Free printable summer review

Geometry Summer Review

A free 40-problem Geometry summer review PDF with 36 geometric diagrams, exact-value practice, proofs, transformations, similarity, trigonometry, circles, coordinate geometry, area, volume, and a complete answer key.

40 problems8 skill areasAbout 110 minutesAnswer key included
Download the free worksheet PDF

What this review measures

Geometry builds a style of reasoning as much as a list of formulas. This expanded packet asks students to read and label diagrams, connect algebra to geometric relationships, justify conclusions, and retain the measurement and coordinate skills that transfer directly into Algebra 2 and later mathematics.

For parents and educators

Ask students to mark every given length, angle, and congruence relationship directly on the diagrams. Exact answers should stay in fraction, radical, or pi form unless a problem explicitly requests a decimal.

Preview the complete review

Every question below appears in the printable packet. Students should show their work and simplify answers unless directed otherwise.

Angles and reasoning

Use angle relationships, parallel lines, and conditional statements.

  1. 1Angle A measures 37\displaystyle 37 degrees. Its complement is angle B. Find m(B)\displaystyle m(B).37 degrees
  2. 2The angle shown and angle C form a linear pair. Find m(C)\displaystyle m(C).118 degrees
  3. 3Two parallel lines are cut by a transversal. The marked angle is 64\displaystyle 64 degrees. Find the corresponding angle at the other intersection.64 degrees
  4. 4A triangle has angles measuring 48\displaystyle 48 degrees and 67\displaystyle 67 degrees. Find the third angle.angles 48 degrees, 67 degrees, and x
  5. 5State the converse of: If a figure is a square, then it is a rectangle. Is the converse always true?square and rectangle relationships

Transformations and congruence

Apply coordinate rules and connect rigid motions to congruence.

  1. 6Translate triangle vertices (1,2)\displaystyle (1,2), (4,2)\displaystyle (4,2), and (1,6)\displaystyle (1,6) by ⟨3,−1⟩\displaystyle \langle3,-1\rangle.translate 3 right and 1 down
  2. 7Reflect (2,5)\displaystyle (2,5) across the x\displaystyle x-axis.
  3. 8Rotate (3,−5)\displaystyle (3,-5) 90 degrees counterclockwise about the origin.
  4. 9Triangle ABC is shown. Translate every vertex 1 unit right and 2 units up. List the image coordinates.ABC
  5. 10Explain why a sequence of translations, rotations, and reflections preserves congruence.rigid motions preserve size and shape

Triangles and similarity

Use congruence, similarity, and proportional segments.

  1. 11Triangles ABC and DEF are similar. AB=8\displaystyle AB=8, DE=12\displaystyle DE=12, and BC=10\displaystyle BC=10. Find EF\displaystyle EF.AB = 8, DE = 12, BC = 10, EF = ?
  2. 12An isosceles triangle has a vertex angle of 38\displaystyle 38 degrees. Find each base angle.vertex angle 38 degrees; congruent legs
  3. 13Can side lengths 4\displaystyle 4, 7\displaystyle 7, and 12\displaystyle 12 form a triangle? Explain.possible sides 4, 7, and 12
  4. 14A segment joins the midpoints of two sides of a triangle. If the third side is 18\displaystyle 18, find the midsegment length.third side 18; midpoint segment x
  5. 15An angle bisector divides the opposite side into lengths 6\displaystyle 6 and 9\displaystyle 9. The adjacent side corresponding to 6\displaystyle 6 is 12\displaystyle 12. Find the other adjacent side.opposite segments 6 and 9; adjacent sides 12 and x
  6. 16Two triangles have two pairs of congruent sides and congruent included angles. Name the congruence theorem.two sides and the included angle are marked

Right triangles and trigonometry

Use the Pythagorean theorem, special triangles, and right-triangle ratios.

  1. 17Find the missing leg of the right triangle.leg 5; hypotenuse 13; missing leg x
  2. 18Relative to angle A, the opposite leg is 9\displaystyle 9 and the adjacent leg is 12\displaystyle 12. Find tan⁡(A)\displaystyle \tan(A).opposite 9; adjacent 12
  3. 19A 20-foot ladder reaches 16\displaystyle 16 feet up a wall. How far is its base from the wall?height 16 feet; ladder 20 feet
  4. 20A 45-45-90 triangle has a leg of length 7\displaystyle 7. Find the exact hypotenuse.45-45-90 triangle; leg 7
  5. 21A ramp rises 3\displaystyle 3 feet over a horizontal run of 4\displaystyle 4 feet. Find the sine, cosine, and tangent of its angle with the ground.rise 3; run 4; ramp 5

Circles

Use circle equations, circumference, area, arcs, and tangent properties.

