Question

1. are the triangles similar?

yes or no?
what is the similarity ratio? if there is no ratio, type none
triangles are similar by
complete the similarity statement. if not similar, type not similar. △qxz - △

Explanation:

Step1: Check Proportional Sides and Vertical Angles

First, identify the sides and the vertical angle (∠Q is common or vertical? Wait, ∠XQZ and ∠NQR are vertical angles, so they are equal. Now check the ratios of the sides:

For sides around ∠Q:
- \( \frac{XQ}{NQ} = \frac{120}{30} = 4 \)
- \( \frac{ZQ}{RQ} = \frac{60}{15} = 4 \)

Wait, wait, maybe I mixed up. Wait, the segments: XQ is 120, NQ is 30; ZQ is 60, RQ is 15. Wait, no, maybe the triangles are △QXZ and △QNR? Wait, no, let's re-express:

Wait, the sides: XQ = 120, NQ = 30; ZQ = 60, RQ = 15; and the included angle ∠XQZ and ∠NQR are vertical angles, so equal. So by SAS similarity, if the ratios of two sides are equal and included angle equal, then similar.

Wait, \( \frac{XQ}{NQ} = \frac{120}{30} = 4 \), \( \frac{ZQ}{RQ} = \frac{60}{15} = 4 \). Wait, no, maybe RQ is 15, ZQ is 60? Wait, no, the diagram: Q to R is 15, Q to Z is 60; Q to X is 120, Q to N is 30. So the sides adjacent to ∠Q (vertical angles) are XQ=120, NQ=30; ZQ=60, RQ=15. So \( \frac{XQ}{NQ} = \frac{120}{30} = 4 \), \( \frac{ZQ}{RQ} = \frac{60}{15} = 4 \). So the ratios are equal, and the included angle (∠XQZ and ∠NQR) is equal (vertical angles). So by SAS similarity, the triangles are similar.

Wait, but wait, maybe the ratio is \( \frac{XQ}{NQ} = 4 \), \( \frac{ZQ}{RQ} = 4 \), so the similarity ratio is 4? Wait, no, maybe I got the triangles reversed. Wait, △QXZ and △QNR? Wait, no, let's check the other way: \( \frac{NQ}{XQ} = \frac{30}{120} = \frac{1}{4} \), \( \frac{RQ}{ZQ} = \frac{15}{60} = \frac{1}{4} \). So the ratio of △QNR to △QXZ is \( \frac{1}{4} \), or △QXZ to △QNR is 4.

Wait, let's confirm the sides:

- XQ = 120, NQ = 30 ⇒ \( \frac{XQ}{NQ} = 4 \)
- ZQ = 60, RQ = 15 ⇒ \( \frac{ZQ}{RQ} = 4 \)

Included angle: ∠XQZ and ∠NQR are vertical angles, so equal. Therefore, by SAS similarity, △QXZ ~ △QNR (wait, no, the vertices: Q is common? Wait, no, ∠XQZ and ∠NQR are vertical angles, so Q is the vertex. So △QXZ and △QNR: Q is the vertex, X and Z are the other vertices of one triangle, N and R are the other of the other. So the sides: QX=120, QN=30; QZ=60, QR=15; included angle ∠XQZ = ∠NQR. So the ratio of QX/QN = 120/30 = 4, QZ/QR = 60/15 = 4. So SAS similarity, so the triangles are similar.

Step2: Determine Similarity Ratio

The similarity ratio is the ratio of corresponding sides. So △QXZ to △QNR: QX/QN = 4, so ratio is 4? Wait, no, wait QN is 30, QX is 120, so QX/QN = 4, so the ratio of △QXZ to △QNR is 4:1, or 4. Wait, but let's check the other way: QN/QX = 30/120 = 1/4, QR/QZ = 15/60 = 1/4. So if △QNR ~ △QXZ, the ratio is 1/4, but if △QXZ ~ △QNR, ratio is 4. Wait, the problem says △QXZ ~ △?, so we need to find the corresponding triangle.

Wait, the sides: QX=120, QZ=60; QN=30, QR=15. So QX corresponds to QN? No, wait, maybe the triangles are △QXZ and △QNR, with QX corresponding to QN, QZ corresponding to QR, and XZ corresponding to NR. So the ratio of QX/QN = 120/30 = 4, QZ/QR = 60/15 = 4, so the ratio is 4.

Step3: Similarity Criterion

Since two sides are in proportion (ratio 4) and included angle (vertical angles) is equal, so by SAS (Side-Angle-Side) similarity criterion.

Step4: Similarity Statement

△QXZ ~ △QNR? Wait, no, wait the vertices: Q is common, X corresponds to N, Z corresponds to R? Wait, no, because QX/QN = 4, QZ/QR = 4, so X corresponds to N, Z corresponds to R, Q corresponds to Q. So △QXZ ~ △QNR? Wait, no, maybe △QXZ ~ △QNR? Wait, no, let's check the order. The similarity statement should have corresponding vertices. So QX corresponds to QN, QZ corresponds to QR, so X corresponds to N, Z corresponds to R, Q corresponds to Q. So △QXZ ~ △QNR? Wait, but let's confirm the angles. The included angle is ∠XQZ and ∠NQR, which are vertical angles, so equal. So yes, SAS similarity.

Wait, but maybe I made a mistake in the side ratios. Wait, 120/30 = 4, 60/15 = 4, so yes, the ratios are equal. So the triangles are similar by SAS similarity, ratio 4, and the similarity statement is △QXZ ~ △QNR? Wait, no, wait the other triangle: △QNR? Wait, the triangle with vertices N, Q, R: NQ=30, QR=15, NR? And △QXZ: QX=120, QZ=60, XZ? So yes, the ratio is 4 (120/30 = 4, 60/15 = 4), so similarity ratio 4, SAS similarity, and △QXZ ~ △QNR? Wait, no, maybe △QXZ ~ △QNR? Wait, no, the order: Q is the common vertex, X and Z are the other two of one triangle, N and R are the other two of the other. So △QXZ ~ △QNR.

Wait, but let's re-express:

- QX = 120, QN = 30 ⇒ QX/QN = 4
- QZ = 60, QR = 15 ⇒ QZ/QR = 4
- ∠XQZ = ∠NQR (vertical angles)

Thus, by SAS similarity, △QXZ ~ △QNR, similarity ratio 4.

Answer:

yes
4
SAS (Side - Angle - Side) similarity
△QNR