Practice
Parallelograms Practice
Fifty original questions on parallelogram and rhombus properties, diagonals, angles, heights, and areas.
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Question 1
Explanation
A parallelogram is defined by two pairs of parallel opposite sides.
- Reasoning
Name only the property guaranteed for every parallelogram.
Question 2
Explanation
Opposite sides inherit both parallelism and congruence.
- Reasoning
Name only the property guaranteed for every parallelogram.
Question 3
Explanation
Parallel-side relationships make each consecutive pair total 180 degrees.
- Reasoning
Name only the property guaranteed for every parallelogram.
Question 4
Explanation
Each diagonal cuts the other into two equal pieces.
- Reasoning
Name only the property guaranteed for every parallelogram.
Question 5
Explanation
Opposite sides are congruent, so solve \(3x+7=5x-9\). This gives \(x=8\) and \(AB=31\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 6
Explanation
Opposite sides are congruent, so solve \(4x+2=6x-12\). This gives \(x=7\) and \(AB=30\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 7
Explanation
Opposite sides are congruent, so solve \(5x-3=2x+18\). This gives \(x=7\) and \(AB=32\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 8
Explanation
Opposite sides are congruent, so solve \(7x+1=4x+25\). This gives \(x=8\) and \(AB=57\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 9
Explanation
Opposite sides are congruent, so solve \(2x+11=6x-5\). This gives \(x=4\) and \(AB=19\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 10
Explanation
Opposite sides are congruent, so solve \(8x-6=3x+29\). This gives \(x=7\) and \(AB=50\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 11
Explanation
Opposite sides are congruent, so solve \(9x+4=5x+32\). This gives \(x=7\) and \(AB=67\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 12
Explanation
Opposite sides are congruent, so solve \(6x+13=10x-7\). This gives \(x=5\) and \(AB=43\).
- Reasoning
Set opposite-side expressions equal, then substitute back for the requested length.
Question 13
Explanation
Consecutive angles are supplementary, so \(180-58=122^\circ\). The opposite angle remains 58 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 14
Explanation
Consecutive angles are supplementary, so \(180-67=113^\circ\). The opposite angle remains 67 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 15
Explanation
Consecutive angles are supplementary, so \(180-74=106^\circ\). The opposite angle remains 74 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 16
Explanation
Consecutive angles are supplementary, so \(180-81=99^\circ\). The opposite angle remains 81 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 17
Explanation
Consecutive angles are supplementary, so \(180-46=134^\circ\). The opposite angle remains 46 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 18
Explanation
Consecutive angles are supplementary, so \(180-103=77^\circ\). The opposite angle remains 103 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 19
Explanation
Consecutive angles are supplementary, so \(180-119=61^\circ\). The opposite angle remains 119 degrees.
- Reasoning
Subtract a known angle from 180 for its neighbors.
Question 20
Explanation
Diagonals bisect each other, so \(AE=CE\). Solving gives \(x=7\), each half is 29, and full \(AC=58\).
- Reasoning
Equate halves on the same diagonal, then double a half for the full diagonal.
Question 21
Explanation
Diagonals bisect each other, so \(AE=CE\). Solving gives \(x=7\), each half is 29, and full \(AC=58\).
- Reasoning
Equate halves on the same diagonal, then double a half for the full diagonal.
Question 22
Explanation
Diagonals bisect each other, so \(AE=CE\). Solving gives \(x=6\), each half is 31, and full \(AC=62\).
- Reasoning
Equate halves on the same diagonal, then double a half for the full diagonal.
Question 23
Explanation
Diagonals bisect each other, so \(AE=CE\). Solving gives \(x=7\), each half is 59, and full \(AC=118\).
- Reasoning
Equate halves on the same diagonal, then double a half for the full diagonal.
Question 24
Explanation
Diagonals bisect each other, so \(AE=CE\). Solving gives \(x=5\), each half is 37, and full \(AC=74\).
- Reasoning
Equate halves on the same diagonal, then double a half for the full diagonal.
Question 25
Explanation
Diagonals bisect each other, so \(AE=CE\). Solving gives \(x=8\), each half is 47, and full \(AC=94\).
- Reasoning
Equate halves on the same diagonal, then double a half for the full diagonal.
Question 26
Explanation
A rhombus inherits parallelogram properties and adds four equal sides, perpendicular diagonals, and opposite-angle bisection.
- Reasoning
Separate inherited properties from rhombus-specific ones.
Question 27
Explanation
A rhombus inherits parallelogram properties and adds four equal sides, perpendicular diagonals, and opposite-angle bisection.
- Reasoning
Separate inherited properties from rhombus-specific ones.
Question 28
Explanation
A rhombus inherits parallelogram properties and adds four equal sides, perpendicular diagonals, and opposite-angle bisection.
- Reasoning
Separate inherited properties from rhombus-specific ones.
Question 29
Explanation
A rhombus inherits parallelogram properties and adds four equal sides, perpendicular diagonals, and opposite-angle bisection.
- Reasoning
Separate inherited properties from rhombus-specific ones.
Question 30
Explanation
A rhombus inherits parallelogram properties and adds four equal sides, perpendicular diagonals, and opposite-angle bisection.
- Reasoning
Separate inherited properties from rhombus-specific ones.
Question 31
Explanation
Use \(A=bh=12(7)=84\).
- Reasoning
Use the perpendicular height rather than a slanted side.
Question 32
Explanation
Use \(A=bh=15(9)=135\).
- Reasoning
Use the perpendicular height rather than a slanted side.
Question 33
Explanation
Use \(A=bh=18(11)=198\).
- Reasoning
Use the perpendicular height rather than a slanted side.
Question 34
Explanation
Use \(A=bh=24(6)=144\).
- Reasoning
Use the perpendicular height rather than a slanted side.
