Question

δrsq ~ ____
part a - determine if the triangles are similar. are the triangles similar?
part b - complete the similarity statement. if they are not similar, type not similar.

Explanation:

Step1: Check Proportions of Sides

First, identify the sides around the vertical angles (∠RSQ and ∠HSG, which are equal). The sides of △RSQ are RS = 49, SQ = 84, and RQ (not given directly, but we use the other triangle's sides: SG = 14, SH = 24, and GH (not given). Let's check the ratios of corresponding sides.

For the sides adjacent to the vertical angles:
Ratio of RS to SH: $\frac{RS}{SH} = \frac{49}{24}$? Wait, no, maybe the other pair. Wait, actually, the triangles are △RSQ and △HSG? Wait, no, let's see the segments. Wait, the lines are R - S - G (length RS = 49, SG = 14) and Q - S - H (length SQ = 84, SH = 24). So the sides around the vertical angles: RS and SG, SQ and SH? Wait, no, vertical angles are ∠RSQ and ∠GS H? Wait, no, ∠RSQ and ∠HSG are vertical angles (since S is the intersection of R - G and Q - H). So the triangles are △RSQ and △HSG? Wait, no, let's label the triangles. Let's see: △RSQ has vertices R, S, Q. △HSG has vertices H, S, G. Wait, maybe the correct correspondence is △RSQ ~ △HSG? Wait, no, let's check the ratios of the sides.

Wait, RS = 49, SG = 14; SQ = 84, SH = 24. Let's check the ratios: $\frac{RS}{SG} = \frac{49}{14} = \frac{7}{2}$, and $\frac{SQ}{SH} = \frac{84}{24} = \frac{7}{2}$. Oh, so the ratios are equal, and the included angle (vertical angles) is equal, so by SAS similarity, the triangles are similar. Now, the correspondence: R corresponds to H? Wait, no, wait: RS and SH? Wait, no, wait, RS is 49, SH is 24? No, wait, I messed up the segments. Wait, the line from R to G: R---S---G, with RS = 49, SG = 14. The line from Q to H: Q---S---H, with SQ = 84, SH = 24. So the two triangles are △RSQ (vertices R, S, Q) and △HSG (vertices H, S, G)? Wait, no, ∠RSQ and ∠HSG are vertical angles, so they are equal. Then, the sides adjacent to the angle: for △RSQ, sides RS = 49 and SQ = 84; for △HSG, sides HS = 24 and SG = 14. Wait, no, that can't be. Wait, maybe the other way: △RSQ and △GS H? Wait, no, let's recast. Wait, maybe the triangles are △RSQ and △HSG, but the ratios: $\frac{RS}{HS} = \frac{49}{24}$, $\frac{SQ}{SG} = \frac{84}{14} = 6$. No, that's not equal. Wait, I made a mistake. Wait, the correct segments: R to S is 49, S to G is 14, so RG = 49 + 14 = 63? No, the problem is about the triangles formed by the intersection. Wait, the two triangles are △RSQ and △HSG? Wait, no, the vertical angles are ∠RSQ and ∠GS H? Wait, no, S is the intersection of R - G and Q - H, so the vertical angles are ∠RSQ and ∠HSG (since R - S - G and Q - S - H, so ∠RSQ and ∠HSG are vertical angles). So the sides: in △RSQ, sides RS = 49, SQ = 84; in △HSG, sides HS = 24, SG = 14. Wait, no, that's not the right correspondence. Wait, maybe the triangles are △RSQ and △GS H? Wait, no, let's check the ratios of the sides that are proportional. Wait, 49 and 24? No, 14 and 49: 14/49 = 2/7; 24 and 84: 24/84 = 2/7. Ah! There we go. So SG = 14, RS = 49; SH = 24, SQ = 84. So the ratio of SG to RS is 14/49 = 2/7, and the ratio of SH to SQ is 24/84 = 2/7. So the sides around the vertical angles (∠SGH and ∠SRQ? Wait, no, vertical angles are ∠RSQ and ∠HSG. Wait, maybe the triangles are △RSQ and △GS H? Wait, no, let's see: ∠RSQ ≅ ∠HSG (vertical angles). Then, SG/RS = 14/49 = 2/7, and SH/SQ = 24/84 = 2/7. So by SAS similarity, △HSG ~ △RSQ? Wait, no, the correspondence: H corresponds to R, S corresponds to S, G corresponds to Q? No, wait, SG corresponds to RS, SH corresponds to SQ. So △HSG ~ △RSQ? Wait, no, the order: if SG/RS = SH/SQ, then the triangles are △HSG and △RSQ, with ∠HSG ≅ ∠RSQ (vertical angles). So the similarity statement is △HSG ~ △RSQ? Wait, but the problem says "△RSQ ~ ______". So we need to find the triangle similar to △RSQ. So from the ratios, SG/RS = 14/49 = 2/7, SH/SQ = 24/84 = 2/7, and ∠HSG = ∠RSQ (vertical angles). So the triangle similar to △RSQ is △HSG? Wait, no, let's check the vertices. Let's list the sides:

