p.1
Classification of Triangles
What are the three classifications of triangles based on interior angles?
Acute, Right, and Obtuse.
p.1
Exterior Angle Theorem
What does the exterior angle theorem state?
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
p.32
Solving for Unknown Angles
What types of angles can be used to find angles at a point?
Complementary, supplementary, or vertically opposite angles.
p.14
Solving for Unknown Angles
How do you find the value of x in the angle diagrams?
By using the properties of angles (e.g., supplementary, complementary).
How can you derive the measure of angle ADC?
Reflex ∠ ADC = 360° − ∠ ADC = 360° − (360° − (sum of the other three angles)).
p.8
Types of Quadrilaterals
What characterizes a convex quadrilateral?
All vertices point outwards and the diagonals lie inside the figure.
p.33
Types of Quadrilaterals
How can you classify solids?
By their number of faces and appropriate names.
p.14
Solving for Unknown Angles
What type of angle is complementary to ∠ QOR?
An angle that adds up to 90° with ∠ QOR.
p.33
Regular Polygons and Their Angles
How do you find the size of each interior angle in a regular polygon?
By using the formula for interior angles based on the number of sides.
p.14
Classification of Triangles
What are two words to describe a triangle that has two equal sides?
Isosceles and acute (if the angles are less than 90°).
p.32
Solving for Unknown Angles
What is the angle sum in a quadrilateral?
The sum of the interior angles in a quadrilateral is always 360 degrees.
p.19
Regular Polygons and Their Angles
How do you find the size of an interior angle in a regular octagon?
Divide the angle sum (1080°) by the number of sides (8), resulting in 135°.
p.8
Types of Quadrilaterals
What are parallelograms?
Quadrilaterals with two pairs of parallel sides.
p.34
Classification of Triangles
In a triangle with angles 25°, 120°, and a°, how would you classify this triangle?
It is an obtuse triangle.
p.5
Classification of Triangles
What type of triangle is △ AOB if it is constructed using a circle and two radius lengths?
Isosceles triangle, because two sides (radii) are equal.
What is the relationship between angle ADC and the other angles in a quadrilateral?
∠ ADC = 360° − (sum of the other three angles).
p.5
Exterior Angle Theorem
What is the reason that ∠ BCD = b°?
Because it is an exterior angle to triangle ABC.
p.19
Properties of Polygons
What is the angle sum of a pentagon?
540° (calculated as (5 - 2) × 180°).
p.8
Types of Quadrilaterals
What is a rectangle?
A parallelogram with all angles equal to 90°.
What are the properties of corresponding angles when two parallel lines are crossed by a transversal?
Corresponding angles are equal.
p.8
Types of Quadrilaterals
What is a rhombus?
A parallelogram with all sides equal.
What is the angle sum property of a quadrilateral?
The sum of the interior angles equals 360°.
What is the angle sum in a triangle?
The sum of the interior angles in a triangle is always 180 degrees.
Why can a triangle not have two right angles?
Because the sum of angles in a triangle must equal 180°.
p.17
Properties of Polygons
What defines a polygon?
Shapes with straight sides that can be convex or non-convex.
p.3
Solving for Unknown Angles
How do you find the value of angle a if the exterior angle is 161° and one interior angle is 90°?
a = 180° - (19° + 90°) = 71°.
p.17
Properties of Polygons
What is the difference between convex and non-convex polygons?
Convex polygons have all vertices pointing outwards, while non-convex polygons have at least one vertex pointing inwards and at least one reflex interior angle.
How can the angle sum of a triangle be used in relation to the angle in a semicircle?
It can be used to prove that ∠ACB is always 90°.
What is the relationship between the values of angles a and c in the context of the semicircle?
The sum of the values of a and c is always equal to the value of angle b.
What is the relationship between alternate angles when two parallel lines are intersected?
Alternate angles are equal.
p.27
Euler's Rule for Polyhedra
How many vertices does a polyhedron have if it has 6 faces and 12 edges?
Use Euler's rule to find the number of vertices.
p.15
Solving for Unknown Angles
What is the angle sum property of a quadrilateral?
The sum of the interior angles of a quadrilateral is 360°.
p.12
Types of Quadrilaterals
Which quadrilaterals have diagonals that bisect each other?
Parallelograms, rectangles, rhombuses, and squares.
What is the significance of the angles a, b, and c in isosceles triangles within the context of the semicircle?
They help demonstrate the relationship between angles in a triangle and the angle in a semicircle.
p.15
Solving for Unknown Angles
What is the value of angle 'a' in the first diagram if it is given as 'a°' and 'b' is '2a°'?
To find 'a', additional information about the angles is needed.
p.27
Euler's Rule for Polyhedra
How do you solve for the number of faces (F) using Euler's rule?
Substitute E and V into the formula and solve for F.
p.24
Euler's Rule for Polyhedra
What is Euler's rule for polyhedra?
E = F + V - 2, where E is edges, F is faces, and V is vertices.
p.8
Types of Quadrilaterals
What is a square?
