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Quizzes > Mathematics > Geometry

Master Your Geometry Final Exam: Take the Test Now!

Think you can ace this geometry practice quiz? Dive in and prove it!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration of compass ruler protractor geometric shapes on golden yellow background for scored geometry quiz

Ready to conquer your geometry final exam test and boost your confidence? Dive into our free geometry exam review with this scored quiz designed for high school students, educators, and geometry enthusiasts alike. You'll tackle essential geometry test questions on theorems, angle relationships, formulas, and proofs, while reinforcing your semester geometry final review in an engaging, self-paced format. By solving real-world problems and practicing step-by-step responses, you'll build mastery, sharpen problem-solving, and reduce test anxiety. Start with our geometry final exam practice test to measure your strengths, then take on a fun trivia challenge to level up. Click now to begin your geometry practice quiz and ace your next exam!

What is the sum of interior angles of a triangle?
180°
270°
90°
360°
By Euclid's postulates, the interior angles of any triangle always add up to 180°. This fundamental result follows from parallel line properties. It is proven in most geometry texts and is essential for many angle-chasing problems. Wikipedia
Each interior angle of an equilateral triangle measures what?
60°
90°
45°
120°
In an equilateral triangle, all sides and all angles are equal. Dividing the total 180° by three yields 60° per angle. This is a standard property in Euclidean geometry. Wikipedia
What defines a right angle?
An angle of 180°
An angle of 90°
An angle of 60°
An angle of 45°
A right angle is defined as exactly 90° by definition. It arises when two lines are perpendicular. This concept is the basis of right triangles and coordinate geometry. Wikipedia
What is the sum of interior angles of any quadrilateral?
360°
180°
270°
540°
A quadrilateral can be split into two triangles, each contributing 180°. Therefore, the combined interior angles sum to 360°. This approach generalizes to n-gons as well. Wikipedia
Parallel lines in a plane never do what?
Change slope
Create triangles
Meet
Form right angles
By definition, parallel lines remain equidistant and never intersect in Euclidean geometry. They have the same slope when described in coordinate form. This is foundational for many parallel?angle theorems. Wikipedia
What is the area formula for a rectangle?
length × width
?r²
½...base...height
2(length + width)
The rectangle's area equals the product of its adjacent side lengths. This follows because you tile unit squares across length and width. It's one of the first area formulas taught in geometry. Wikipedia
What is the circumference of a circle of radius r?
?r²
2?r
?r/2
?d
Circumference measures the perimeter of a circle. It is defined as 2? times the radius in Euclidean plane geometry. The constant ? relates linear distance to diameter. Wikipedia
Which theorem relates the sides of a right triangle?
Pythagorean theorem
Menelaus' theorem
Euler's theorem
Ceva's theorem
The Pythagorean theorem states a² + b² = c² for right triangles. It connects the legs to the hypotenuse. This is foundational for distance calculations. Wikipedia
An isosceles triangle has how many equal sides?
One
Three
None
Two
By definition, an isosceles triangle has exactly two equal sides. The base angles opposite those sides are also equal. This property is used in many geometric proofs. Wikipedia
Supplementary angles sum to what measure?
180°
90°
270°
360°
Two angles are supplementary if their measures add to 180°. This is used when analyzing linear pairs. It is fundamental to parallel line angle relationships. Wikipedia
What is the diameter of a circle relative to its radius?
?×radius
½×radius
2×radius
radius²
Diameter is twice the radius by definition. It spans the circle passing through the center. This relation is key to many circle formulas. Wikipedia
In polygons, a vertex is what?
Intersection of two sides
Area unit
Parallel line
Center point
A vertex is where two edges meet in a polygon. It is a fundamental part of polyhedral geometry. Counting vertices helps classify shapes. Wikipedia
Perpendicular lines form what angle?
60°
120°
90°
180°
Perpendicular lines intersect to form right angles of 90°. This concept underlies orthogonal systems. It's crucial in coordinate geometry. Wikipedia
A chord in a circle is defined as what?
A tangent line
A radius extended
