Squares and Rhombi Worksheet for 10th Grade | Lesson Planet - Free Printable
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Step-by-step solution for: Squares and Rhombi Worksheet for 10th Grade | Lesson Planet
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Step-by-step solution for: Squares and Rhombi Worksheet for 10th Grade | Lesson Planet
Here are the solutions to the problems on the worksheet, broken down step by step.
Rules:
* Rectangle: A parallelogram with 4 right angles ($90^\circ$).
* Rhombus: A parallelogram with 4 congruent (equal length) sides.
* Square: A parallelogram that is BOTH a rectangle and a rhombus (4 right angles AND 4 equal sides).
1. The shape has markings showing all 4 sides are equal (rhombus property) and all 4 corners are right angles (rectangle property).
* Answer: Square (It has 4 right angles and 4 congruent sides).
2. The shape has markings showing all 4 sides are equal and all 4 corners are right angles.
* Answer: Square (It has 4 right angles and 4 congruent sides).
3. The shape has markings showing all 4 sides are equal, but the angles are not marked as right angles.
* Answer: Rhombus (It has 4 congruent sides).
4. The shape has markings showing opposite sides are parallel and one angle is a right angle. In a parallelogram, if one angle is $90^\circ$, all of them are. There are no markings showing all sides are equal.
* Answer: Rectangle (It is a parallelogram with a right angle).
5. The shape has markings showing all 4 sides are equal. The angles are not right angles.
* Answer: Rhombus (It has 4 congruent sides).
---
Key Properties of a Rhombus:
1. Opposite angles are equal.
2. Consecutive angles add up to $180^\circ$.
3. Diagonals bisect the vertex angles (cut them in half).
4. Diagonals intersect at $90^\circ$ (right angles).
5. The diagonals create four small right triangles inside.
6.
* Angle 2 is opposite the $54^\circ$ angle, so $\angle 2 = 54^\circ$.
* The diagonal cuts the corner angle in half. So, $\angle 3$ is half of $\angle 2$. $\angle 3 = 54 / 2 = 27^\circ$.
* The diagonals cross at $90^\circ$. In the bottom triangle, angles must add to $180^\circ$. $\angle 4 + \angle 3 + 90 = 180$. So, $\angle 4 = 90 - 27 = 63^\circ$.
* Angle 1 is alternate interior to angle 4 (or part of the other bisected corner), so $\angle 1 = 63^\circ$.
* Answer: $m\angle 1 = 63^\circ, m\angle 2 = 54^\circ, m\angle 3 = 27^\circ, m\angle 4 = 63^\circ$.
7.
* The diagonal creates an isosceles triangle on the left because two sides of a rhombus are equal. The base angles are equal. One is given as $68^\circ$, so $\angle 1 = 68^\circ$.
* Angles in a triangle add to $180^\circ$. The top angle of that left triangle is $180 - 68 - 68 = 44^\circ$.
* The diagonal bisects the corner, so $\angle 2$ is also $44^\circ$.
* The diagonals cross at $90^\circ$. In the top right small triangle: $\angle 3 + \angle 2 + 90 = 180$. $\angle 3 = 90 - 44 = 46^\circ$.
* Angle 4 is alternate interior to the $68^\circ$ angle at the bottom left, so $\angle 4 = 68^\circ$.
* Answer: $m\angle 1 = 68^\circ, m\angle 2 = 44^\circ, m\angle 3 = 46^\circ, m\angle 4 = 68^\circ$.
8.
* We are given a partial angle of $20^\circ$. Since the diagonal bisects the corner, the full angle at D is $40^\circ$.
* Consecutive angles add to $180^\circ$. So angle $DAB = 180 - 40 = 140^\circ$.
* Angle 1 is half of angle $DAB$. $\angle 1 = 140 / 2 = 70^\circ$.
* Angle 2 is the other half of angle $DAB$. $\angle 2 = 70^\circ$.
* The diagonals cross at $90^\circ$. In the top triangle: $\angle 3 + \angle 2 + 90 = 180$. $\angle 3 = 90 - 70 = 20^\circ$.
* Angle 4 is alternate interior to the $20^\circ$ angle at D. $\angle 4 = 20^\circ$.