  1. 22Write the equation of a circle centered at (2,−3)\displaystyle (2,-3) with radius 5\displaystyle 5.r = 5
  2. 23Find the exact circumference of the circle.d = 14
  3. 24Find the exact area of the circle.r = 6
  4. 25A 60\displaystyle 60-degree central angle intercepts an arc in a circle of radius 9\displaystyle 9. Find the exact arc length.r = 960 degrees
  5. 26A radius meets a tangent at the point of tangency. What angle do they form?r

Area and volume

Solve two-dimensional and three-dimensional measurement problems with correct units.

  1. 27Find the area of the rectangle.127
  2. 28Find the area of the trapezoid.bases 10 and 16; height 7
  3. 29A rhombus has diagonals 10\displaystyle 10 and 24\displaystyle 24. Find its area.diagonals 10 and 24
  4. 30A square has diagonal length 10\displaystyle 10. Find its exact area.diagonal 10
  5. 31Find the volume of the rectangular prism.8 cm3 cm5 cm
  6. 32Find the surface area of the rectangular prism.6 in.5 in.4 in.

Coordinate geometry

Use slope, midpoint, distance, and area to verify geometric relationships.

  1. 33Points A and B are shown. Find the exact distance between them.AB
  2. 34Find the midpoint of (−7,5)\displaystyle (-7,5) and (9,−3)\displaystyle (9,-3).
  3. 35Do points A, B, C, and D form a parallelogram? Justify with slopes.ABCD
  4. 36A segment has midpoint (4,−1)\displaystyle (4,-1) and one endpoint (9,6)\displaystyle (9,6). Find the other endpoint.
  5. 37Triangle ABC is shown. Classify it by angle type and find its area.ABC

Challenge

Combine proof, circle, coordinate, and area ideas.

  1. 38A chord is 8\displaystyle 8 units from a circle's center. The radius is 10\displaystyle 10. Find the chord length.r = 10
  2. 39Explain why the diagonals of a rectangle are congruent using coordinates.vertices (0,0), (a,0), (a,b), (0,b)
  3. 40A circle is inscribed in a square with side length 10\displaystyle 10. Find the exact area inside the square but outside the circle.side 10
View the complete answer key

Open-ended explanations may use different wording while showing the same reasoning.

Angles and reasoning

1. 53\displaystyle 53 degrees

2. 62\displaystyle 62 degrees

3. 64\displaystyle 64 degrees

4. 65\displaystyle 65 degrees

5. If a figure is a rectangle, then it is a square; the converse is false

Transformations and congruence

6. (4,1)\displaystyle (4,1), (7,1)\displaystyle (7,1), and (4,5)\displaystyle (4,5)

7. (2,−5)\displaystyle (2,-5)

8. (5,3)\displaystyle (5,3)

9. (3,4)\displaystyle (3,4), (7,4)\displaystyle (7,4), and (5,7)\displaystyle (5,7)

10. Rigid motions preserve every distance and angle measure

Triangles and similarity

11. 15\displaystyle 15

12. 71\displaystyle 71 degrees

13. No; 4+7<12\displaystyle 4+7<12

14. 9\displaystyle 9

15. 18\displaystyle 18

16. SAS congruence

Right triangles and trigonometry

17. 12\displaystyle 12

18. 34\displaystyle \frac{3}{4}

19. 12\displaystyle 12 feet

20. 72\displaystyle 7\sqrt{2}

21. sin⁡(A)=35\displaystyle \sin(A)=\frac{3}{5}, cos⁡(A)=45\displaystyle \cos(A)=\frac{4}{5}, tan⁡(A)=34\displaystyle \tan(A)=\frac{3}{4}

Circles

22. (x−2)2+(y+3)2=25\displaystyle (x-2)^{2}+(y+3)^{2}=25

23. 14π\displaystyle 14\pi units

24. 36π\displaystyle 36\pi square units

25. 3π\displaystyle 3\pi units

26. 90\displaystyle 90 degrees

Area and volume

27. 84\displaystyle 84 square units

28. 91\displaystyle 91 square units

29. 120\displaystyle 120 square units

30. 50\displaystyle 50 square units

31. 120\displaystyle 120 cubic centimeters

32. 148\displaystyle 148 square inches

Coordinate geometry

33. 10\displaystyle 10 units

34. (1,1)\displaystyle (1,1)

35. Yes; opposite sides have slopes 12\displaystyle \frac{1}{2} and −2\displaystyle -2

36. (−1,−8)\displaystyle (-1,-8)

37. Right triangle; area 6\displaystyle 6 square units

Challenge

38. 12\displaystyle 12 units

39. Place vertices at (0,0)\displaystyle (0,0), (a,0)\displaystyle (a,0), (a,b)\displaystyle (a,b), and (0,b)\displaystyle (0,b). Both diagonals have length a2+b2\displaystyle \sqrt{a^{2}+b^{2}}

40. 100−25π\displaystyle 100-25\pi square units

All summer math reviewsAlgebra 1 Summer ReviewAlgebra 2 Summer ReviewPractice one specific skillTry a readiness quiz