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Florida Geometry EOC study materials
Florida Geometry EOC study resources

Florida Geometry EOC practice questions

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Question 1Logic, Relationships, and Theorems

In a deductive argument, a conclusion is valid when it follows necessarily from the premises. Which statement best describes a valid argument whose premises are all true?

  1. Its conclusion must also be true. Correct answer.
  2. Its conclusion is true only if observed.
  3. Its conclusion is probably true.
  4. Its conclusion may be true or false.

Explanation

Validity guarantees that true premises force a true conclusion; this is the defining property of a sound deductive argument. Probability and observation belong to inductive reasoning, not deduction.

Question 2Logic, Relationships, and Theorems

A conditional statement has the form "If P, then Q." Which statement is its converse?

  1. If not P, then not Q.
  2. If Q, then P. Correct answer.
  3. If not Q, then not P.
  4. P and not Q.

Explanation

The converse swaps the hypothesis and conclusion, giving "If Q, then P." The other forms are the contrapositive, inverse, and negation of the conditional.

Question 3Logic, Relationships, and Theorems

Which statement is the contrapositive of "If a figure is a square, then it has four right angles"?

  1. If a figure does not have four right angles, then it is not a square. Correct answer.
  2. A figure is a square and has four right angles.
  3. If a figure has four right angles, then it is a square.
  4. If a figure is not a square, then it does not have four right angles.

Explanation

The contrapositive negates both parts and reverses their order: "If not Q, then not P." A statement and its contrapositive are logically equivalent.

Question 4Logic, Relationships, and Theorems

Consider the statement "If it rains, then the ground is wet." Which related statement is logically equivalent to it?

  1. If the ground is wet, then it rained.
  2. If the ground is not wet, then it did not rain. Correct answer.
  3. If it does not rain, then the ground is not wet.
  4. If it rains, then the ground is dry.

Explanation

Only the contrapositive is logically equivalent to the original conditional. It negates both clauses and reverses their order.

Question 5Logic, Relationships, and Theorems

Which pair of statements forms a biconditional that is true?

  1. A figure is a square if and only if it has four sides.
  2. A figure is a rhombus if and only if it has four sides.
  3. A figure is a rectangle if and only if it has four sides.
  4. A figure is a triangle if and only if it has three sides. Correct answer.

Explanation

A biconditional requires both the conditional and its converse to be true. Having three sides both defines and is defined by being a triangle, so both directions hold.

Question 6Logic, Relationships, and Theorems

A proof shows that assuming a statement is false leads to a contradiction. Which proof technique is being used?

  1. Direct proof
  2. Proof by construction
  3. Proof by counterexample
  4. Proof by contradiction Correct answer.

Explanation

Proof by contradiction assumes the negation of the claim and derives an impossibility, which forces the original claim to be true. Direct proof instead reasons forward from given facts.

Question 7Logic, Relationships, and Theorems

A student wants to disprove the claim "All quadrilaterals have diagonals that bisect each other." What is the most efficient approach?

  1. Assume the claim is true and derive a result.
  2. Prove the claim for a square.
  3. Find one quadrilateral whose diagonals do not bisect each other. Correct answer.
  4. Show the claim holds for every rectangle.

Explanation

A single counterexample is enough to disprove a universal statement. A quadrilateral such as a generic trapezoid whose diagonals do not bisect demonstrates the claim is false.

Question 8Logic, Relationships, and Theorems

In a two-column proof, which item belongs in the right-hand column?

  1. A diagram of the figure
  2. The statement being proved
  3. The given information
  4. The reason justifying each statement Correct answer.

Explanation

Two-column proofs list statements on the left and the justification for each statement on the right. The reasons cite definitions, postulates, and previously proven theorems.

Question 9Logic, Relationships, and Theorems

Two lines are cut by a transversal. Which condition guarantees that the two lines are parallel?

  1. A pair of same-side interior angles are congruent.
  2. A pair of corresponding angles are supplementary.
  3. A pair of vertical angles are congruent.
  4. A pair of alternate interior angles are congruent. Correct answer.

Explanation

If alternate interior angles are congruent, the lines must be parallel. This is the converse of the alternate interior angles theorem.

Question 10Logic, Relationships, and Theorems

In triangle ABC, the exterior angle at vertex C measures 118 degrees and one remote interior angle measures 47 degrees. What is the measure of the other remote interior angle?

  1. 47 degrees
  2. 133 degrees
  3. 118 degrees
  4. 71 degrees Correct answer.

Explanation

The Exterior Angle Theorem states that an exterior angle equals the sum of the two remote interior angles. So the missing angle is 118 minus 47, which is 71 degrees.