* Answer: $m\angle 1 = 70^\circ, m\angle 2 = 70^\circ, m\angle 3 = 20^\circ, m\angle 4 = 20^\circ$.
9.
* Angle 2 is opposite the $52^\circ$ angle, so $\angle 2 = 52^\circ$.
* The diagonal bisects the angle, so $\angle 1$ is half of $52^\circ$. $\angle 1 = 26^\circ$.
* Diagonals cross at $90^\circ$. In the left triangle: $\angle 3 + \angle 1 + 90 = 180$. $\angle 3 = 90 - 26 = 64^\circ$.
* Angle 4 is alternate interior to angle 3. $\angle 4 = 64^\circ$.
* Answer: $m\angle 1 = 26^\circ, m\angle 2 = 52^\circ, m\angle 3 = 64^\circ, m\angle 4 = 64^\circ$.
10.
* Angle 4 is opposite the $59^\circ$ angle, so $\angle 4 = 59^\circ$.
* The diagonal bisects the angle, so $\angle 3$ is half of $59^\circ$. $\angle 3 = 29.5^\circ$.
* Diagonals cross at $90^\circ$. In the bottom triangle: $\angle 2 + \angle 3 + 90 = 180$. $\angle 2 = 90 - 29.5 = 60.5^\circ$.
* Angle 1 is alternate interior to angle 2. $\angle 1 = 60.5^\circ$.
* Answer: $m\angle 1 = 60.5^\circ, m\angle 2 = 60.5^\circ, m\angle 3 = 29.5^\circ, m\angle 4 = 59^\circ$.
11.
* This looks identical to Problem 9.
* Angle 2 is opposite $52^\circ$, so $\angle 2 = 52^\circ$.
* Angle 1 is half of $52^\circ$, so $\angle 1 = 26^\circ$.
* Angle 3 is $90 - 26 = 64^\circ$.
* Angle 4 is alternate interior to angle 3, so $\angle 4 = 64^\circ$.
* Answer: $m\angle 1 = 26^\circ, m\angle 2 = 52^\circ, m\angle 3 = 64^\circ, m\angle 4 = 64^\circ$.
12.
* We are given a partial angle of $39^\circ$. The diagonal bisects the corner, so the full angle is $78^\circ$.
* Consecutive angles add to $180^\circ$. The adjacent corner angle is $180 - 78 = 102^\circ$.
* Angle 2 is half of that $102^\circ$ angle. $\angle 2 = 51^\circ$.
* Diagonals cross at $90^\circ$. In the top triangle: $\angle 1 + \angle 2 + 90 = 180$. $\angle 1 = 90 - 51 = 39^\circ$.
* Angle 3 is alternate interior to the given $39^\circ$ angle. $\angle 3 = 39^\circ$.
* Angle 4 is alternate interior to angle 1. $\angle 4 = 39^\circ$. (Wait, let's recheck. Angle 4 is part of the bottom right corner. The bottom right corner is opposite the top left ($102^\circ$). So angle 4 is half of $102^\circ$, which is $51^\circ$? No, look at the position. Angle 4 is adjacent to the side. Let's use the triangle method. Bottom triangle has angle 3 ($39^\circ$) and center ($90^\circ$). So the other angle is $90-39=51^\circ$. That is the full half-angle. Angle 4 is labeled as the angle between the diagonal and the side. Yes, $\angle 4 = 51^\circ$.)
* Correction: Let's trace carefully.
* Given angle at left tip is split into $39^\circ$ and $39^\circ$. Total left angle = $78^\circ$.
* Top-left angle + Left angle = $180^\circ$. Top-left angle = $102^\circ$.
* Diagonal bisects Top-left. So $\angle 2 = 51^\circ$.
* Triangle containing $\angle 1$ and $\angle 2$ is a right triangle. $\angle 1 = 90 - 51 = 39^\circ$.
* $\angle 3$ is alternate interior to the top part of the left angle ($39^\circ$). So $\angle 3 = 39^\circ$.
* $\angle 4$ is in the bottom right. The bottom right angle is opposite the top left ($102^\circ$). The diagonal bisects it. So $\angle 4 = 51^\circ$.
* Answer: $m\angle 1 = 39^\circ, m\angle 2 = 51^\circ, m\angle 3 = 39^\circ, m\angle 4 = 51^\circ$.
