SOLUTION: Geometry Parallelogram worksheet - Studypool - Free Printable

Explanation:
Let’s go step by step through the worksheet. We’ll focus on the parts that are clearly answerable using standard parallelogram properties.

I. Complete each statement (basic properties):

1. In a parallelogram, opposite sides are congruent and parallel.
→ Because definition of parallelogram: both pairs of opposite sides are parallel; and a theorem says they’re also equal in length.

2. In a parallelogram, consecutive angles are supplementary (i.e., add to 180°).
→ Because consecutive angles lie on the same side of a transversal between two parallel lines — same-side interior angles are supplementary.

3. In a parallelogram, diagonals bisect each other, which means they split each other in half.
→ This is a key property: the point where diagonals intersect divides each diagonal into two equal segments.

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II. Using Parallelogram DCBA
Note: The labeling is DCBA, so vertices go D → C → B → A (counterclockwise or clockwise). From the diagram (even though we can’t see it, the labels match typical order), we assume:
- AB is opposite CD
- AD is opposite BC
- Diagonals AC and BD intersect at E
- Angles labeled: ∠1 near C, ∠2 near C (maybe part of triangle), ∠3 near A, ∠4 near A — but from standard labeling in such problems:
- ∠1 = ∠ACB (or angle at C in triangle BEC)
- ∠2 = ∠DCA (angle at C in triangle DEC)
- ∠3 = ∠CAB (angle at A in triangle AEB)
- ∠4 = ∠DAC

But more reliably, we use known properties:

4. If AD = 20, then BC = 20
→ Opposite sides equal: AD = BC.

5. If AB = 13, then DC = 13
→ AB = DC (opposite sides).

6. If DB = 22, then DE = 11
→ Diagonals bisect each other, so E is midpoint of DB ⇒ DE = EB = 22/2 = 11.

7. If AE = 18, then AC = 36
→ E is midpoint of AC ⇒ AC = 2 × AE = 36.

8. If m∠ADC = 115°, then m∠ABC = 115°
→ Opposite angles in parallelogram are congruent. ∠ADC and ∠ABC are opposite.

9. If m∠DAB = 75°, m∠ADC = 105°
→ Consecutive angles are supplementary: ∠DAB + ∠ADC = 180° ⇒ ∠ADC = 180 − 75 = 105°.

10. If m∠1 = 30°, then m∠4 = 30°
→ Assuming ∠1 and ∠4 are alternate interior angles formed by diagonal AC crossing parallel sides AB ∥ DC, or by symmetry: In parallelogram, triangles AED and CEB are congruent, and angles at A and C corresponding are equal. Also, if ∠1 = ∠ACB = 30°, then since AB ∥ DC, ∠DAC = ∠ACB (alternate interior) ⇒ ∠4 = 30°.

11. If m∠AED = 72°, m∠DEC = 108°
→ Points A–E–C are collinear (diagonal AC), so ∠AED and ∠DEC are adjacent angles forming a straight line along BD? Wait — actually, E is intersection of diagonals. So angles around point E sum to 360°. But ∠AED and ∠DEC are *adjacent* angles sharing ray ED, with rays EA and EC opposite (since A–E–C is a line). So ∠AED and ∠DEC are supplementary (they form a linear pair along line AC). So ∠DEC = 180° − 72° = 108°.

12. If m∠ADC = 130°, and m∠1 = 35°, find m∠2.
Let’s interpret: In parallelogram ABCD (DCBA same), ∠ADC = 130° is angle at D between AD and DC. Diagonal AC splits ∠ADC into two angles: ∠1 and ∠2 (as labeled near D? Wait — in the diagram, ∠1 and ∠2 are near point C, not D). Hmm — but problem says “m∠1 = 35°”, and asks for m∠2, given ∠ADC = 130°. Likely ∠1 and ∠2 are the two parts into which diagonal BD or AC splits angle at D or C.

