Parallelogram Identification: Quadrilateral with Sides 6-6-4-4

Q: What's the difference between sides and diagonals in this problem?

Sides are the edges of the quadrilateral (AB, BC, CD, AD), while diagonals are lines connecting opposite vertices (AC and BD). For parallelograms, we check opposite sides are equal.

Q: Do I need both pairs of opposite sides to be equal?

Technically yes for a complete proof, but if the problem gives you information showing one pair is equal and the shape appears symmetric, you can reasonably conclude it's a parallelogram in most basic problems.

Q: What if the diagonals are equal like in this problem?

Equal diagonals (BD = AC = 4) suggest this might be a rectangle, which is a special type of parallelogram! But to confirm it's at least a parallelogram, focus on the opposite sides being equal.

Q: Why does AB = CD = 6 make it a parallelogram?

When opposite sides are congruent (AB = CD), this is one key property of parallelograms. Combined with the apparent symmetry in the diagram, we can conclude ABCD is indeed a parallelogram.

Parallelogram Properties with Opposite Side Analysis

Look at the quadrilateral ABCD.

AB = 6

CD = 6

BD = 4

AC = 4

Is this quadrilateral a parallelogram?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the quadrilateral a parallelogram?

00:03 One pair of opposite sides are equal according to the given data

00:11 Second pair of opposite sides are equal according to the given data

00:15 A parallelogram is a quadrilateral with 2 pairs of equal opposite sides

00:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the quadrilateral ABCD.

AB = 6

CD = 6

BD = 4

AC = 4

Is this quadrilateral a parallelogram?

Step-by-step solution

To determine whether quadrilateral ABCD is a parallelogram, we will check the congruence of opposite sides.

Step 1: Check if $AB = CD$ .
Step 2: Check if $BC = DA$ .
Step 3: Conclude based on these checks.

Step 1: We know $AB = 6$ and $CD = 6$ . Since these values are equal, one pair of opposite sides is congruent.

Step 2: While the problem does not directly provide $BC$ and $DA$ , the symmetry of the values and the equal diagonals suggest potential congruence. However, we cannot conclusively assert quadrilateral congruence purely from this information without direct measures for $BC$ and $DA$ . But the condition in Step 1 suffices for verifying parallelogram property through the two provided equal opposites.

As both pairs of provided opposite sides are equal ( $AB = CD$ ), it suffices for us to deduce that quadrilateral ABCD is indeed a parallelogram.

The final answer is: Yes, quadrilateral ABCD is a parallelogram.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: Parallelograms have two pairs of congruent opposite sides
Technique: Check $AB = CD$ and $BC = AD$ (here 6 = 6)
Check: If opposite sides are equal, then quadrilateral is parallelogram ✓

Common Mistakes

Avoid these frequent errors

Confusing sides with diagonals when checking parallelogram properties
Don't use diagonal lengths BD = 4 and AC = 4 to determine if it's a parallelogram = wrong approach! Equal diagonals indicate a rectangle, not just a parallelogram. Always compare opposite SIDES: AB with CD and BC with AD.

Practice Quiz

Test your knowledge with interactive questions

Shown below is the quadrilateral ABCD.

AB = 15 and CD = 13.

BD = 6 and AC = 4

Is it possible to conclude that this quadrilateral is a parallelogram?

FAQ

Everything you need to know about this question

What's the difference between sides and diagonals in this problem?

Sides are the edges of the quadrilateral (AB, BC, CD, AD), while diagonals are lines connecting opposite vertices (AC and BD). For parallelograms, we check opposite sides are equal.

Do I need both pairs of opposite sides to be equal?

Technically yes for a complete proof, but if the problem gives you information showing one pair is equal and the shape appears symmetric, you can reasonably conclude it's a parallelogram in most basic problems.

What if the diagonals are equal like in this problem?

Equal diagonals (BD = AC = 4) suggest this might be a rectangle, which is a special type of parallelogram! But to confirm it's at least a parallelogram, focus on the opposite sides being equal.

Why does AB = CD = 6 make it a parallelogram?

When opposite sides are congruent (AB = CD), this is one key property of parallelograms. Combined with the apparent symmetry in the diagram, we can conclude ABCD is indeed a parallelogram.

Could this quadrilateral be something other than a parallelogram?

With opposite sides equal (AB = CD = 6) and equal diagonals, this is definitely at least a parallelogram. The equal diagonals actually suggest it might be a rectangle, which is a special type of parallelogram!

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