13.
* We are given a partial angle of $49^\circ$. The diagonal bisects the corner, so the full bottom-right angle is $98^\circ$.
* Opposite angles are equal, so the top-left angle is also $98^\circ$.
* Angle 2 is half of the top-left angle. $\angle 2 = 49^\circ$.
* Consecutive angles add to $180^\circ$. Bottom-left angle = $180 - 98 = 82^\circ$.
* Angle 3 is half of the bottom-left angle. $\angle 3 = 41^\circ$.
* Diagonals cross at $90^\circ$. In the bottom triangle: $\angle 4 + \angle 3 + 90 = 180$. $\angle 4 = 90 - 41 = 49^\circ$.
* Angle 1 is alternate interior to angle 4? No, Angle 1 is half of the top-right angle. Top-right is opposite bottom-left ($82^\circ$). So $\angle 1 = 41^\circ$.
* Answer: $m\angle 1 = 41^\circ, m\angle 2 = 49^\circ, m\angle 3 = 41^\circ, m\angle 4 = 49^\circ$.
---
Rule: The diagonals of a rectangle are equal in length. Set the expressions for the diagonals equal to each other to solve for $x$. Then plug $x$ back in to find the length.
14. $HJ = x$ and $IK = 2x - 7$
* Set equal: $x = 2x - 7$
* Subtract $x$ from both sides: $0 = x - 7$
* Add 7 to both sides: $x = 7$
* Length: $HJ = 7$. (Check: $IK = 2(7)-7 = 7$).
* Answer: $x = 7$, Length = $7$.
15. $HJ = 3x + 5$ and $IK = 5x - 9$
* Set equal: $3x + 5 = 5x - 9$
* Subtract $3x$ from both sides: $5 = 2x - 9$
* Add 9 to both sides: $14 = 2x$
* Divide by 2: $x = 7$
* Length: $HJ = 3(7) + 5 = 21 + 5 = 26$.
* Answer: $x = 7$, Length = $26$.
16. $HJ = 3x + 7$ and $IK = 6x - 11$
* Set equal: $3x + 7 = 6x - 11$
* Subtract $3x$ from both sides: $7 = 3x - 11$
* Add 11 to both sides: $18 = 3x$
* Divide by 3: $x = 6$
* Length: $HJ = 3(6) + 7 = 18 + 7 = 25$.
* Answer: $x = 6$, Length = $25$.
17. $HJ = 19 + 2x$ and $IK = 3x + 22$
* Set equal: $19 + 2x = 3x + 22$
* Subtract $2x$ from both sides: $19 = x + 22$
* Subtract 22 from both sides: $-3 = x$
* Length: $HJ = 19 + 2(-3) = 19 - 6 = 13$.
* Check: $IK = 3(-3) + 22 = -9 + 22 = 13$.
* Answer: $x = -3$, Length = $13$.
Final Answer:
1. Square
2. Square
3. Rhombus
4. Rectangle
5. Rhombus
6. $m\angle 1=63^\circ, m\angle 2=54^\circ, m\angle 3=27^\circ, m\angle 4=63^\circ$
7. $m\angle 1=68^\circ, m\angle 2=44^\circ, m\angle 3=46^\circ, m\angle 4=68^\circ$
8. $m\angle 1=70^\circ, m\angle 2=70^\circ, m\angle 3=20^\circ, m\angle 4=20^\circ$
9. $m\angle 1=26^\circ, m\angle 2=52^\circ, m\angle 3=64^\circ, m\angle 4=64^\circ$
10. $m\angle 1=60.5^\circ, m\angle 2=60.5^\circ, m\angle 3=29.5^\circ, m\angle 4=59^\circ$
11. $m\angle 1=26^\circ, m\angle 2=52^\circ, m\angle 3=64^\circ, m\angle 4=64^\circ$
12. $m\angle 1=39^\circ, m\angle 2=51^\circ, m\angle 3=39^\circ, m\angle 4=51^\circ$
13. $m\angle 1=41^\circ, m\angle 2=49^\circ, m\angle 3=41^\circ, m\angle 4=49^\circ$
14. $x=7$, Length $= 7$
15. $x=7$, Length $= 26$
16. $x=6$, Length $= 25$
17. $x=-3$, Length $= 13$
Part 1: Identify the Shape (Problems 1–5)
Rules:
* Rectangle: A parallelogram with 4 right angles ($90^\circ$).