Question 11Logic, Relationships, and Theorems

A triangle has two angles measuring 65 degrees and 65 degrees. Which theorem most directly explains why the sides opposite those angles are congruent?

  1. Triangle Angle-Sum Theorem
  2. Exterior Angle Theorem
  3. Isosceles Triangle Theorem Correct answer.
  4. Triangle Inequality Theorem

Explanation

The Isosceles Triangle Theorem states that if two angles of a triangle are congruent, the sides opposite them are congruent. Equal base angles imply equal opposite sides.

Question 12Logic, Relationships, and Theorems

Which set of segment lengths can form a triangle?

  1. 1, 2, 4
  2. 5, 7, 11 Correct answer.
  3. 2, 3, 6
  4. 4, 5, 9

Explanation

The Triangle Inequality Theorem requires the sum of any two sides to exceed the third. For 5, 7, and 11, every pairwise sum exceeds the remaining side, so a triangle is possible.

Question 13Logic, Relationships, and Theorems

Two triangles are congruent by the SAS criterion. Which pair of parts must be congruent in addition to two pairs of corresponding sides?

  1. The perimeters of the triangles
  2. The included angles between those sides Correct answer.
  3. The third pair of sides
  4. A pair of non-included angles

Explanation

SAS requires two sides and the included angle (the angle formed by those sides) to be congruent. Without the included angle, congruence is not guaranteed.

Question 14Logic, Relationships, and Theorems

In triangles ABC and DEF, angle A is congruent to angle D, angle B is congruent to angle E, and side AB is congruent to side DE. Which congruence criterion applies?

  1. ASA Correct answer.
  2. AAA
  3. SSS
  4. SAS

Explanation

Two angles and the included side are congruent, which is the ASA criterion. The side AB lies between angles A and B, matching side DE between angles D and E.

Question 15Logic, Relationships, and Theorems

Which statement about similar triangles is always true?

  1. They have congruent corresponding sides.
  2. Their corresponding angles are congruent. Correct answer.
  3. They have equal areas.
  4. Their perimeters are equal.

Explanation

Similar triangles have congruent corresponding angles and proportional corresponding sides. Equal areas, equal sides, and equal perimeters belong to congruent triangles, not similar ones.

Question 16Logic, Relationships, and Theorems

Two triangles are similar with a scale factor of 3 from the smaller to the larger. If the smaller triangle has an area of 10 square units, what is the area of the larger triangle?

  1. 90 square units Correct answer.
  2. 120 square units
  3. 30 square units
  4. 60 square units

Explanation

Areas of similar figures scale by the square of the linear scale factor. Since 3 squared is 9, the larger area is 10 times 9, or 90 square units.

Question 17Logic, Relationships, and Theorems

A line segment joins the midpoints of two sides of a triangle. Which statement is guaranteed by the Triangle Midsegment Theorem?

  1. The segment bisects the third side.
  2. The segment equals the third side in length.
  3. The segment is perpendicular to the third side.
  4. The segment is parallel to the third side and half its length. Correct answer.

Explanation

The Triangle Midsegment Theorem states that a midsegment is parallel to the third side and half as long. This follows from the proportionality of the two sides.

Question 18Logic, Relationships, and Theorems

In a right triangle, the legs measure 6 and 8. What is the length of the hypotenuse?

  1. 48
  2. 10 Correct answer.
  3. 12
  4. 14

Explanation

By the Pythagorean Theorem, the hypotenuse squared equals 6 squared plus 8 squared, which is 36 plus 64, or 100. The hypotenuse is the square root of 100, which is 10.

Question 19Logic, Relationships, and Theorems

A triangle has sides 5, 12, and 13. Which classification is correct?

  1. Isosceles triangle
  2. Obtuse triangle
  3. Acute triangle
  4. Right triangle Correct answer.

Explanation

The converse of the Pythagorean Theorem applies: since 5 squared plus 12 squared equals 13 squared (25 plus 144 equals 169), the triangle is right. The right angle is opposite the longest side.

Question 20Logic, Relationships, and Theorems

In a 30-60-90 triangle, the side opposite the 30-degree angle measures 7. What is the length of the hypotenuse?

  1. 7 times the square root of 2
  2. 7 times the square root of 3
  3. 7
  4. 14 Correct answer.

Explanation

In a 30-60-90 triangle, the hypotenuse is twice the shorter leg, which is the side opposite the 30-degree angle. Twice 7 is 14.

Question 21Logic, Relationships, and Theorems

A quadrilateral has both pairs of opposite sides congruent. Which conclusion follows from this alone?

