Geometry Proofs 9th Grade) : r/HomeworkHelp - Free Printable

Let’s work through each triangle congruence problem step by step.

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Problem 1: ΔADE ≅ ΔBEC

We are told these two triangles are congruent. That means their corresponding sides and angles match up exactly — in the same order as written.

So:

- A corresponds to B
- D corresponds to E
- E corresponds to C

Therefore:

Sides:
- AD ↔ BE
- DE ↔ EC
- AE ↔ BC

Angles:
- ∠A ↔ ∠B
- ∠D ↔ ∠E (that’s ∠ADE ↔ ∠BEC)
- ∠E ↔ ∠C (that’s ∠AED ↔ ∠BCE)

But looking at the worksheet, it’s asking you to fill in blanks for each triangle’s parts.

For ΔADE:
- Side AD → matches with side BE in ΔBEC
- Angle A → matches with angle B in ΔBEC
- Side DE → matches with side EC in ΔBEC
- Angle D → matches with angle E (∠BEC) in ΔBEC
- Side AE → matches with side BC in ΔBEC
- Angle E (∠AED) → matches with angle C (∠BCE) in ΔBEC

Similarly for ΔBEC, reverse the mapping.

The student already wrote:
- Side AD → correct
- Angle A → correct
- Side BC → that’s actually the matching side for AE, not AD — so this might be a mistake.
Wait — let’s recheck.

Actually, since ΔADE ≅ ΔBEC:

Correspondence:
A → B
D → E
E → C

So:

ΔADE sides:
AD → corresponds to BE
DE → corresponds to EC
AE → corresponds to BC

ΔADE angles:
∠A → ∠B
∠D → ∠E (i.e., ∠BEC)
∠E (∠AED) → ∠C (∠BCE)

So for the worksheet:

Under ΔADE:
- Side AD → should match with BE (but they wrote AD — maybe they meant to write what it matches? Wait, no — looking at the layout, it seems like they’re listing the parts of ΔADE and then what they correspond to in ΔBEC.

Actually, rereading the worksheet structure:

It says:

ΔADE Side ___ Side ___ Side ___
Angle ___ Angle ___ Angle ___
ΔBEC Side ___ Side ___ Side ___
Angle ___ Angle ___ Angle ___

And above it says ΔADE ≅ ΔBEC.

So likely, they want you to list the three sides and three angles of ΔADE, then the corresponding ones in ΔBEC.

But the way it's laid out, it might be pairing them.

Looking at the student’s answer:

They wrote under ΔADE:
- Side AD
- Angle A
Then under ΔBEC:
- Side BC
- Angle B

That suggests they are matching AD with BC? But according to congruence ΔADE ≅ ΔBEC, AD should match BE, not BC.

Wait — perhaps the student made an error.

Let me double-check the diagram.

In the first diagram: Two triangles sharing vertex E, with AD and BC marked as equal (single tick marks). Also, vertical angles at E are equal.

So likely, the congruence is based on SAS or ASA.

But the problem states ΔADE ≅ ΔBEC — so we must go by the letter order.

Thus:

A ↔ B
D ↔ E
E ↔ C

So:

Side AD ↔ Side BE
Side DE ↔ Side EC
Side AE ↔ Side BC

Angle at A ↔ Angle at B
Angle at D ↔ Angle at E (in triangle BEC, that’s angle at E, which is ∠BEC)
Angle at E (in ADE, ∠AED) ↔ Angle at C (in BEC, ∠BCE)

So for the worksheet:

For ΔADE:
- Side 1: AD → corresponds to BE in ΔBEC
- Side 2: DE → corresponds to EC in ΔBEC
- Side 3: AE → corresponds to BC in ΔBEC

- Angle 1: ∠A → corresponds to ∠B in ΔBEC
- Angle 2: ∠D → corresponds to ∠BEC in ΔBEC
- Angle 3: ∠AED → corresponds to ∠BCE in ΔBEC

For ΔBEC:
- Side 1: BE → corresponds to AD in ΔADE
- Side 2: EC → corresponds to DE in ΔADE
- Side 3: BC → corresponds to AE in ΔADE

- Angle 1: ∠B → corresponds to ∠A in ΔADE
- Angle 2: ∠BEC → corresponds to ∠D in ΔADE
- Angle 3: ∠BCE → corresponds to ∠AED in ΔADE

The student wrote:
- Side AD (correct for ΔADE)
- Angle A (correct)
- Side BC (this is for AE, not AD — so if they’re listing sides of ΔADE, BC is not a side of ΔADE; it’s a side of ΔBEC. So probably they meant to write the corresponding side, but the blank is under ΔADE, so it should be a side of ΔADE.

I think there’s confusion in how the worksheet is structured.

Looking again: The worksheet has:

ΔADE Side ___ Side ___ Side ___
Angle ___ Angle ___ Angle ___
ΔBEC Side ___ Side ___ Side ___
Angle ___ Angle ___ Angle ___

And the congruence is given as ΔADE ≅ ΔBEC.

So likely, they want you to list the three sides and three angles of ΔADE in the top row, and the corresponding parts of ΔBEC in the bottom row.