* Rhombus: A parallelogram with 4 congruent (equal length) sides.
* Square: A parallelogram that is BOTH a rectangle and a rhombus (4 right angles AND 4 equal sides).
1. The shape has markings showing all 4 sides are equal (rhombus property) and all 4 corners are right angles (rectangle property).
* Answer: Square (It has 4 right angles and 4 congruent sides).
2. The shape has markings showing all 4 sides are equal and all 4 corners are right angles.
* Answer: Square (It has 4 right angles and 4 congruent sides).
3. The shape has markings showing all 4 sides are equal, but the angles are not marked as right angles.
* Answer: Rhombus (It has 4 congruent sides).
4. The shape has markings showing opposite sides are parallel and one angle is a right angle. In a parallelogram, if one angle is $90^\circ$, all of them are. There are no markings showing all sides are equal.
* Answer: Rectangle (It is a parallelogram with a right angle).
5. The shape has markings showing all 4 sides are equal. The angles are not right angles.
* Answer: Rhombus (It has 4 congruent sides).
---
Part 2: Find Angle Measures in Rhombuses (Problems 6–13)
Key Properties of a Rhombus:
1. Opposite angles are equal.
2. Consecutive angles add up to $180^\circ$.
3. Diagonals bisect the vertex angles (cut them in half).
4. Diagonals intersect at $90^\circ$ (right angles).
5. The diagonals create four small right triangles inside.
6.
* Angle 2 is opposite the $54^\circ$ angle, so $\angle 2 = 54^\circ$.
* The diagonal cuts the corner angle in half. So, $\angle 3$ is half of $\angle 2$. $\angle 3 = 54 / 2 = 27^\circ$.
* The diagonals cross at $90^\circ$. In the bottom triangle, angles must add to $180^\circ$. $\angle 4 + \angle 3 + 90 = 180$. So, $\angle 4 = 90 - 27 = 63^\circ$.
* Angle 1 is alternate interior to angle 4 (or part of the other bisected corner), so $\angle 1 = 63^\circ$.
* Answer: $m\angle 1 = 63^\circ, m\angle 2 = 54^\circ, m\angle 3 = 27^\circ, m\angle 4 = 63^\circ$.
7.
* The diagonal creates an isosceles triangle on the left because two sides of a rhombus are equal. The base angles are equal. One is given as $68^\circ$, so $\angle 1 = 68^\circ$.
* Angles in a triangle add to $180^\circ$. The top angle of that left triangle is $180 - 68 - 68 = 44^\circ$.
* The diagonal bisects the corner, so $\angle 2$ is also $44^\circ$.
* The diagonals cross at $90^\circ$. In the top right small triangle: $\angle 3 + \angle 2 + 90 = 180$. $\angle 3 = 90 - 44 = 46^\circ$.
* Angle 4 is alternate interior to the $68^\circ$ angle at the bottom left, so $\angle 4 = 68^\circ$.
* Answer: $m\angle 1 = 68^\circ, m\angle 2 = 44^\circ, m\angle 3 = 46^\circ, m\angle 4 = 68^\circ$.
8.
* We are given a partial angle of $20^\circ$. Since the diagonal bisects the corner, the full angle at D is $40^\circ$.
* Consecutive angles add to $180^\circ$. So angle $DAB = 180 - 40 = 140^\circ$.
* Angle 1 is half of angle $DAB$. $\angle 1 = 140 / 2 = 70^\circ$.
* Angle 2 is the other half of angle $DAB$. $\angle 2 = 70^\circ$.
* The diagonals cross at $90^\circ$. In the top triangle: $\angle 3 + \angle 2 + 90 = 180$. $\angle 3 = 90 - 70 = 20^\circ$.
* Angle 4 is alternate interior to the $20^\circ$ angle at D. $\angle 4 = 20^\circ$.
* Answer: $m\angle 1 = 70^\circ, m\angle 2 = 70^\circ, m\angle 3 = 20^\circ, m\angle 4 = 20^\circ$.
9.
* Angle 2 is opposite the $52^\circ$ angle, so $\angle 2 = 52^\circ$.
