Congruent Triangles ABC and CDA: Finding Equal Angles

Q: How do I know which vertices correspond in congruent triangles?

Look at the order of letters in the congruence statement! In $ ABC \cong CDA $, the first letters match (A↔C), second letters match (B↔D), and third letters match (C↔A).

Q: Why does angle BAC equal angle DCA and not angle CAD?

The middle letter shows the vertex of the angle! Angle BAC has vertex A, and since A corresponds to C, we need the angle at vertex C, which is angle DCA.

Q: What if the triangles are written in different orders?

The order matters! $ ABC \cong CDA $ is different from $ ABC \cong ACD $. Always match vertices in the exact order they're written in the congruence statement.

Q: Can I just look at the diagram to find equal angles?

No! The diagram might be misleading or not drawn to scale. Always use the congruence statement to determine which vertices correspond, then find the matching angles.

Q: How do I write angle names correctly?

Use three letters with the vertex in the middle. For angle at vertex A between rays AB and AC, write it as $ \angle BAC $ or $ \angle CAB $.

Congruent Triangle Correspondence with Vertex Matching

Triangles ABC and CDA are congruent.

Which angle is equal to angle BAC?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 In two triangles that overlap, which angle matches angle B. A. C?

00:12 We know the triangles overlap based on the info given.

00:17 Let's look at all corresponding angles step by step.

00:40 Remember, when an angle uses three letters, the middle one is the corner point.

00:52 So, we'll now find matching angles for these triangles.

00:57 And that's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Triangles ABC and CDA are congruent.

Which angle is equal to angle BAC?

Step-by-step solution

We observe the order of the letters in the congruent triangles and write the matches (from left to right).

$ABC=CDA$

That is:

Angle A is equal to angle C.

Angle B is equal to angle D.

Angle C is equal to angle A.

From this, it is deduced that angle BAC (where the letter A is in the middle) is equal to angle C — that is, to angle DCA (where the letter C is in the middle).

Final Answer

Key Points to Remember

Essential concepts to master this topic

Vertex Order: In $ABC \cong CDA$ , match A↔C, B↔D, C↔A
Angle Matching: Angle BAC equals angle DCA because A and C correspond
Check Position: Middle letter shows vertex angle: BAC has A at vertex ✓

Common Mistakes

Avoid these frequent errors

Matching triangles by position instead of vertex order
Don't just look at which angles appear similar visually = wrong correspondences! The diagram position doesn't determine congruence relationships. Always match vertices based on the exact order given in the congruence statement ABC ≅ CDA.

Practice Quiz

Test your knowledge with interactive questions

Are the triangles in the figure congruent?

If so, according to which theorem?

FAQ

Everything you need to know about this question

How do I know which vertices correspond in congruent triangles?

Look at the order of letters in the congruence statement! In $ABC \cong CDA$ , the first letters match (A↔C), second letters match (B↔D), and third letters match (C↔A).

Why does angle BAC equal angle DCA and not angle CAD?

The middle letter shows the vertex of the angle! Angle BAC has vertex A, and since A corresponds to C, we need the angle at vertex C, which is angle DCA.

What if the triangles are written in different orders?

The order matters! $ABC \cong CDA$ is different from $ABC \cong ACD$ . Always match vertices in the exact order they're written in the congruence statement.

Can I just look at the diagram to find equal angles?

No! The diagram might be misleading or not drawn to scale. Always use the congruence statement to determine which vertices correspond, then find the matching angles.

How do I write angle names correctly?

Use three letters with the vertex in the middle. For angle at vertex A between rays AB and AC, write it as $\angle BAC$ or $\angle CAB$ .

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