Congruence Statements Worksheet - Practice identifying corresponding parts and completing congruence statements for triangles.

Let’s go step by step to solve each congruence statement. Remember: when two triangles are congruent, their corresponding parts (angles and sides) match up in the same order as the triangle names.

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A) Complete each congruence statement.

1) △DEF ≅ △XYZ
→ So D ↔ X, E ↔ Y, F ↔ Z
We’re asked for EF̄ — that’s side between E and F.
In △XYZ, the matching side is between Y and Z → ȲZ̄
✔ EF̄ ≅ ȲZ̄

2) △LMN ≅ △PQR
→ L ↔ P, M ↔ Q, N ↔ R
We’re asked for ∠M — that’s angle at vertex M.
Matching angle in △PQR is at vertex Q → ∠Q
✔ ∠M ≅ ∠Q

3) △ABC ≅ △FGH
→ A ↔ F, B ↔ G, C ↔ H
We’re asked for ∠F — that’s angle at vertex F.
Matching angle in △ABC is at vertex A → ∠A
✔ ∠F ≅ ∠A

4) △STU ≅ △XYZ
→ S ↔ X, T ↔ Y, U ↔ Z
We’re asked for ST̄ — side between S and T.
Matching side in △XYZ is between X and Y → XȲ
✔ ST̄ ≅ XȲ

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B) Complete each congruence statement using diagrams.

5) Diagram shows △QRS and △UVW with markings:
- QR has one tick, UV has one tick → QR ≅ UV
- RS has two ticks, VW has two ticks → RS ≅ VW
- QS has three ticks, UW has three ticks → QS ≅ UW
Also angles: ∠R and ∠V both have double arcs → ∠R ≅ ∠V
So correspondence: Q↔U, R↔V, S↔W
We’re asked for ∠Q → matches ∠U
✔ ∠Q ≅ ∠U

6) Diagram shows right triangles ABC and EDC:
- Right angles at B and D → ∠B ∠D
- AB and ED both have one tick → AB ≅ ED
- BC and DC both have two ticks → BC ≅ DC
- AC and EC both have three ticks → AC ≅ EC
So correspondence: A↔E, B↔D, C↔C
We’re asked for AB̄ → matches ĒD
✔ AB̄ ĒD̄

7) Diagram shows triangles EGF and HGI sharing point G.
Markings:
- EG and HG both have one tick → EG ≅ HG
- FG and IG both have two ticks → FG ≅ IG
- Angles at G: ∠EGF and ∠HGI are vertical angles → always congruent
So correspondence: E↔H, G↔G, F↔I
We’re asked for HĪ → that’s side from H to I.
In other triangle, matching side is from E to F → ĒF̄
Wait — let’s check:
Triangle EGF ≅ Triangle HGI?
Actually, looking at markings:
EG ≅ HG, FG ≅ IG, and included angle at G is shared/vertical → so SAS congruence.
Correspondence: E↔H, G↔G, F↔I
So side HI corresponds to side EF
But question asks for HĪ — which is same as IH̄, but direction doesn’t matter for congruence.
Matching side in first triangle is ĒF̄
✔ HĪ ≅ ĒF̄

8) Diagram shows quadrilateral split into triangles XYZ and RPQ? Wait — actually looks like triangles XYZ and RQP or something. Let’s look carefully.

Points: X, Y, Z form top triangle; R, P, Q form bottom. Markings:

- XY and RQ both have one tick → XY ≅ RQ
- YZ and QP both have two ticks → YZ ≅ QP
- XZ and RP both have three ticks → XZ ≅ RP
Angles: ∠Y and ∠Q both have double arcs → ∠Y ≅ ∠Q
So correspondence: X↔R, Y↔Q, Z↔P
We’re asked for ∠X → matches ∠R
✔ ∠X ≅ ∠R

9) Triangles QRS and BCD are congruent.
We need corresponding part of SQ̄.
First, write congruence: △QRS ≅ △BCD
So Q↔B, R↔C, S↔D
SQ̄ is side from S to Q → in second triangle, that’s from D to B → DB̄
Note: order matters for naming, but segment DB̄ is same as BD̄ — we usually write it as BD̄ unless specified.
But since SQ̄ goes from S to Q, and S↔D, Q↔B, then corresponding segment is from D to B → DB̄
However, in geometry, segments are unordered, so DB̄ ≡ BD̄. But to match direction, sometimes we keep order.
Looking at standard practice: if △QRS ≅ △BCD, then side QS corresponds to side DB.
Because Q↔B, S↔D → so QS ↔ BD
But the question says “corresponding part of SQ̄” — SQ̄ is same as QS̄, just reversed.
Still, corresponding segment is BD̄ (or DB̄).
Typically, we write it as BD̄ to match the order of the triangle name.
Since Q↔B and S↔D, then SQ̄ corresponds to DB̄.
But in most textbooks, they’d say BD̄. Let’s think:
If you map Q→B and S→D, then vector SQ maps to vector DB.
But for congruence of segments, direction doesn’t matter.
The answer should be BD̄ or DB̄ — either is acceptable, but let’s see the pattern.
In previous problems, we matched based on vertex correspondence.
S↔D, Q↔B → so segment SQ corresponds to segment DB.
But often written as BD̄. To be safe, let’s write it as BD̄ since B comes before D in the triangle name BCD.
Actually, no — the correspondence is fixed by the triangle naming.
△QRS ≅ △BCD means:
Q ↔ B
R ↔ C
S ↔ D
Therefore, side QS (from Q to S) corresponds to side BD (from B to D).
Side SQ is the same segment, so its corresponding part is still BD̄.
✔ BD̄

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Final Answer:
1) ȲZ̄
2) ∠Q
3) ∠A
4) XȲ
5) ∠U
6) ĒD̄
7) ĒF̄
8) ∠R
9) BD̄