* The diagonal bisects the angle, so $\angle 1$ is half of $52^\circ$. $\angle 1 = 26^\circ$.
* Diagonals cross at $90^\circ$. In the left triangle: $\angle 3 + \angle 1 + 90 = 180$. $\angle 3 = 90 - 26 = 64^\circ$.
* Angle 4 is alternate interior to angle 3. $\angle 4 = 64^\circ$.
* Answer: $m\angle 1 = 26^\circ, m\angle 2 = 52^\circ, m\angle 3 = 64^\circ, m\angle 4 = 64^\circ$.
10.
* Angle 4 is opposite the $59^\circ$ angle, so $\angle 4 = 59^\circ$.
* The diagonal bisects the angle, so $\angle 3$ is half of $59^\circ$. $\angle 3 = 29.5^\circ$.
* Diagonals cross at $90^\circ$. In the bottom triangle: $\angle 2 + \angle 3 + 90 = 180$. $\angle 2 = 90 - 29.5 = 60.5^\circ$.
* Angle 1 is alternate interior to angle 2. $\angle 1 = 60.5^\circ$.
* Answer: $m\angle 1 = 60.5^\circ, m\angle 2 = 60.5^\circ, m\angle 3 = 29.5^\circ, m\angle 4 = 59^\circ$.
11.
* This looks identical to Problem 9.
* Angle 2 is opposite $52^\circ$, so $\angle 2 = 52^\circ$.
* Angle 1 is half of $52^\circ$, so $\angle 1 = 26^\circ$.
* Angle 3 is $90 - 26 = 64^\circ$.
* Angle 4 is alternate interior to angle 3, so $\angle 4 = 64^\circ$.
* Answer: $m\angle 1 = 26^\circ, m\angle 2 = 52^\circ, m\angle 3 = 64^\circ, m\angle 4 = 64^\circ$.
12.
* We are given a partial angle of $39^\circ$. The diagonal bisects the corner, so the full angle is $78^\circ$.
* Consecutive angles add to $180^\circ$. The adjacent corner angle is $180 - 78 = 102^\circ$.
* Angle 2 is half of that $102^\circ$ angle. $\angle 2 = 51^\circ$.
* Diagonals cross at $90^\circ$. In the top triangle: $\angle 1 + \angle 2 + 90 = 180$. $\angle 1 = 90 - 51 = 39^\circ$.
* Angle 3 is alternate interior to the given $39^\circ$ angle. $\angle 3 = 39^\circ$.
* Angle 4 is alternate interior to angle 1. $\angle 4 = 39^\circ$. (Wait, let's recheck. Angle 4 is part of the bottom right corner. The bottom right corner is opposite the top left ($102^\circ$). So angle 4 is half of $102^\circ$, which is $51^\circ$? No, look at the position. Angle 4 is adjacent to the side. Let's use the triangle method. Bottom triangle has angle 3 ($39^\circ$) and center ($90^\circ$). So the other angle is $90-39=51^\circ$. That is the full half-angle. Angle 4 is labeled as the angle between the diagonal and the side. Yes, $\angle 4 = 51^\circ$.)
* Correction: Let's trace carefully.
* Given angle at left tip is split into $39^\circ$ and $39^\circ$. Total left angle = $78^\circ$.
* Top-left angle + Left angle = $180^\circ$. Top-left angle = $102^\circ$.
* Diagonal bisects Top-left. So $\angle 2 = 51^\circ$.
* Triangle containing $\angle 1$ and $\angle 2$ is a right triangle. $\angle 1 = 90 - 51 = 39^\circ$.
* $\angle 3$ is alternate interior to the top part of the left angle ($39^\circ$). So $\angle 3 = 39^\circ$.
* $\angle 4$ is in the bottom right. The bottom right angle is opposite the top left ($102^\circ$). The diagonal bisects it. So $\angle 4 = 51^\circ$.
* Answer: $m\angle 1 = 39^\circ, m\angle 2 = 51^\circ, m\angle 3 = 39^\circ, m\angle 4 = 51^\circ$.
13.
* We are given a partial angle of $49^\circ$. The diagonal bisects the corner, so the full bottom-right angle is $98^\circ$.