But the student wrote under ΔADE: Side AD, Angle A — which are parts of ΔADE, good.

Then under ΔBEC: Side BC, Angle B — which are parts of ΔBEC, good.

But they didn’t fill all blanks.

Perhaps the exercise is to identify corresponding parts.

Given that, for Problem 1, the correct corresponding pairs are:

AD ↔ BE
DE ↔ EC
AE ↔ BC
∠A ↔ ∠B
∠D ↔ ∠BEC
∠AED ↔ ∠BCE

Now, Problem 2: ΔADF ≅ ΔBCE

Diagram: Rectangle ABCD, with points E and F on DC, and lines from A to F and B to E intersecting at G.

Congruence: ΔADF ≅ ΔBCE

So correspondence:

A ↔ B
D ↔ C
F ↔ E

So:

Sides:
AD ↔ BC
DF ↔ CE
AF ↔ BE

Angles:
∠A ↔ ∠B
∠D ↔ ∠C
∠F ↔ ∠E

Since it’s a rectangle, AD = BC, and if DF = CE, then yes.

Problem 3: ΔABE ≅ ΔCBD

Diagram: Triangle ABC, with D and E on AC, and BD and BE drawn.

Congruence: ΔABE ≅ ΔCBD

Correspondence:

A ↔ C
B ↔ B
E ↔ D

So:

Sides:
AB ↔ CB
BE ↔ BD
AE ↔ CD

Angles:
∠A ↔ ∠C
∠B (in ABE, that’s ∠ABE) ↔ ∠B (in CBD, that’s ∠CBD)
∠E ↔ ∠D

Note: Vertex B is common, but in different positions.

In ΔABE, vertices A,B,E
In ΔCBD, vertices C,B,D

So A→C, B→B, E→D

Thus:

Side AB ↔ Side CB
Side BE ↔ Side BD
Side AE ↔ Side CD

Angle at A ↔ Angle at C
Angle at B (∠ABE) ↔ Angle at B (∠CBD)
Angle at E (∠AEB) ↔ Angle at D (∠CDB)

Now, to provide the final answer as per the worksheet format.

Since the user didn't specify which problem to solve, and the image has three problems, I'll assume we need to complete all three as per the congruence statements.

But the instruction says "solve the problem", and the image shows three separate problems.

Perhaps the main task is to fill in the blanks for each.

Given that, I'll provide the completed correspondences for each.

But to keep it simple, since the student might just need the answers for the blanks, I'll list them.

However, the response format requires a Final Answer section.

Perhaps for each problem, list the corresponding parts.

But to avoid overcomplicating, I'll focus on the first problem as an example, but the user might expect all.

Looking back at the user's message, they said "solve the problem", and the image has three, but perhaps it's one worksheet with three parts.

I think I should provide the answers for all three.

But to fit the format, I'll summarize.

Final Answer for Problem 1 (ΔADE ≅ ΔBEC):

Corresponding parts:
- Side AD corresponds to Side BE
- Side DE corresponds to Side EC
- Side AE corresponds to Side BC
- Angle A corresponds to Angle B
- Angle D corresponds to Angle BEC
- Angle AED corresponds to Angle BCE

For Problem 2 (ΔADF ≅ ΔBCE):

- Side AD corresponds to Side BC
- Side DF corresponds to Side CE
- Side AF corresponds to Side BE
- Angle A corresponds to Angle B
- Angle D corresponds to Angle C
- Angle F corresponds to Angle E

For Problem 3 (ΔABE ≅ ΔCBD):

- Side AB corresponds to Side CB
- Side BE corresponds to Side BD
- Side AE corresponds to Side CD
- Angle A corresponds to Angle C
- Angle ABE corresponds to Angle CBD
- Angle AEB corresponds to Angle CDB

But since the worksheet has blanks, perhaps they want the letters filled in.

For instance, in Problem 1, under ΔADE, Side: AD, then the corresponding side in ΔBEC is BE, but the blank is for the part of ΔADE, so it should be AD, DE, AE for sides, and angles A, D, E for angles.

I think I'm overcomplicating.

Let me look at the student's attempt:

They wrote for ΔADE: Side AD, Angle A
For ΔBEC: Side BC, Angle B

Which is incorrect because AD corresponds to BE, not BC.

So for accuracy, I'll provide the correct correspondence.

Perhaps the final answer is the list of corresponding parts.

To adhere to the format, I'll box the key correspondences.

But since there are three problems, I'll state them clearly.

Final Answer:

For ΔADE ≅ ΔBEC:
AD ↔ BE, DE ↔ EC, AE ↔ BC, ∠A ↔ ∠B, ∠D ↔ ∠BEC, ∠AED ↔ ∠BCE

For ΔADF ≅ ΔBCE:
AD ↔ BC, DF ↔ CE, AF ↔ BE, ∠A ↔ ∠B, ∠D ↔ ∠C, ∠F ↔ ∠E

For ΔABE ≅ ΔCBD:
AB ↔ CB, BE ↔ BD, AE ↔ CD, ∠A ↔ ∠C, ∠ABE ↔ ∠CBD, ∠AEB ↔ ∠CDB

This should be the accurate solution.