A rhombus with all angles equal to 90° or a rectangle with all sides equal.
p.24
Euler's Rule for Polyhedra
What is the relationship between edges, faces, and vertices in a polyhedron?
The number of edges (E) is equal to the number of faces (F) plus the number of vertices (V) minus 2.
Does the angle sum of a polygon have a limit as the number of sides increases?
No, it increases to infinity as n increases.
p.15
Solving for Unknown Angles
How can you find the value of 'w' in a quadrilateral with angles 'w°', '32°', '75°', and '30°'?
Set up the equation: w + 32 + 75 + 30 = 360 and solve for 'w'.
p.28
Euler's Rule for Polyhedra
How many edges does a polyhedron with 16 faces and 12 vertices have?
Using Euler's rule: E = F + V - 2, so E = 16 + 12 - 2 = 26.
p.14
Solving for Unknown Angles
What type of angle is supplementary to ∠ POT?
An angle that adds up to 180° with ∠ POT.
p.24
Euler's Rule for Polyhedra
What is a polyhedron?
A closed solid with flat surfaces (faces), vertices, and edges.
p.25
Euler's Rule for Polyhedra
What defines the cross-section of a prism?
The two identical (congruent) ends.
p.20
Properties of Polygons
How do you determine the number of sides of a polygon from its angle sum?
Use the formula: Number of sides = (Angle sum / 180) + 2.
p.9
Types of Quadrilaterals
What are the characteristics of a square?
All sides are of equal length, diagonals intersect at right angles.
p.27
Classification of Triangles
What type of solid is classified by the number of faces?
Examples include octahedron, hexahedron, etc.
What is the angle sum of a triangle?
The angle sum of a triangle is 180°.
p.21
Regular Polygons and Their Angles
What size does each interior angle of a regular polygon approach as n increases?
Each interior angle approaches 180°.
p.15
Solving for Unknown Angles
What is the relationship between angles 'a', '63°', and '125°' in a triangle?
The sum of angles in a triangle is 180°, so a + 63 + 125 = 180.
p.21
Regular Polygons and Their Angles
What is the formula for the size of an interior angle in terms of the angle sum S and the number of sides n?
Each interior angle = S/n.
p.21
Regular Polygons and Their Angles
What is the formula for the size of an interior angle in terms of n only?
Each interior angle = (180(n-2))/n.
What is the angle sum of a hexagon?
720° (calculated as (6 - 2) × 180°).
p.9
Types of Quadrilaterals
What is true about a trapezium?
There are not always two pairs of parallel sides.
p.22
Applications of Geometry in Design
If the number zero is added and is six times more likely to be landed on, what will be the sector angle for zero?
The sector angle for zero will be 90°; the other numbers will have a sector angle of 10°.
p.22
Applications of Geometry in Design
What are the requirements for Spinning Wheel C?
Must include two reflex angles.
p.32
Exterior Angle Theorem
What theorem is used to find unknown angles outside a triangle?
The exterior angle theorem.
p.8
Types of Quadrilaterals
What is a defining feature of non-convex quadrilaterals?
One vertex points inwards and has one reflex interior angle.
p.19
Properties of Polygons
What is the angle sum of a hexagon?
720° (calculated as (6 - 2) × 180°).
p.2
Exterior Angle Theorem
What is the relationship between the exterior angle and the interior angles in a triangle?
The exterior angle is equal to the sum of the two opposite interior angles.
What is the angle sum formula for a polygon with n sides?
The angle sum is (n - 2) × 180°.
p.9
Types of Quadrilaterals
What are the properties of a parallelogram?
There are two pairs of equal length and parallel sides.
p.30
Types of Quadrilaterals
What is the challenge in problem 1a?
Remove 2 matchsticks to form 2 squares.
p.25
Euler's Rule for Polyhedra
What is the relationship between the number of faces, vertices, and edges in a polyhedron?
A polyhedron has faces, vertices, and edges.
p.35
Regular Polygons and Their Angles
How do you calculate the size of an interior angle of a regular pentagon?
The size of an interior angle of a regular pentagon is 108°.
p.31
Solving for Unknown Angles
What is the definition of complementary angles?
Two angles that sum to 90°.
p.31
Solving for Unknown Angles
What is the definition of supplementary angles?
Two angles that sum to 180°.
p.32
Solving for Unknown Angles
How can angles be found using parallel lines?
By applying the properties of corresponding, alternate interior, and same-side interior angles.
p.33
Classification of Triangles
What is a heptahedron?
A solid with seven faces.
p.33
Euler's Rule for Polyhedra
What does Euler's rule help determine?
The relationship between the number of faces, edges, and vertices in a polyhedron.
How do you find the value of angle a in the second triangle example?
By solving 2a + 26 = 180.
What can be concluded about angles ∠ BAC and ∠ ACD in triangle ABC?
