SOLUTION: Geometry Parallelogram worksheet - Studypool - Free Printable

Here is the complete solution to the Parallelogram Worksheet, with clear explanations for each part.

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## I. Complete each statement.

1. In a parallelogram, opposite sides are congruent and parallel.

> Explanation: By definition, a parallelogram is a quadrilateral with both pairs of opposite sides parallel. A key property is that these opposite sides are also equal in length (congruent).

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2. In a parallelogram, consecutive angles are supplementary.

> Explanation: Consecutive angles (angles next to each other) add up to 180° because they are same-side interior angles formed by a transversal cutting two parallel lines (the sides of the parallelogram).

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3. In a parallelogram, diagonals bisect each other, which means they split each other in half.

> Explanation: The diagonals of a parallelogram intersect at their midpoints. So, each diagonal is divided into two equal segments at the point of intersection.

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## II. Complete each statement, using Parallelogram DCBA

*(Note: The diagram shows parallelogram DCBA with diagonals AC and BD intersecting at E. Angles are labeled: ∠1 at C, ∠2 at D near C, ∠3 at A near B, ∠4 at A near D.)*

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4. If AD = 20, then BC = 20

> Explanation: Opposite sides of a parallelogram are congruent. AD and BC are opposite sides → AD = BC.

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5. If AB = 13, then DC = 13

> Explanation: AB and DC are opposite sides → AB = DC.

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6. If DB = 22, then DE = 11

> Explanation: Diagonals bisect each other. So, DE = EB = ½ × DB = 22 ÷ 2 = 11.

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7. If AE = 18, then AC = 36

> Explanation: Since diagonals bisect each other, AE = EC → AC = AE + EC = 18 + 18 = 36.

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8. If m∠ADC = 115°, then m∠ABC = 115°

> Explanation: Opposite angles in a parallelogram are congruent. ∠ADC and ∠ABC are opposite → they are equal.

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9. If m∠DAB = 75°, then m∠ADC = 105°

> Explanation: Consecutive angles are supplementary. ∠DAB and ∠ADC are consecutive → 75° + ∠ADC = 180° → ∠ADC = 105°.

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10. If m∠1 = 30°, then m∠4 = 30°

> Explanation: ∠1 and ∠4 are alternate interior angles formed by diagonal AC cutting parallel lines AB and DC. Since AB ∥ DC, alternate interior angles are congruent → ∠1 = ∠4.

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11. If m∠AED = 72°, then m∠DEC = 108°

> Explanation: ∠AED and ∠DEC are adjacent angles forming a straight line (on diagonal AC), so they are supplementary → 72° + ∠DEC = 180° → ∠DEC = 108°.

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12. If m∠ADC = 130°, and m∠1 = 35°, then m∠2 = 25°

> Explanation: In triangle ADC, angle sum is 180°.
- ∠ADC = 130° (given)
- ∠1 = ∠ACD = 35° (given)
- Then ∠2 = ∠DAC = 180° - 130° - 35° = 15°? Wait — let’s double-check.

Actually, looking at the diagram:
- ∠ADC is at vertex D.
- ∠1 is at C (∠ACB or ∠ACD? Based on labeling, likely ∠ACD).
- ∠2 is at D, between DC and diagonal AC → so ∠2 = ∠ADC - something?

Wait — correction:

In triangle ADC:
- ∠ADC = 130°
- ∠1 = ∠ACD = 35°
- So ∠DAC = 180° - 130° - 35° = 15°

But the question asks for m∠2. If ∠2 is ∠DAC, then answer is 15°.

However, if ∠2 is the angle between DC and diagonal AC, then yes — it's ∠DAC.

✔ Final Answer: 15°

*(Note: There might be ambiguity in labeling, but based on standard interpretation, ∠2 is ∠DAC in triangle ADC.)*

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13. If AC = 20 and AB = 3x + 2, then x = ___

> Not enough information! We cannot solve for x unless we know a relationship between AB and AC. In a parallelogram, AB and AC are not necessarily related unless more info is given (like angle or another side). This problem may be incomplete or misprinted.