  1. The quadrilateral is a square.
  2. The quadrilateral is a rhombus.
  3. The quadrilateral is a rectangle.
  4. The quadrilateral is a parallelogram. Correct answer.

Explanation

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Additional conditions would be needed to classify it more specifically.

Question 22Logic, Relationships, and Theorems

In a parallelogram, one angle measures 72 degrees. What is the measure of an angle adjacent to it?

  1. 72 degrees
  2. 288 degrees
  3. 108 degrees Correct answer.
  4. 18 degrees

Explanation

Consecutive angles of a parallelogram are supplementary because they are same-side interior angles between parallel lines. So the adjacent angle is 180 minus 72, which is 108 degrees.

Question 23Logic, Relationships, and Theorems

A quadrilateral has diagonals that bisect each other and are also perpendicular. Which classification is guaranteed?

  1. Rectangle
  2. Isosceles trapezoid
  3. Square
  4. Rhombus Correct answer.

Explanation

Diagonals that bisect each other make the figure a parallelogram, and perpendicular diagonals in a parallelogram make it a rhombus. A square would also require congruent diagonals.

Question 24Logic, Relationships, and Theorems

A circle has a radius of 10. A chord is 6 units from the center. What is the length of the chord?

  1. 16 Correct answer.
  2. 8
  3. 12
  4. 20

Explanation

A radius drawn perpendicular to a chord bisects it. The perpendicular from the center, the radius, and half the chord form a right triangle with legs 6 and half the chord and hypotenuse 10. Half the chord is the square root of 100 minus 36, which is 8, so the chord is 16.

Question 25Congruence and Similarity

Triangle ABC is congruent to triangle DEF. Which statement must be true?

  1. Side AB is congruent to side EF
  2. Side AC is congruent to side DE
  3. Angle A is congruent to angle D Correct answer.
  4. Angle A is congruent to angle E

Explanation

In a congruence statement, corresponding vertices are listed in matching order, so A corresponds to D, B to E, and C to F. Corresponding angles and sides are therefore congruent.

Question 26Congruence and Similarity

Two triangles are congruent by the SSS congruence criterion. Which set of information is sufficient to establish that congruence?

  1. One pair of sides and two pairs of angles are congruent
  2. Two pairs of sides and a non-included angle are congruent
  3. All three pairs of corresponding angles are congruent
  4. All three pairs of corresponding sides are congruent Correct answer.

Explanation

SSS requires the three pairs of corresponding sides to be congruent, which forces the triangles to be congruent. Angle-only information determines shape but not size.

Question 27Congruence and Similarity

In triangles PQR and STU, angle P is congruent to angle S, side PQ is congruent to side ST, and angle Q is congruent to angle T. Which congruence criterion applies?

  1. SSS
  2. HL
  3. ASA Correct answer.
  4. SAS

Explanation

The congruent parts are two angles with the included side between them, which is the ASA criterion. The side lies between the two named angles in each triangle.

Question 28Congruence and Similarity

Triangle ABC has a right angle at C. Triangle DEF has a right angle at F. If hypotenuse AB is congruent to hypotenuse DE and leg AC is congruent to leg DF, which criterion proves the triangles congruent?

  1. HL Correct answer.
  2. SSA
  3. AAS
  4. SAS

Explanation

HL (hypotenuse-leg) applies to right triangles when the hypotenuse and one leg of one triangle are congruent to the hypotenuse and corresponding leg of the other.

Question 29Congruence and Similarity

Which transformation always produces a figure congruent to the original figure?

  1. A rotation Correct answer.
  2. A dilation with scale factor 3
  3. A vertical compression
  4. A horizontal stretch

Explanation

Rigid motions (translations, rotations, and reflections) preserve both side lengths and angle measures, so they map a figure onto a congruent figure. Dilations and stretches change size or shape.

Question 30Congruence and Similarity

A sequence of transformations maps triangle ABC onto triangle A'B'C'. The sequence is a translation followed by a reflection. What is the relationship between the two triangles?

  1. They are similar but not congruent
  2. They are neither similar nor congruent
  3. They are congruent only if the translation is zero
  4. They are congruent Correct answer.

Explanation

Translations and reflections are rigid motions, and any composition of rigid motions preserves distances and angles. Therefore the image triangle is congruent to the original.

Question 31Congruence and Similarity

Triangle ABC is similar to triangle DEF with a scale factor of 2 from ABC to DEF. If side AB has length 5, what is the length of the corresponding side DE?

  1. 10 Correct answer.
  2. 5
  3. 2.5
  4. 7

Explanation

In similar figures, corresponding sides are proportional by the scale factor. Multiplying AB by the scale factor 2 gives DE = 5 × 2 = 10.