Question 35
Explanation
Use \(A=bh=21(13)=273\).
- Reasoning
Use the perpendicular height rather than a slanted side.
Question 36
Explanation
Use full diagonals in \(A=\tfrac12d_1d_2=\tfrac12(10)(14)=70\).
- Reasoning
Confirm both given lengths are full diagonals before applying the one-half product.
Question 37
Explanation
Use full diagonals in \(A=\tfrac12d_1d_2=\tfrac12(12)(18)=108\).
- Reasoning
Confirm both given lengths are full diagonals before applying the one-half product.
Question 38
Explanation
Use full diagonals in \(A=\tfrac12d_1d_2=\tfrac12(16)(22)=176\).
- Reasoning
Confirm both given lengths are full diagonals before applying the one-half product.
Question 39
Explanation
Use full diagonals in \(A=\tfrac12d_1d_2=\tfrac12(20)(26)=260\).
- Reasoning
Confirm both given lengths are full diagonals before applying the one-half product.
Question 40
Explanation
Use full diagonals in \(A=\tfrac12d_1d_2=\tfrac12(24)(30)=360\).
- Reasoning
Confirm both given lengths are full diagonals before applying the one-half product.
Question 41
Explanation
The altitude is \(\sqrt{10^2-6^2}=8\). Then \(A=18(8)=144\).
- Reasoning
Find the perpendicular altitude from the attached right triangle before using A=bh.
Question 42
Explanation
The altitude is \(\sqrt{13^2-5^2}=12\). Then \(A=24(12)=288\).
- Reasoning
Find the perpendicular altitude from the attached right triangle before using A=bh.
Question 43
Explanation
The altitude is \(\sqrt{17^2-8^2}=15\). Then \(A=20(15)=300\).
- Reasoning
Find the perpendicular altitude from the attached right triangle before using A=bh.
Question 44
Explanation
The altitude is \(\sqrt{25^2-7^2}=24\). Then \(A=28(24)=672\).
- Reasoning
Find the perpendicular altitude from the attached right triangle before using A=bh.
Question 45
Explanation
The altitude is \(\sqrt{17^2-15^2}=8\). Then \(A=30(8)=240\).
- Reasoning
Find the perpendicular altitude from the attached right triangle before using A=bh.
Question 46
Explanation
A central right triangle gives side \(\sqrt{9^2+12^2}=15\). Full diagonals are 18 and 24, so area is \(\tfrac12(18)(24)=216\).
- Reasoning
Use half-diagonals for Pythagorean length but full diagonals for area.
Question 47
Explanation
Supplementary angles give \(8x+4=180\), so \(x=22\). The measures are \(73^\circ\) and \(107^\circ\).
- Reasoning
Set consecutive expressions to 180, solve, and substitute into both.
Question 48
Explanation
Parallelogram area uses perpendicular distance between the base lines; a slanted side is not automatically that distance.
- Reasoning
Locate or derive a right-angle altitude before multiplying.
Question 49
Explanation
Bisection is universal for parallelograms; congruent diagonals are guaranteed for rectangles and squares, not general parallelograms.
- Reasoning
Do not import a rectangle property into a general parallelogram.
Question 50
Explanation
The other half-diagonal is \(\sqrt{13^2-5^2}=12\), so its full diagonal is 24. The first full diagonal is 10, giving area \(\tfrac12(10)(24)=120\).
- Reasoning
Use perpendicular half-diagonals as right-triangle legs, then double both before area.
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Questions to review
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- Question 1Parallelogram propertiesEasy
- Question 2Parallelogram propertiesEasy
- Question 3Parallelogram propertiesEasy
- Question 4Parallelogram propertiesEasy
- Question 5Algebraic opposite sidesEasy
- Question 6Algebraic opposite sidesEasy
- Question 7Algebraic opposite sidesEasy
- Question 8Algebraic opposite sidesEasy
- Question 9Algebraic opposite sidesEasy
- Question 10Algebraic opposite sidesEasy
- Question 11Algebraic opposite sidesEasy
- Question 12Algebraic opposite sidesEasy
- Question 13Parallelogram angle relationshipsEasy
- Question 14Parallelogram angle relationshipsEasy
- Question 15Parallelogram angle relationshipsEasy
- Question 16Parallelogram angle relationshipsMedium
- Question 17Parallelogram angle relationshipsMedium
- Question 18Parallelogram angle relationshipsMedium
- Question 19Parallelogram angle relationshipsMedium
- Question 20Diagonal bisectionMedium
- Question 21Diagonal bisectionMedium
- Question 22Diagonal bisectionMedium
- Question 23Diagonal bisectionMedium
- Question 24Diagonal bisectionMedium
- Question 25Diagonal bisectionMedium
- Question 26Rhombus propertiesMedium
- Question 27Rhombus propertiesMedium
- Question 28Rhombus propertiesMedium
- Question 29Rhombus propertiesMedium
- Question 30Rhombus propertiesMedium
- Question 31Parallelogram areaMedium
- Question 32Parallelogram areaMedium
- Question 33Parallelogram areaMedium
- Question 34Parallelogram areaMedium
- Question 35Parallelogram areaMedium
- Question 36Rhombus areaMedium
- Question 37Rhombus areaMedium
- Question 38Rhombus areaMedium
- Question 39Rhombus areaMedium
- Question 40Rhombus areaMedium
- Question 41Height from right trianglesHard
- Question 42Height from right trianglesHard
- Question 43Height from right trianglesHard
- Question 44Height from right trianglesHard
- Question 45Height from right trianglesHard
- Question 46Rhombus half-diagonalsHard
- Question 47Multi-step angle algebraHard
- Question 48Area error analysisHard
- Question 49Property error analysisHard
- Question 50Rhombus synthesisHard