Parallelogram ABCD: Comparing Triangles Formed by Diagonal AD

Triangle Congruence with Parallelogram Properties

Look at the parallelogram ABCD below.

AAABBBDDDCCC

What can be said about triangles ACD and ABD?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 What can be said about triangles ACD and ABD
00:03 The quadrilateral is a parallelogram according to the given
00:06 Opposite sides are equal in a parallelogram (S)
00:21 Opposite angles are equal in a parallelogram (A)
00:29 The triangles are congruent according to SAS theorem
00:49 Congruent triangles have equal areas
01:01 Congruent triangles have equal perimeters
01:19 Congruent triangles are necessarily similar
01:23 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the parallelogram ABCD below.

AAABBBDDDCCC

What can be said about triangles ACD and ABD?

2

Step-by-step solution

According to the side-angle-side theorem, the triangles are similar and coincide with each other:

AC = BD (Any pair of opposite sides of a parallelogram are equal)

Angle C is equal to angle B.

AB = CD (Any pair of opposite sides of the parallelogram are equal)

Therefore, all of the answers are correct.

3

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic
  • SAS Theorem: Use Side-Angle-Side to prove triangle congruence
  • Technique: Apply AC = BD and AB = CD from parallelogram properties
  • Check: Verify all corresponding sides and angles match exactly ✓

Common Mistakes

Avoid these frequent errors
  • Claiming triangles are only similar, not congruent
    Don't assume triangles formed by parallelogram diagonals are just similar = missing full congruence! Similar means same shape but different sizes. Always use parallelogram properties to prove exact congruence with equal sides and angles.

Practice Quiz

Test your knowledge with interactive questions

True or false?

One of the angles in a rectangle may be an acute angle.

FAQ

Everything you need to know about this question

How can triangles ACD and ABD be congruent when they look different?

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Congruent triangles have the same size and shape, but can be positioned differently! Triangle ABD can be rotated or reflected to match triangle ACD perfectly.

Why are all three answers (similar, equal areas, congruent) correct?

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When triangles are congruent, they automatically have the same areas and perimeters, and they're also similar! Congruence is the strongest relationship - it includes all the others.

What parallelogram properties help prove congruence?

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In parallelograms:

  • Opposite sides are equal: AC = BD and AB = CD
  • Opposite angles are equal: ∠C = ∠B
These give us the SAS conditions for congruence!

Do the triangles share any sides or angles?

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Yes! Both triangles share the diagonal AD as a common side. This shared side, plus the parallelogram properties, makes proving congruence much easier.

Can I use other congruence theorems besides SAS?

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Absolutely! You could also use SSS (all three sides equal) since AC = BD, AB = CD, and AD = AD. The parallelogram gives you multiple ways to prove congruence.

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