* Opposite angles are equal, so the top-left angle is also $98^\circ$.
* Angle 2 is half of the top-left angle. $\angle 2 = 49^\circ$.
* Consecutive angles add to $180^\circ$. Bottom-left angle = $180 - 98 = 82^\circ$.
* Angle 3 is half of the bottom-left angle. $\angle 3 = 41^\circ$.
* Diagonals cross at $90^\circ$. In the bottom triangle: $\angle 4 + \angle 3 + 90 = 180$. $\angle 4 = 90 - 41 = 49^\circ$.
* Angle 1 is alternate interior to angle 4? No, Angle 1 is half of the top-right angle. Top-right is opposite bottom-left ($82^\circ$). So $\angle 1 = 41^\circ$.
* Answer: $m\angle 1 = 41^\circ, m\angle 2 = 49^\circ, m\angle 3 = 41^\circ, m\angle 4 = 49^\circ$.
---
Part 3: Algebra with Rectangles (Problems 14–17)
Rule: The diagonals of a rectangle are equal in length. Set the expressions for the diagonals equal to each other to solve for $x$. Then plug $x$ back in to find the length.
14. $HJ = x$ and $IK = 2x - 7$
* Set equal: $x = 2x - 7$
* Subtract $x$ from both sides: $0 = x - 7$
* Add 7 to both sides: $x = 7$
* Length: $HJ = 7$. (Check: $IK = 2(7)-7 = 7$).
* Answer: $x = 7$, Length = $7$.
15. $HJ = 3x + 5$ and $IK = 5x - 9$
* Set equal: $3x + 5 = 5x - 9$
* Subtract $3x$ from both sides: $5 = 2x - 9$
* Add 9 to both sides: $14 = 2x$
* Divide by 2: $x = 7$
* Length: $HJ = 3(7) + 5 = 21 + 5 = 26$.
* Answer: $x = 7$, Length = $26$.
16. $HJ = 3x + 7$ and $IK = 6x - 11$
* Set equal: $3x + 7 = 6x - 11$
* Subtract $3x$ from both sides: $7 = 3x - 11$
* Add 11 to both sides: $18 = 3x$
* Divide by 3: $x = 6$
* Length: $HJ = 3(6) + 7 = 18 + 7 = 25$.
* Answer: $x = 6$, Length = $25$.
17. $HJ = 19 + 2x$ and $IK = 3x + 22$
* Set equal: $19 + 2x = 3x + 22$
* Subtract $2x$ from both sides: $19 = x + 22$
* Subtract 22 from both sides: $-3 = x$
* Length: $HJ = 19 + 2(-3) = 19 - 6 = 13$.
* Check: $IK = 3(-3) + 22 = -9 + 22 = 13$.
* Answer: $x = -3$, Length = $13$.
Final Answer:
1. Square
2. Square
3. Rhombus
4. Rectangle
5. Rhombus
6. $m\angle 1=63^\circ, m\angle 2=54^\circ, m\angle 3=27^\circ, m\angle 4=63^\circ$
7. $m\angle 1=68^\circ, m\angle 2=44^\circ, m\angle 3=46^\circ, m\angle 4=68^\circ$
8. $m\angle 1=70^\circ, m\angle 2=70^\circ, m\angle 3=20^\circ, m\angle 4=20^\circ$
9. $m\angle 1=26^\circ, m\angle 2=52^\circ, m\angle 3=64^\circ, m\angle 4=64^\circ$
10. $m\angle 1=60.5^\circ, m\angle 2=60.5^\circ, m\angle 3=29.5^\circ, m\angle 4=59^\circ$
11. $m\angle 1=26^\circ, m\angle 2=52^\circ, m\angle 3=64^\circ, m\angle 4=64^\circ$
12. $m\angle 1=39^\circ, m\angle 2=51^\circ, m\angle 3=39^\circ, m\angle 4=51^\circ$
13. $m\angle 1=41^\circ, m\angle 2=49^\circ, m\angle 3=41^\circ, m\angle 4=49^\circ$
14. $x=7$, Length $= 7$
15. $x=7$, Length $= 26$
16. $x=6$, Length $= 25$
17. $x=-3$, Length $= 13$
Parent Tip: Review the logic above to help your child master the concept of rectangle rhombus square worksheet.