Alternative interpretation (more consistent with typical worksheets): In triangle DEC or BEC, ∠1 and ∠2 are angles at C, and ∠ADC = 130° is angle at D. Since AB ∥ DC, and AD is transversal, ∠DAB = 180 − 130 = 50°. Also, diagonal AC creates triangles. But maybe simpler: In parallelogram, diagonal AC creates two triangles: ΔADC and ΔABC. In ΔADC, we know ∠ADC = 130°, and if ∠1 = ∠DCA = 35°, then remaining angle ∠DAC = 180 − 130 − 35 = 15°. But question asks for ∠2 — likely ∠DAC = ∠2? Or maybe ∠2 = ∠DAC = 15°.

Wait — look at problem 10: m∠1 = 30°, then m∠4 = ? We answered 30°, assuming alternate interior. So ∠1 and ∠4 are alternate interior angles across diagonal AC with AB ∥ DC. So ∠1 is at C (∠DCA), ∠4 is at A (∠CAB or ∠DAC?). Actually, ∠4 is marked near A, between DA and AC — that’s ∠DAC. And ∠1 is between DC and AC — that’s ∠DCA. Since AB ∥ DC, and AC is transversal, ∠DAC and ∠DCA are *not* alternate interior — those would be ∠DAC and ∠BCA. Hmm.

Better: Use triangle ADC. Angles in triangle ADC:
∠ADC = 130° (given)
∠DCA = ∠1 = 35°
Then ∠DAC = 180 − 130 − 35 = 15°
If ∠2 is ∠DAC, then answer is 15°.

This matches common worksheet layout: ∠1 and ∠2 are the two angles at C and A created by diagonal AC in triangle ADC. But label shows ∠1 and ∠2 both at point C in the diagram (the small numbers 1 and 2 near C), while 3 and 4 near A. So likely:
- ∠1 = ∠ACB (in triangle ABC)
- ∠2 = ∠DCA (in triangle ADC)
But then why give ∠ADC = 130°? To find other angles.

Actually, problem 12: “If m∠ADC = 130°, and m∠1 = 35°, m∠2 = ___” — and in the diagram, ∠1 and ∠2 are the two angles that make up ∠DCB? No.

Let’s try another route: In parallelogram, consecutive angles supplementary. So ∠ADC = 130° ⇒ ∠DAB = 50°. Diagonal AC splits ∠DAB into ∠3 and ∠4. If ∠1 = 35°, and due to alternate interior, ∠4 = ∠1 = 35° (as in Q10), then ∠3 = 50° − 35° = 15°. But they ask for ∠2, not ∠3.

Given ambiguity, but in most standard versions of this worksheet (widely circulated), question 12 has:
∠ADC = 130°, ∠1 = 35°, and ∠1 and ∠2 are the two angles that diagonal AC makes at point C, i.e., ∠DCA = ∠1 = 35°, and ∠BCA = ∠2. Since AB ∥ DC, ∠BCA = ∠DAC (alternate interior). In triangle ADC: ∠DAC = 180 − 130 − 35 = 15°, so ∠2 = 15°.

Yes, answer is 15°.

Now, questions 13–14 involve algebraic expressions — but they’re cut off / blurry. Since the instruction says “Solve the problem accurately”, and only clear items are 1–12, we’ll answer those.

Also Section III has many blanks with a second diagram — too blurry to decode reliably. Since the user didn’t specify which parts to solve, and only clear ones are I and II (1–12), we’ll provide answers for 1–12.

Let me list all final answers clearly:

1. congruent, parallel
2. supplementary
3. bisect, half
4. 20
5. 13
6. 11
7. 36
8. 115°
9. 105°
10. 30°
11. 108°
12. 15°

These are consistent with parallelogram properties and careful angle reasoning.

Final Answer:
1. congruent, parallel
2. supplementary
3. bisect, half
4. 20
5. 13
6. 11
7. 36
8. 115°
9. 105°
10. 30°
11. 108°
12. 15°