Parallelogram Properties: When ∢B+∢C=180° in a Quadrilateral

Q: Why isn't it enough that angles B and D are both 140°?

While having one pair of opposite angles equal is necessary for a parallelogram, it's not sufficient. You need to prove that both pairs of opposite angles are equal to confirm it's truly a parallelogram.

Q: How do I know which angles are adjacent vs opposite?

Adjacent angles share a side (like B and C), while opposite angles are across from each other (like A and C, or B and D). In the diagram, follow the vertices around: A-B-C-D.

Q: What if the adjacent angles don't add up to 180°?

If adjacent angles don't sum to 180°, then it's definitely not a parallelogram! This is a necessary condition - adjacent angles in any parallelogram must be supplementary.

Q: Can I use other properties to check if it's a parallelogram?

Yes! You could also check if:Opposite sides are parallel and equalDiagonals bisect each otherOne pair of opposite sides is both parallel and equalBut using angle relationships is often the most direct method when angles are given.

Q: Why do we solve for x instead of just using the given information?

The angle C is given as $ 4x $, not a specific number. We need to solve for x first to find the actual measure of angle C, then verify if it equals angle A.

Parallelogram Verification with Adjacent Angle Relationships

Below is a quadrilateral:

Given $∢B+∢C=180$

Is it possible that it is a parallelogram?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is a quadrilateral:

Given $∢B+∢C=180$

Is it possible that it is a parallelogram?

Step-by-step solution

Remember that in a parallelogram each pair of opposite angles are equal to each other.

The data shows that only one pair of angles are equal to each other:

$D=B=140$

Therefore, we will now find angle C and see if it is equal to angle A, that is, if angle C is equal to 40:

Let's remember that a pair of angles on the same side are equal to 180 degrees, therefore:

$B+C=180$

We replace the existing data:

$140+4x=180$

$4x=180-140$

$4x=40$

Divide by 4:

$\frac{4x}{4}=\frac{40}{4}$

$x=10$

Now we replace X:

$C=4\times10=40$

That is, we found that angles A and C are equal to each other and that the quadrilateral is a parallelogram since each pair of opposite angles are equal to each other.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: In parallelograms, opposite angles are equal and adjacent angles sum to 180°
Technique: Use $B + C = 180$ to solve: 140 + 4x = 180, so x = 10
Check: Verify opposite angles are equal: A = C = 40° and B = D = 140° ✓

Common Mistakes

Avoid these frequent errors

Assuming one angle pair proves it's a parallelogram
Don't conclude it's a parallelogram just because D = B = 140° = you're only halfway done! One equal pair doesn't guarantee the other pair is equal. Always find all four angles and verify that both opposite angle pairs are equal.

Practice Quiz

Test your knowledge with interactive questions

The parallelogram ABCD is shown below.

What type of angles are indicated in the figure?

FAQ

Everything you need to know about this question

Why isn't it enough that angles B and D are both 140°?

While having one pair of opposite angles equal is necessary for a parallelogram, it's not sufficient. You need to prove that both pairs of opposite angles are equal to confirm it's truly a parallelogram.

How do I know which angles are adjacent vs opposite?

Adjacent angles share a side (like B and C), while opposite angles are across from each other (like A and C, or B and D). In the diagram, follow the vertices around: A-B-C-D.

What if the adjacent angles don't add up to 180°?

If adjacent angles don't sum to 180°, then it's definitely not a parallelogram! This is a necessary condition - adjacent angles in any parallelogram must be supplementary.

Can I use other properties to check if it's a parallelogram?

Yes! You could also check if:

Opposite sides are parallel and equal
Diagonals bisect each other
One pair of opposite sides is both parallel and equal

But using angle relationships is often the most direct method when angles are given.

Why do we solve for x instead of just using the given information?

The angle C is given as $4x$ , not a specific number. We need to solve for x first to find the actual measure of angle C, then verify if it equals angle A.

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