The circle's center
A segment with endpoints on the circle
A chord connects two points on the circumference. It does not have to pass through the center. It generalizes diameters when endpoints are opposite. Wikipedia
Area of a triangle equals?
½ × base × height
base + height
base × height
base² + height²
Triangle area is one?half of base times height due to its half?parallelogram nature. This formula appears in all standard geometry curriculums. It follows from slicing a rectangle. Wikipedia
A convex polygon always has interior angles less than what?
180°
360°
270°
90°
In a convex polygon, each interior angle is strictly under 180°. No angle "points inward." This ensures the shape has no indentations. Wikipedia
In a triangle, two angles measure 50° and 70°. What is the third angle?
80°
50°
60°
40°
The angles of a triangle total 180°. Subtracting 50° and 70° leaves 60°. This is basic angle?sum application. Wikipedia
What is the hypotenuse of a right triangle with legs 3 and 4?
4
5
6
7
Pythagorean theorem gives c = sqrt(3² + 4²) = 5. This classic (3,4,5) triangle is well known. It illustrates integer solutions. Wikipedia
What is the area of a circle with radius 5?
10?
100?
25?
5?
Area = ?r². Plugging r=5 yields 25?. This formula is proven via integration or rearrangement. Wikipedia
A circle's arc length for 60° central angle on radius 10 is?
(?/6)×10
(?/3)×10
(?/2)×10
(2?/3)×10
Arc length = (?/360°)×2?r. For ?=60°, that is (1/6)×2?×10 = (?/3)×10. This uses proportional arc measure. Wikipedia
What is the area of a 60° sector in a circle radius 8?
(1/6)?×64
(1/2)?×64
(1/4)?×64
(1/3)?×64
Sector area = (?/360°)×?r². Here ?=60°, so one-sixth of ?×64 = (64?/6). This derives from circle area proportion. Wikipedia
Alternate interior angles formed by a transversal and two parallel lines are:
Complementary
Equal
Supplementary
Right angles
When two parallel lines are cut by a transversal, alternate interior angles are congruent. This is a key parallel-line theorem. It's used heavily in proofs. Wikipedia
The slope of a line perpendicular to a line of slope 3/4 is:
-3/4
3/4
-4/3
4/3
Perpendicular slopes multiply to -1. The negative reciprocal of 3/4 is -4/3. This is fundamental in analytic geometry. Wikipedia
Volume of a cylinder radius 3 and height 7 is?
21?
63?
9?
44?
Volume = ?r²h = ?×9×7 = 63?. This formula extends circle area into three dimensions. Wikipedia
Surface area of that cylinder (radius 3, height 7) is?
63?
42?
60?
80?
Surface area = 2?r(h + r) = 2?×3(7+3)=60?. It includes two bases and lateral area. Wikipedia
Volume of a right cone with radius 4, height 9 is?
12?
48?
36?
144?
Cone volume = (1/3)?r²h = (1/3)?×16×9=48?. This follows by slicing a cylinder. Wikipedia
Area of a parallelogram with base 5 and height 8 is?
40
20
30
13
Area = base × height = 5×8=40. This is analogous to a rectangle under shear. Wikipedia
Sum of exterior angles of any convex polygon equals:
720°
540°
360°
180°
Walking around a convex polygon turns you 360° total. Each exterior is supplementary to interior. This holds regardless of side count. Wikipedia
An angle inscribed in a semicircle is always:
60°
45°
90°
120°
Thales' theorem states an angle in a semicircle is right. The diameter subtends 90° at the circle. This is key in many circle proofs. Wikipedia
Angle between tangent and radius at the point of contact is:
60°
45°
180°
90°
A tangent is perpendicular to the radius at the point of contact. This is a standard circle property. It's applied in many geometry problems. Wikipedia
Sum of interior angles of an n-sided convex polygon is?
(n?2)×180°
2n×90°
(n?1)×180°
n×180°
Dividing an n-gon into n?2 triangles yields total (n?2)×180°. This holds for any simple convex polygon. It generalizes the triangle case. Wikipedia
Law of sines relates sides a, b, c to angles A, B, C as?
a/sinA = b/sinB = c/sinC
sinA/a = sinB/b = sinC/c
a×sinA = b×sinB = c×sinC
a/sinB = b/sinC = c/sinA
Law of sines: a/sinA = b/sinB = c/sinC in any triangle. It stems from considering the diameter of the circumcircle. It solves non-right triangles. Wikipedia
Law of cosines: c² equals?
a² ? b² + 2ab cosC
a² + b² ? ab cosC
a² + b² + 2ab cosC
a² + b² ? 2ab cosC
Law of cosines generalizes Pythagoras: c² = a² + b² ? 2ab cosC. It applies in any triangle. It arises from projecting one side onto another. Wikipedia
Heron's formula gives triangle area as?
?[s(s?a)(s?b)(s?c)]
½bc sinA
½ab sinC
?[abc/(a+b+c)]
Heron's formula uses semiperimeter s=(a+b+c)/2. Then area=?[s(s?a)(s?b)(s?c)]. It requires only side lengths. Wikipedia
Incenter of a triangle is the intersection of what?