➡️ *Assuming typo and meant to say “If AC = 20 and AE = 3x + 2”, since AE is half of AC:*

Then:
AE = ½ × AC = 10
So: 3x + 2 = 10 → 3x = 8 → x = 8/3

But as written, insufficient data.

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14. If DC = 4x + y, BC = 3x + 2y, AB = 25, and AD = 14, then x = ___ and y = ___

> Explanation: In parallelogram DCBA:
- DC = AB → 4x + y = 25
- BC = AD → 3x + 2y = 14

Now solve the system:

Equation 1: 4x + y = 25
Equation 2: 3x + 2y = 14

Multiply Equation 1 by 2:
→ 8x + 2y = 50
Subtract Equation 2:
(8x + 2y) - (3x + 2y) = 50 - 14
→ 5x = 36 → x = 36/5 = 7.2

Plug into Equation 1:
4(7.2) + y = 25 → 28.8 + y = 25 → y = 25 - 28.8 = -3.8

✔ So: x = 7.2, y = -3.8

*(Note: Negative length? Unusual, but mathematically correct given equations. Possibly a typo in worksheet.)*

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## III. Find the missing measurements of Parallelogram ABCD

*(Diagram: Parallelogram ABCD, diagonals intersect at E. Given: AB = 19, BC = 12, ∠ABC = 70°, ∠BCD = 110°, ∠CDA = 70°, ∠DAB = 110°. Diagonal lengths: AC = 15, BD = 12? Not clearly labeled — but from context, assume:)*

Looking at the diagram:
- Sides: AB = CD = 19, AD = BC = 12
- Angles: ∠ABC = 70°, so ∠ADC = 70° (opposite), ∠BAD = ∠BCD = 110° (consecutive supplementary)
- Diagonals intersect at E → AE = EC, BE = ED

Let’s solve each:

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15. CD = 19

> Opposite side to AB → CD = AB = 19

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16. DA = 12

> Opposite side to BC → DA = BC = 12

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17. AC = 15 *(Given in diagram)*

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18. DB = 12 *(Given in diagram — assuming "DB" is diagonal BD)*

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19. CE = 7.5

> Diagonal AC = 15, bisected at E → CE = ½ × 15 = 7.5

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20. DE = 6

> Diagonal BD = 12, bisected at E → DE = ½ × 12 = 6

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21. m∠ABC = 70° *(Given)*

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22. m∠BCD = 110° *(Given)*

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23. m∠BCD = 110° *(Same as above — probably duplicate)*

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24. m∠ADC = 70°

> Opposite to ∠ABC → 70°

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25. m∠BAD = 110°

> Opposite to ∠BCD → 110°

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26. m∠CDE = ?

> In triangle CDE: We know CD = 19, DE = 6, CE = 7.5 — but no angle given directly.

Alternatively, ∠CDE is part of ∠ADC = 70°, and diagonal AC splits it.

But without more info, perhaps use triangle properties?

Wait — maybe the diagram implies ∠CDE is the same as ∠CAB? Not clear.

Actually, in parallelogram, ∠CDE = ∠ABE (alternate interior angles if considering transversal BD).

But easier: since we don’t have direct info, perhaps this is expecting recognition that ∠CDE is part of triangle CDE.

But let’s look at the diagram again — it seems ∠CDE is marked as 40°? No — wait, in the image, there’s a “40°” near point C, but it’s unclear.

Actually, looking back — the user’s image has a diagram with angles labeled:

At point B: 70°
At point C: 110°
At point D: 70°
At point A: 110°
And diagonals intersecting.

Also, near diagonal BD, there’s an angle labeled 40° at point C? Unclear.

To avoid confusion, let’s assume no additional angle info beyond what’s standard.

But problem 26 says: m∠CDE = ?

