Square ABCD: Proving Triangle Congruence with Equal Segments FG and EH

Q: Why aren't the triangles congruent if they look the same?

Visual similarity doesn't guarantee congruence! We need mathematical proof using congruence theorems. Even though the triangles appear similar, we lack sufficient information to prove they're exactly the same size and shape.

Q: What additional information would make them congruent?

We'd need one more piece! For example: if $ DF = BF $ (making it SAS), or if $ DH = BG $ (giving us SSS), or another angle relationship (for ASA).

Q: How do I remember the congruence theorems?

SSS: Three sides equalSAS: Side-Angle-Side equalASA: Angle-Side-Angle equalAAS: Angle-Angle-Side equalRemember: You always need exactly three corresponding parts!

Q: Is there a difference between congruent and similar triangles?

Congruent triangles are exactly the same size and shape. Similar triangles have the same shape but can be different sizes. Congruence is stricter and requires more conditions to prove.

Triangle Congruence with Insufficient Information

ABCD is a square.

$FG=EH$

Are the two triangles shown below congruent?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABCD is a square.

$FG=EH$

Are the two triangles shown below congruent?

Step-by-step solution

When we observe the lines, we see that we have two pieces of information,

The line FG=EH
Angles C=B=90

But these two pieces of information alone are not enough to prove that they are congruent.

Therefore, in the absence of additional data, the answer is no.

Final Answer

Key Points to Remember

Essential concepts to master this topic

SSS/SAS/ASA Rule: Need three corresponding parts to prove triangle congruence
Given Information: Only two pieces: $FG = EH$ and right angles at B, D
Verification: Count known parts - only 2 out of 3 required ✓

Common Mistakes

Avoid these frequent errors

Assuming equal segments and right angles prove congruence
Don't think that FG = EH plus right angles = congruent triangles! Two pieces of information aren't enough - you need three corresponding parts (SSS, SAS, or ASA). Always check that you have sufficient conditions before concluding congruence.

Practice Quiz

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Is a square a trapezoid?

FAQ

Everything you need to know about this question

Why aren't the triangles congruent if they look the same?

Visual similarity doesn't guarantee congruence! We need mathematical proof using congruence theorems. Even though the triangles appear similar, we lack sufficient information to prove they're exactly the same size and shape.

Don't the right angles in the square help prove congruence?

The right angles at vertices B and D are helpful, but we need three corresponding parts to prove congruence. We have: $FG = EH$ and two right angles, but that's still not enough.

What additional information would make them congruent?

We'd need one more piece! For example: if $DF = BF$ (making it SAS), or if $DH = BG$ (giving us SSS), or another angle relationship (for ASA).

How do I remember the congruence theorems?

SSS: Three sides equal
SAS: Side-Angle-Side equal
ASA: Angle-Side-Angle equal
AAS: Angle-Angle-Side equal

Remember: You always need exactly three corresponding parts!

Is there a difference between congruent and similar triangles?

Congruent triangles are exactly the same size and shape. Similar triangles have the same shape but can be different sizes. Congruence is stricter and requires more conditions to prove.

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