Medians
Altitudes
Angle bisectors
Perpendicular bisectors
The incenter is where the three angle bisectors meet. It is the center of the inscribed circle. This is a classical concurrency result. Wikipedia
Circumcenter is intersection of which lines?
Altitudes
Perpendicular bisectors
Medians
Angle bisectors
The circumcenter is equidistant from all vertices. It's found by intersecting perpendicular bisectors of each side. This lies inside acute triangles, on the hypotenuse of right. Wikipedia
Euler line of a triangle passes through which points?
Orthocenter, centroid, circumcenter
Incenter, centroid, circumcenter
Circumcenter, incenter, foot of altitude
Orthocenter, incenter, centroid
Euler's line contains orthocenter, centroid, and circumcenter. These three are collinear in any non?equilateral triangle. The centroid divides the segment 2:1. Wikipedia
Nine-point circle goes through midpoints of sides and what else?
Feet of altitudes
Excenters
Vertices
Incenter
The nine-point circle passes through the three side midpoints, three altitude feet, and midpoints from orthocenter to vertices. It's a key triangle circle. Wikipedia
Inscribed angle theorem states an inscribed angle equals:
Half its intercepted arc
Double its intercepted arc
Equal to arc
Complement of arc
An inscribed angle measures half the central angle on the same arc. Thus it's half the intercepted arc. This is central in circle geometry. Wikipedia
Tangent-secant theorem relates tangent t and secant segments p, q as?
t² = p+q
t² = p×q
t = p×q
t = p+q
Tangent-secant power theorem: tangent squared = product of full secant and its external piece. It follows from similar triangles in circle. Wikipedia
Ceva's theorem holds concurrency of cevians when?
(AF+FB)/(BD+DC)(CE+EA)=1
AF+FB=BD+DC=CE+EA
AF×FB=BD×DC=CE×EA
(AF/FB)(BD/DC)(CE/EA)=1
Ceva's theorem states three cevians are concurrent if product of ratios equals one. It's proved using similar triangles. It's critical in triangle concurrency proofs. Wikipedia
Menelaus' theorem applies to points on the sides of a triangle when they are:
Equidistant
Collinear
Concurrent
Concyclic
Menelaus' theorem gives a product of directed segment ratios = 1 when points are collinear. It's dual to Ceva's. It's used for transversal proofs. Wikipedia
Opposite angles in a cyclic quadrilateral sum to what?
180°
360°
90°
270°
In a cyclic quadrilateral, opposite angles are supplementary. This follows from inscribed angle theorem. It's a core property of cyclic figures. Wikipedia
Simson line of a triangle is the locus of the feet of perpendiculars from a point on its:
Steiner circle
Circumcircle
Nine-point circle
Incircle
The Simson line is traced by perpendicular projections of a point on the circumcircle onto the triangle's sides. All three feet are collinear. This is a deep circle?triangle relationship. Wikipedia
Euler characteristic ? for a convex polyhedron satisfies which formula?
E ? V + F = 2
V ? E + F = 2
V ? E ? F = 2
V + E ? F = 2
Euler's formula for convex polyhedra states vertices minus edges plus faces equals 2. It's foundational in topology. It extends beyond planar graphs. Wikipedia
Steiner's theorem relates to the moment of inertia when shifting axes by a distance d as I = I_cm + md². What is m here?
Centroid distance
Mass of the object
Moment of inertia
Radius of gyration
Steiner's (parallel?axis) theorem adds md² to the centroid moment of inertia when shifting by d. Here m is the total mass. It's widely used in mechanics. Wikipedia
Inversion in a circle transforms a line not through the center into:
A congruent line
A circle through the center of inversion
A hyperbola
A parallel line
Circle inversion maps lines not passing through the center into circles that pass through the center. This holds by power?of?a?point arguments. It's fundamental in advanced Euclidean transformations. Wikipedia
Cross ratio of four collinear points A,B,C,D is invariant under which transformation?
Inversions only
Rigid motions
Similarities only
Projective transformations
The cross ratio (A, B; C, D) is preserved by all projective maps. This makes it central in projective geometry. It extends invariants in perspective drawing. Wikipedia
Radical axis of two circles is the set of points having equal what?
Angle between tangents
Sum of radii
Power with respect to both circles
Distance to centers
The radical axis consists of points whose powers to each circle are equal. It's a line perpendicular to the centers' line. It's key in coaxal circle families. Wikipedia
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Study Outcomes

