Here are 100 multiple choice geometry questions following the specified format: Which of the following shapes has the most lines of symmetry? a) Rectangle b) Circle c) Equilateral triangle d) Square The correct answer is: d Rationale: A square has 4 lines of symmetry passing through opposite vertices and midpoints of opposite sides. If two parallel lines are intersected by a third line, what is the relationship between the alternate interior angles? a) They are equal b) They are supplementary c) They are complementary d) They are vertical angles The correct answer is: a Rationale: Alternate interior angles formed by parallel lines intersected by a transversal are equal. What is the sum of the interior angles of a hexagon? a) 720° b) 900° c) 540° d) 1080° The correct answer is: a Rationale: The sum of interior angles of any n-sided polygon is (n-2)*180°. For a hexagon, n=6, so (6-2)*180° = 720°. If a rectangle has a length of 6 units and a width of 4 units, what is its area? a) 10 square units b) 24 square units c) 16 square units d) 20 square units The correct answer is: b Rationale: Area of a rectangle = length x width = 6 x 4 = 24 square units. Two circles intersect at two points. What is the maximum number of common tangents to the two circles? a) 2 b) 4 c) 6 d) 8 The correct answer is: b Rationale: The maximum number of common tangents to two intersecting circles is 4. In a right triangle, if one leg is 5 units and the hypotenuse is 13 units, what is the other leg? a) 10 units b) 8 units c) 12 units d) 7 units The correct answer is: b Rationale: Using Pythagorean theorem, if hypotenuse=13 and one leg=5, other leg = √(13^2 - 5^2) = 8. What is the measure of an exterior angle of a regular hexagon? a) 120° b) 90° c) 135° d) 60° The correct answer is: a Rationale: Exterior angle of any regular polygon = 360°/n where n is number of sides. For a hexagon, n=6, so 360°/6 = 120°. A circle is inscribed in a square. What fraction of the square's area is the area of the circle? a) π/4 b) π/2 c) π/3 d) π The correct answer is: a Rationale: Area of circle = πr^2 and area of square with side length a = a^2. The largest circle inscribed in a square has radius a/2, so area ratio is π(a/2)^2/a^2 = π/4. If a triangle has sides of lengths 5, 12, and 13, what kind of triangle is it? a) Obtuse b) Right c) Acute d) Equilateral The correct answer is: a Rationale: A triangle is obtuse if one angle is greater than 90°. For a triangle with sides 5, 12, 13, applying the triangle inequality shows the longest side 13 must subtend an obtuse angle. Two circles are externally tangent. The larger circle has a radius of 8 units. What is the maximum radius of the smaller circle? a) 4 units b) 2 units c) 6 units d) 8 units The correct answer is: b Rationale: When two circles are externally tangent, the sum of radii equals the distance between centers. The maximum radius of the smaller circle is when the centers coincide, which is 8 units. If the area of a triangle is 24 square units and its base is 8 units, what is its height? a) 3 units b) 6 units c) 4 units d) 5 units The correct answer is: c Rationale: The area of a triangle is (1/2) * base * height. Substituting the given values, we get 24 = (1/2)8height, so height = 6 units. A square is inscribed in a circle. If the circle's diameter is 10 units, what is the length of a side of the square? a) 7.07 units b) 5 units c) 6.67 units d) 8 units The correct answer is: b Rationale: The side length of the maximum square inscribed in a circle equals the radius of the circle. Given diameter is 10, so radius is 5 units. The sum of the radii of two circles is 20 cm and the distance between their centers is 36 cm. What is the length of their common internal tangent? a) 8 cm b) 12 cm c) 16 cm d) 4 cm The correct answer is: a Rationale: Let the radii be r1 and r2. The common internal tangent length is |r1 - r2|. Given r1 + r2 = 20 and center distance 36, r1 - r2 must be 16. So |r1 - r2| = 8 cm. A cone has a base radius of 6 inches and a slant height of 10 inches. What is its volume? a) 120π cubic inches b) 180π cubic inches c) 60π cubic inches d) 240π cubic inches The correct answer is: a Rationale: Volume of a cone = (1/3) π r^2 h, where r is base radius and h is height. Substituting given values, volume = (1/3) π (6)^2 (10) = 120π cubic inches. In a semicircle, an angle is inscribed. What is the measure of the angle's supplement? a) 120° b) 90° c) 180° d) 60° The correct answer is: b Rationale: An inscribed angle in a semicircle has measure 90°. Its supplement is 180° - 90° = 90°. Two lines intersect to form an angle of 120°. What is the measure of an alternate interior angle? a) 30° b) 60° c) 120° d) 90° The correct answer is: b Rationale: Alternate interior angles formed by intersecting lines sum to 180°. If one angle is 120°, the alternate interior angle must be 180° - 120° = 60°. A sphere has a surface area of 36π square units. What is its volume? a) 16π cubic units b) 24π cubic units c) 36π cubic units d) 12π cubic units The correct answer is: b Rationale: Surface area of a sphere = 4πr^2. Volume = (4/3)πr^3. Solving for r from surface area, we get r = 3. Substituting in volume formula gives (4/3)π(3)^3 = 24π cubic units. Two circles have radii 3 cm and 4 cm. What is the length of their common external tangent? a) 7 cm b) 5 cm c) 1 cm d) 3 cm The correct answer is: a Rationale: Length of common external tangent is the sum of radii, which is 3 + 4 = 7 cm. What is the volume of a sphere with diameter 10 units? a) (125π)/3 cubic units b) (500π)/3 cubic units c) 100π cubic units d) 25π cubic units The correct answer is: b Rationale: Diameter = 10 units, so radius = 5 units. Volume of a sphere = (4/3)π(radius)^3 = (4/3)π(5)^3 = (500π)/3 cubic units.