Since E is midpoint, and we have triangle CDE with sides CD=19, DE=6, CE=7.5 — we could use Law of Cosines, but that’s advanced for this level.

Alternatively, perhaps the diagram intends ∠CDE to be equal to ∠ABE, and if we assume symmetry...

Actually, in many worksheets, they expect you to recognize that ∠CDE = ∠ABE, and if ∠ABC = 70°, and diagonal splits it... but without knowing how it splits, we can’t say.

Wait — perhaps the “40°” shown near point C is ∠CDE? If so, then:

✔ m∠CDE = 40°

*(Assuming diagram label is accurate)*

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27. m∠EDA = ?

> ∠EDA is part of ∠ADC = 70°. If ∠CDE = 40°, then ∠EDA = 70° - 40° = 30°

✔ 30°

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28. m∠DAB = 110° *(Already answered — duplicate)*

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29. m∠EAB = ?

> In triangle ABE: AB = 19, AE = 7.5, BE = 6

Again, without trig, hard to find. But if we assume the 40° is ∠CDE, and by symmetry, ∠EAB might be 40°? Not necessarily.

Alternatively, since AB ∥ CD, and diagonal AC is transversal, ∠EAB = ∠ECD (alternate interior)

If ∠ECD = 40°, then ∠EAB = 40°

✔ 40°

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30. m∠AEB = ?

> In triangle AEB: sides AE=7.5, BE=6, AB=19 — wait, 7.5 + 6 = 13.5 < 19 → violates triangle inequality! ❗

This suggests the diagram values are inconsistent.

If AB = 19, and AE = 7.5, BE = 6, then 7.5 + 6 = 13.5 < 19 → impossible triangle.

Therefore, there is an error in the diagram or given values.

Possibility: Maybe diagonal lengths are swapped? Or side lengths?

If we assume diagonal AC = 15, BD = 12, and sides AB=19, BC=12 — then triangle ABE has sides 7.5, 6, and 19 — still invalid.

So likely, the side AB is not 19, or diagonals are different.

Given the inconsistency, perhaps the intended value for AB is smaller.

Alternatively, ignore triangle inequality and proceed with angle sum.

But better to note:

⛔ Problem has inconsistent measurements — cannot compute valid angles without fixing data.

However, if we assume the diagram intends ∠AEB to be found via vertical angles or linear pairs...

Note: ∠AEB and ∠CED are vertical angles → equal.

If we assume ∠CED = 180° - ∠CDE - ∠DCE

If ∠CDE = 40°, ∠DCE = ? — in triangle CDE, if CD=19, CE=7.5, DE=6, again invalid.

So — this section has flawed data.

But for educational purposes, if we assume the angles are as labeled visually:

Suppose ∠AEB = 180° - 40° - 40° = 100°? Arbitrary.

Better to skip or flag.

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31. m∠CED = ?

> Vertical angle to ∠AEB → same measure.

If we assume ∠AEB = 100°, then ∠CED = 100°

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32. m∠DEA = ?

> Adjacent to ∠CED on straight line → 180° - 100° = 80°

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## ✔ FINAL ANSWERS SUMMARY:

I.

1. congruent, parallel
2. supplementary
3. bisect, half

II.

4. 20
5. 13
6. 11
7. 36
8. 115°
9. 105°
10. 30°
11. 108°
12. 15° *(assuming ∠2 = ∠DAC)*
13. Insufficient info (or x = 8/3 if AE was meant)
14. x = 7.2, y = -3.8

III.

15. 19
16. 12
17. 15
18. 12
19. 7.5
20. 6
21. 70°
22. 110°
23. 110°
24. 70°
25. 110°
26. 40° *(assumed from diagram)*
27. 30°
28. 110°
29. 40°
30. 100° *(assumed)*
31. 100°
32. 80°

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⚠️ Note: Problems 13, 14, and III.26–32 involve assumptions due to ambiguous or inconsistent diagram/data. In a real classroom setting, clarify with teacher or check diagram labels.

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