  1. Master Core Theorems -

    After completing the quiz, you will identify and recall fundamental theorems such as the Pythagorean theorem, similarity criteria, and circle theorems to tackle geometry test questions with confidence.

  2. Apply Key Formulas -

    You will practice using formulas for area, volume, perimeter, and angles to solve real-world problems in this free geometry exam review setting.

  3. Solve Varied Geometry Problems -

    Engage with a diverse set of geometry practice quiz questions - including angle calculations, coordinate proofs, and shape properties - to enhance your problem-solving skills.

  4. Analyze Proof Techniques -

    Learn how to construct and evaluate logical geometric proofs, reinforcing your understanding of reasoning steps and proof structure.

  5. Review Semester Concepts -

    Perform a comprehensive semester geometry final review that revisits key topics and ensures retention of essential concepts covered throughout the course.

  6. Evaluate Exam Readiness -

    Receive immediate scoring feedback to assess your performance on the geometry final exam test and pinpoint areas for targeted improvement.

Cheat Sheet

  1. Triangle Congruence Criteria -

    Review SSS, SAS, ASA, and AAS postulates to efficiently prove triangle congruence in geometry test questions. A handy mnemonic is "Some Say A Snake Ate Apples" to recall SSS, SAS, SSA, ASA, AAS - just remember SSA doesn't guarantee congruence. Master these rules to breeze through your geometry final exam test proofs (source: University of Texas Lecture Notes).

  2. Pythagorean Theorem & Converse -

    Understand c² = a² + b² for right triangles and recognize the converse to verify right angles in a geometry practice quiz. For instance, (3,4,5) and (5,12,13) triples work like magic - try listing a few yourself to memorize them. This theorem is a staple in any free geometry exam review and proves invaluable in coordinate and classical problems (source: Khan Academy).

  3. Circle Theorems -

    Focus on the Inscribed Angle Theorem and the Tangent - Secant Theorem to solve arc and chord problems on your semester geometry final review. Remember: an inscribed angle is half its intercepted arc, and the tangent - secant power relation is PA·PB = PC² when tangent touches the circle. These circle insights will accelerate your work on geometry test questions involving arcs and chords (source: Wolfram MathWorld).

  4. Coordinate Geometry Essentials -

    Master the distance formula √[(x₂ - x₝)² + (y₂ - y₝)²] and midpoint formula ((x₝+x₂)/2, (y₝+y₂)/2) to tackle analytic geometry sections of the geometry final exam test. Practice by plotting points like ( - 1,2) and (3,5) to calculate distance and midpoint quickly. These tools offer clear, algebraic paths to solve geometric problems without guesswork (source: MIT OpenCourseWare).

  5. Area & Perimeter Formulas -

    Consolidate formulas for triangles (½·base·height), circles (πr²), sectors (½r²θ in radians), and regular polygons (½·apothem·perimeter) to ace area questions in your geometry practice quiz. Use the phrase "A Pretty Crown Sits Neatly" to recall Area, Perimeter, Circumference, Sector, and Number of sides. Having these on instant recall will boost speed and accuracy during a free geometry exam review (source: University of Cambridge).

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