Question 32Congruence and Similarity

Two triangles are similar. Which statement about them must be true?

  1. Corresponding angles are proportional and corresponding sides are congruent
  2. Corresponding angles are congruent and corresponding sides are proportional Correct answer.
  3. All corresponding angles are supplementary
  4. All corresponding sides are congruent

Explanation

Similarity means the triangles have the same shape: corresponding angles are congruent and corresponding side lengths are in a constant ratio.

Question 33Congruence and Similarity

In triangles ABC and DEF, angle A is congruent to angle D and angle B is congruent to angle E. Which conclusion follows?

  1. The triangles are similar only if a pair of sides is also congruent
  2. The triangles are congruent by ASA
  3. No conclusion can be drawn from two pairs of angles
  4. The triangles are similar by AA similarity Correct answer.

Explanation

The AA criterion states that if two angles of one triangle are congruent to two angles of another, the triangles are similar. The third pair of angles is then automatically congruent.

Question 34Congruence and Similarity

A triangle has side lengths 6, 8, and 10. A second triangle has side lengths 9, 12, and 15. Which statement is true?

  1. The triangles are not similar because the side lengths differ
  2. The triangles are congruent because both are right triangles
  3. The triangles are similar because all three side ratios equal 1.5 Correct answer.
  4. The triangles are similar because the differences of corresponding sides are equal

Explanation

Check the ratios of corresponding sides: 9/6 = 12/8 = 15/10 = 1.5. Since all three ratios are equal, the triangles are similar by SSS similarity.

Question 35Congruence and Similarity

In triangle ABC, point D lies on side AB and point E lies on side AC, with DE parallel to BC. If AD = 4, DB = 6, and AE = 6, what is the length of EC?

  1. 4
  2. 2.4
  3. 9 Correct answer.
  4. 10

Explanation

By the triangle proportionality theorem, a line parallel to one side divides the other two sides proportionally: AD/DB = AE/EC. So 4/6 = 6/EC, giving 4·EC = 36 and EC = 9.

Question 36Congruence and Similarity

Triangles ABC and DEF are similar, with AB = 8, BC = 12, and DE = 12. What is the length of EF?

  1. 9
  2. 8
  3. 18 Correct answer.
  4. 16

Explanation

The scale factor from ABC to DEF is DE/AB = 12/8 = 1.5. Applying it to BC gives EF = 12 × 1.5 = 18.

Question 37Congruence and Similarity

A 5-foot-tall person casts a shadow 8 feet long at the same time a nearby tree casts a shadow 32 feet long. How tall is the tree?

  1. 20 feet Correct answer.
  2. 51.2 feet
  3. 25 feet
  4. 12 feet

Explanation

The person and the tree form similar right triangles with the sun's rays as corresponding angles. So height/shadow is constant: 5/8 = h/32, giving h = 5 × 32 ÷ 8 = 20 feet.

Question 38Congruence and Similarity

In right triangle ABC, the altitude from the right angle at C to hypotenuse AB meets AB at point D. Which relationship is true?

  1. AC² = AD · DB
  2. CD² = AD · DB Correct answer.
  3. CD² = AD + DB
  4. CD = AD · DB

Explanation

The geometric mean theorem for a right triangle states that the altitude to the hypotenuse is the geometric mean of the two segments it creates: CD² = AD · DB.

Question 39Congruence and Similarity

Triangle ABC is rotated 90 degrees counterclockwise about the origin and then translated 3 units right. The image is triangle A'B'C'. Which statement is true?

  1. Triangle A'B'C' has a different perimeter from triangle ABC
  2. Triangle A'B'C' is similar but not congruent to triangle ABC
  3. Triangle A'B'C' is congruent to triangle ABC Correct answer.
  4. Triangle A'B'C' has larger angles than triangle ABC

Explanation

Rotations and translations are rigid motions that preserve all side lengths and angle measures. Any composition of rigid motions therefore maps a figure onto a congruent figure.

Question 40Congruence and Similarity

Which pair of triangles can be proven congruent using the SAS criterion?

  1. Three angles of one triangle congruent to three angles of the other
  2. Two sides and the included angle of one triangle congruent to two sides and the included angle of the other Correct answer.
  3. Two sides and a non-included angle of one triangle congruent to two sides and a non-included angle of the other
  4. Two angles and a non-included side of one triangle congruent to two angles and a non-included side of the other

Explanation

SAS requires two congruent sides with the angle between them also congruent. The included angle is essential to the criterion.

