by Side - Angle - Side (SAS) congruence criterion. The congruence statement is \(\triangle YHI\cong\triangle JHI\)

For problem 15:

Step1: Identify vertical angles

\(\angle XVW\) and \(\angle JVK\) are vertical angles, so \(\angle XVW=\angle JVK\)

Step2: Match sides and angles

By observing the side markings (equal sides) and using the Angle - Side - Angle (ASA) or Side - Angle - Side (SAS) congruence (depending on which sides and angles we consider, but from the side markings \(XW = JK\), \(XV = JV\), and \(\angle XVW=\angle JVK\)), the congruence statement is \(\triangle XWV\cong\triangle JKV\)

For problem 16:

Step1: Identify vertical angles

\(\angle RPQ\) and \(\angle WPV\) are vertical angles, so \(\angle RPQ=\angle WPV\)

Step2: Match sides and angles

By side - side - side (if all corresponding sides are equal from markings) or using the Angle - Side - Angle (since \(\angle R=\angle W\) (from markings), \(RQ = WV\) (side markings), \(\angle RPQ=\angle WPV\)), the congruence statement is \(\triangle RQP\cong\triangle WVP\)

Answer:

1. \(\triangle YHI\cong\triangle JHI\)
2. \(\triangle XWV\cong\triangle JKV\)
3. \(\triangle RQP\cong\triangle WVP\)