In △RSQ: sides RS = 49, SQ = 84, and RQ (hypotenuse? Not necessarily, but the included angle is ∠RSQ.

In △HSG: sides HS = 24, SG = 14, and HG (hypotenuse). The included angle is ∠HSG, which is equal to ∠RSQ (vertical angles). And the ratio of HS to RS? No, wait, SG/RS = 14/49 = 2/7, SH/SQ = 24/84 = 2/7. So the sides adjacent to the equal angles are SG and RS, SH and SQ. So the triangles are △HSG and △RSQ, with SG corresponding to RS, SH corresponding to SQ, and ∠HSG corresponding to ∠RSQ. Therefore, the similarity statement is △RSQ ~ △HSG? Wait, no, the order of the letters should correspond to the equal angles and proportional sides. So R corresponds to H? S corresponds to S? Q corresponds to G? No, that doesn't make sense. Wait, maybe I mixed up the segments. Let's re-express the lengths:

- RS = 49, SG = 14 ⇒ RS/SG = 49/14 = 7/2
- SQ = 84, SH = 24 ⇒ SQ/SH = 84/24 = 7/2

Ah! So RS/SG = SQ/SH = 7/2, and ∠RSQ = ∠GS H (vertical angles). Wait, ∠RSQ and ∠GS H? No, ∠RSQ and ∠HSG are vertical angles. Wait, maybe the triangles are △RSQ and △GS H? No, let's use the correct vertical angles: when two lines intersect, the vertical angles are equal. So lines R - G and Q - H intersect at S, so ∠RSQ and ∠HSG are vertical angles (equal). Then, the sides:

- For △RSQ: sides RS (length 49) and SQ (length 84)
- For △HSG: sides HS (length 24) and SG (length 14)

Wait, no, that's not the right pairs. Wait, RS is adjacent to ∠RSQ, and SG is adjacent to ∠HSG (since ∠HSG is the vertical angle, so SG is adjacent to ∠HSG). Similarly, SQ is adjacent to ∠RSQ, and SH is adjacent to ∠HSG. So RS and SG are adjacent to the vertical angles, SQ and SH are adjacent to the vertical angles. So the ratio of RS to SG is 49/14 = 7/2, and the ratio of SQ to SH is 84/24 = 7/2. So by SAS similarity, △RSQ ~ △GS H? Wait, no, the vertices: R corresponds to G, S corresponds to S, Q corresponds to H? Because RS (R to S) corresponds to GS (G to S), SQ (S to Q) corresponds to SH (S to H), and ∠RSQ corresponds to ∠GSH (vertical angles). Yes! So ∠RSQ ≅ ∠GSH (vertical angles), RS/GS = 49/14 = 7/2, SQ/SH = 84/24 = 7/2. Therefore, by SAS similarity, △RSQ ~ △GSH.

Step2: Confirm the Similarity Statement

So the triangle similar to △RSQ is △GSH (or △HSG? Wait, no, the order of the letters: R corresponds to G, S corresponds to S, Q corresponds to H. So △RSQ ~ △GSH.

Answer:

$\triangle GSH$ (or $\triangle HSG$? Wait, let's check the ratios again. Wait, RS = 49, GS = 14; SQ = 84, SH = 24. So RS/GS = 49/14 = 7/2, SQ/SH = 84/24 = 7/2. And ∠RSQ = ∠GSH (vertical angles). So the similarity statement is $\triangle RSQ \sim \triangle GSH$.