Question 41Congruence and Similarity

A dilation centered at the origin with scale factor 3/2 maps triangle ABC onto triangle A'B'C'. If triangle ABC has a side of length 10, what is the length of the corresponding side of triangle A'B'C'?

  1. 11.5
  2. 13
  3. 6.67
  4. 15 Correct answer.

Explanation

A dilation multiplies all lengths by the scale factor. Multiplying 10 by 3/2 gives 15.

Question 42Congruence and Similarity

Two similar triangles have a scale factor of 3 from the smaller to the larger. If the smaller triangle has an area of 12 square units, what is the area of the larger triangle?

  1. 48 square units
  2. 36 square units
  3. 108 square units Correct answer.
  4. 72 square units

Explanation

When figures are similar with scale factor k, areas are multiplied by k². Here k = 3, so the area ratio is 9, and 12 × 9 = 108.

Question 43Congruence and Similarity

In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD = 3, DB = 5, and BC = 16, what is the length of DE?

  1. 6 Correct answer.
  2. 10
  3. 12.8
  4. 9.6

Explanation

Because DE is parallel to BC, triangle ADE is similar to triangle ABC by AA similarity. The ratio AD/AB = 3/8, so DE = (3/8)(16) = 6.

Question 44Congruence and Similarity

In the coordinate plane, triangle ABC has vertices A(0,0), B(4,0), and C(0,3). Triangle DEF has vertices D(0,0), E(4,0), and F(0,3). Which single rigid motion maps ABC exactly onto DEF?

  1. A 90-degree rotation about the origin
  2. A dilation by scale factor 2 centered at the origin
  3. A reflection across the line y = x
  4. The identity transformation, which leaves each point fixed Correct answer.

Explanation

The vertices of ABC and DEF coincide exactly, so the identity transformation maps one triangle onto the other. Congruent figures can always be matched by some rigid motion.

Question 45Congruence and Similarity

Triangles ABC and DEF are congruent. Which statement about their corresponding parts must be true?

  1. The angles are proportional to the side lengths
  2. The perimeters are equal and the areas are equal Correct answer.
  3. The perimeters are proportional but the areas are equal
  4. The areas are proportional to the scale factor

Explanation

Congruent triangles have all corresponding side lengths and angle measures equal. Equal side lengths give equal perimeters, and congruent figures have equal areas.

Question 46Congruence and Similarity

In triangle ABC, point D lies on AB and point E lies on AC. It is known that AD/AB = AE/AC. Which conclusion follows?

  1. DE is parallel to BC Correct answer.
  2. DE is perpendicular to BC
  3. Triangle ADE is congruent to triangle ABC
  4. DE is the median to BC

Explanation

This is the converse of the triangle proportionality theorem: if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Question 47Congruence and Similarity

In right triangle ABC, the right angle is at C, and the altitude to hypotenuse AB meets AB at D. If AD = 9 and DB = 4, what is the length of AC?

  1. √13
  2. 13
  3. 6
  4. 3√13 Correct answer.

Explanation

By the leg theorem, AC² = AB · AD. Since AB = 9 + 4 = 13, AC² = 13 × 9 = 117, so AC = √117 = 3√13.

Question 48Congruence and Similarity

A dilation centered at the origin maps a segment of length 4 onto a segment of length 10. Which statement about the image of a triangle under this dilation is correct?

  1. Its angles are unchanged and its area is multiplied by 2.5
  2. Its angles are unchanged and its area is multiplied by 6.25 Correct answer.
  3. Its angles are enlarged and its area is multiplied by 2.5
  4. Its angles are enlarged and its area is multiplied by 6.25

Explanation

A dilation preserves angle measures and multiplies lengths by the scale factor k = 10/4 = 2.5. Areas are multiplied by k² = 6.25.

Question 49Congruence and Similarity

Two triangles are congruent. A side of the first triangle has length 7 and the corresponding side of the second triangle has length 2x + 1. An angle of the first triangle measures 48° and the corresponding angle of the second measures 3y. What are the values of x and y?

  1. x = 4 and y = 48
  2. x = 3 and y = 16 Correct answer.
  3. x = 3 and y = 48
  4. x = 4 and y = 16

Explanation

Corresponding parts of congruent triangles are equal. Solve 2x + 1 = 7 to get x = 3, and 3y = 48 to get y = 16.

Question 50Congruence and Similarity

Which transformation always produces a congruent figure?

  1. Translation Correct answer.
  2. Vertical stretch
  3. Dilation
  4. Horizontal stretch

Explanation

Rigid motions (translations, rotations, reflections) preserve side lengths and angle measures, so the image is congruent to the preimage. Dilations and stretches change size or shape and are not rigid motions.