"Similar Triangles Relay Race" worksheet featuring four problems where students use similar triangles to find missing values, with answers provided at the bottom.

This is a “Similar Triangles Relay Race” — meaning you must solve the problems in order (1 → 2 → 3 → 4), and each problem uses the answer from the previous one to fill in a blank (the “□” symbol).

Let’s solve them step by step.

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🔹 Problem 1: Given ΔABC ~ ΔDEF, find x.

Triangles ABC and DEF are similar. So corresponding sides are proportional.

From the diagram:

- AB = 9, AC = x - 2
- DF = x + 6, DE = 18

We need to match corresponding sides. Since the triangles are named ΔABC ~ ΔDEF, the correspondence is:

> A ↔ D, B ↔ E, C ↔ F

So:

- AB corresponds to DE → 9 ↔ 18
- AC corresponds to DF → (x - 2) ↔ (x + 6)

Set up proportion:

AB / DE = AC / DF

→ 9 / 18 = (x - 2) / (x + 6)

Simplify 9/18 = 1/2:

→ 1/2 = (x - 2)/(x + 6)

Cross-multiply:

→ 1*(x + 6) = 2*(x - 2)

→ x + 6 = 2x - 4

→ 6 + 4 = 2x - x

→ x = 10

✔ Answer for Problem 1: x = 10

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🔹 Problem 2: Given ΔLMN ~ ΔJKN, find x.

Note: The square box “□” in side LJ should be filled with the answer from Problem 1 → 10

So, side LJ = 2x + 10

Given:

- LM = 63, MN = 45
- JK = 35, KN = 45? Wait — look at diagram:

Actually, from diagram:

- In ΔLMN: LM = 63, LN = ? (but we have LJ = 2x + □, and JN = 45)
- In ΔJKN: JK = 35, KN = 45, JN = 45? That can’t be — let’s re-express.

Actually, since ΔLMN ~ ΔJKN, and J is on LN, K is on MN.

Correspondence: L ↔ J, M ↔ K, N ↔ N (common vertex)

So:

- LM ↔ JK → 63 ↔ 35
- LN ↔ JN → (LJ + JN) ↔ JN → (2x + 10 + 45) ↔ 45? Wait — that doesn’t make sense.

Wait — perhaps better to use:

Since ΔLMN ~ ΔJKN, and they share angle N, so:

Corresponding sides:

- LM / JK = LN / JN = MN / KN

We know:

- LM = 63, JK = 35
- MN = 45, KN = 20? Wait — diagram says:

In ΔLMN: LM=63, MN=45, and LN is split into LJ and JN, with JN=45

In ΔJKN: JK=35, KN=20, JN=45

Wait — that would mean:

ΔLMN has sides: LM=63, MN=45, LN = LJ + JN = (2x + 10) + 45 = 2x + 55

ΔJKN has sides: JK=35, KN=20, JN=45

But 35, 20, 45 — check if proportional to 63, 45, ?

Check ratio: LM/JK = 63/35 = 9/5

Then MN/KN = 45/20 = 9/4 → not equal → contradiction?

Wait — maybe I misread the diagram.

Looking again — in ΔJKN, side KN is labeled 20, and side JN is labeled 45.

In ΔLMN, side MN is labeled 45, and side LN is split into LJ = 2x + □ and JN = 45.

So, since ΔLMN ~ ΔJKN, and angle N is common, the correspondence is:

L ↔ J, M ↔ K, N ↔ N

So:

- LM ↔ JK → 63 ↔ 35
- MN ↔ KN → 45 ↔ 20
- LN ↔ JN → (LJ + JN) ↔ JN → (2x + 10 + 45) ↔ 45

Wait — that would imply:

Ratio of similarity = LM / JK = 63 / 35 = 9/5

Also, MN / KN = 45 / 20 = 9/4 → not same → inconsistency?

Hmm… perhaps the correspondence is different.

Alternative: Maybe ΔLMN ~ ΔJKN with correspondence L↔J, M↔N, N↔K? That seems unlikely.

Wait — perhaps it's ΔLMN ~ ΔJK N — but JKN is triangle with vertices J, K, N.

Another possibility: The side labeled “20” is MK? No — diagram shows “20” next to segment from K to N.

Wait — perhaps I misread the labels.

Let me assume that the correct correspondence gives us:

LM / JK = LN / JN

Because both triangles share angle N, and L-J-N is straight line, M-K-N is straight line.

So:

LM = 63, JK = 35

LN = LJ + JN = (2x + 10) + 45 = 2x + 55

JN = 45

So:

63 / 35 = (2x + 55) / 45

Simplify 63/35 = 9/5

So:

9/5 = (2x + 55)/45

Multiply both sides by 45:

9/5 * 45 = 2x + 55

→ 9 * 9 = 2x + 55

→ 81 = 2x + 55

→ 2x = 26

→ x = 13

✔ Answer for Problem 2: x = 13

(And this matches the given answer key!)

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🔹 Problem 3: Given ΔPQT ~ ΔRQS, find x.

The square box “□” in side QS should be filled with the answer from Problem 2 → 13

So, QS = x + 13

Given:

- PT = 52, QT = 8x - 1
- RS = 24, QS = x + 13

Triangles PQT and RQS are similar.

Looking at diagram: They share angle Q, and points P-Q-R and T-Q-S are lines.

So likely correspondence: P ↔ R, Q ↔ Q, T ↔ S

Thus:

- PQ ↔ RQ
- QT ↔ QS
- PT ↔ RS

So set up proportion using PT / RS = QT / QS

PT = 52, RS = 24

QT = 8x - 1, QS = x + 13

So:

52 / 24 = (8x - 1) / (x + 13)

Simplify 52/24 = 13/6

So:

13/6 = (8x - 1)/(x + 13)

Cross-multiply:

13(x + 13) = 6(8x - 1)

→ 13x + 169 = 48x - 6

→ 169 + 6 = 48x - 13x

→ 175 = 35x

→ x = 5

✔ Answer for Problem 3: x = 5

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🔹 Problem 4: Given ΔWXY ~ ΔUTV, find x.

The square box “□” in side TV should be filled with the answer from Problem 3 → 5

So, TV = 5x - 3

Given:

- WY = 10, WX = x + 5
- UV = 30, TV = 5x - 3

Triangles WXY ~ UTV

Correspondence: W ↔ U, X ↔ T, Y ↔ V

So:

- WY ↔ UV → 10 ↔ 30
- WX ↔ UT → (x + 5) ↔ (5x - 3)

Set up proportion:

WY / UV = WX / UT

→ 10 / 30 = (x + 5) / (5x - 3)

Simplify 10/30 = 1/3

So:

1/3 = (x + 5)/(5x - 3)

Cross-multiply:

1*(5x - 3) = 3*(x + 5)

→ 5x - 3 = 3x + 15

→ 5x - 3x = 15 + 3

→ 2x = 18

→ x = 9

✔ Answer for Problem 4: x = 9

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## ✔ Final Answers:

1. x = 10
2. x = 13
3. x = 5
4. x = 9

Matches the provided answer key!

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## 🧠 Summary of Strategy:

- Use similar triangles → corresponding sides proportional
- Fill in the “□” in each subsequent problem with the previous problem’s answer
- Set up proportions carefully based on vertex correspondence
- Solve linear equations

Great relay race! Each problem depends on the last — perfect for group work or passing papers around.