They are supplementary, meaning they add up to 180°.
p.20
Regular Polygons and Their Angles
What is the formula to find the interior angle of a regular polygon?
The formula is (n-2) * 180° / n, where n is the number of sides.
p.24
Euler's Rule for Polyhedra
How can polyhedra be named?
By their number of faces, such as tetrahedron (4 faces) or hexahedron (6 faces).
p.19
Properties of Polygons
What is the angle sum of a nonagon?
1260° (calculated as (9 - 2) × 180°).
p.19
Properties of Polygons
What is the angle sum of a 15-sided polygon?
2340° (calculated as (15 - 2) × 180°).
What is the sum of co-interior angles when two parallel lines are crossed?
Co-interior angles are supplementary (sum to 180°).
p.9
Types of Quadrilaterals
What is true about the diagonals of a rectangle?
Diagonals are equal in length.
p.24
Euler's Rule for Polyhedra
What are some examples of polyhedra?
Tetrahedron, hexahedron, pentagonal pyramid.
p.27
Types of Quadrilaterals
What is a hexagonal prism?
A type of prism with hexagonal bases.
p.31
Properties of Polygons
How do you calculate the sum of interior angles of a polygon?
S = (n - 2) × 180°, where n is the number of sides.
p.31
Exterior Angle Theorem
What is the relationship between exterior angles of a triangle?
An exterior angle is equal to the sum of the two opposite interior angles.
p.37
Regular Polygons and Their Angles
What is the size of each interior angle of a regular polygon with 26 sides?
173° (to the nearest degree).
What is the angle relationship in vertically opposite angles?
Vertically opposite angles are equal.
p.30
Solving for Unknown Angles
What is the task in problem 3?
Find the value of x in the diagrams.
p.29
Convex vs Non-Convex Shapes
What defines a convex solid?
All interior angles are less than 180° and all diagonals are drawn inside the shape.
What does the theorem about the angle in a semicircle state?
The angle ∠ACB in a semicircle is always 90° where AB is a diameter.
p.20
Solving for Unknown Angles
How do you find the value of 'a' in polygons?
By using the properties of angles in polygons and solving for 'a' based on given angles.
p.15
Solving for Unknown Angles
In a quadrilateral with angles 'x°', '107°', and '92°', how can you find the value of 'x'?
Use the angle sum property of quadrilaterals, which states that the sum of the angles is 360°.
p.28
Euler's Rule for Polyhedra
What is Euler's rule for polyhedra?
F + V - E = 2, where F is the number of faces, V is the number of vertices, and E is the number of edges.
p.8
Types of Quadrilaterals
What is a trapezium?
A quadrilateral with at least one pair of parallel sides.
p.11
Properties of Polygons
What is the significance of properties in solving for pronumerals in quadrilaterals?
Properties help determine relationships between sides and angles, aiding in calculations.
p.11
Solving for Unknown Angles
In a quadrilateral with angles a°, 88°, 115°, and 96°, how do you find 'a'?
Set up the equation a + 88° + 115° + 96° = 360°.
p.3
Solving for Unknown Angles
If one angle in a triangle is 127° and another is 17°, what is the value of angle a?
a = 180° - (127° + 17°) = 36°.
p.23
Properties of Polygons
How many isosceles triangles are formed inside a pentagon when creating a five-pointed star?
Five isosceles triangles.
p.28
Classification of Triangles
Are all solids with curved surfaces cylinders?
False, not all solids with curved surfaces are cylinders.
p.28
Classification of Triangles
Are there polyhedra with 3 surfaces?
False, polyhedra must have at least 4 faces.
p.22
Applications of Geometry in Design
What are the requirements for Spinning Wheel B?
Must include at least one acute, one obtuse, and one reflex angle.
p.3
Exterior Angle Theorem
What does the exterior angle theorem state for a triangle?
The exterior angle is equal to the sum of the two opposite interior angles.
p.24
Euler's Rule for Polyhedra
What does the term 'polyhedron' mean?
It comes from Greek words meaning 'many faces'.
p.16
Properties of Polygons
What is the origin of the word 'polygon'?
It comes from the Greek words 'poly', meaning 'many', and 'gonia', meaning 'angles'.
p.25
Euler's Rule for Polyhedra
What are the other faces of a prism if they are rectangles?
The solid is called a right prism.
p.29
Euler's Rule for Polyhedra
What are two other names for a solid with six rectangular faces?
Cuboid and rectangular prism.
p.25
Euler's Rule for Polyhedra
What is a cube?
A hexahedron with six square faces.
p.29
Euler's Rule for Polyhedra
What is the name of a pyramid with a base that has 10 sides?
Decagonal pyramid or 10-sided pyramid.
p.9
Types of Quadrilaterals
What defines a rhombus?
All sides are of equal length and diagonals intersect at right angles.
p.24
Euler's Rule for Polyhedra
What is the significance of counting faces, vertices, and edges in polyhedra?
It helps to verify Euler's rule and understand the properties of polyhedra.
p.35
Exterior Angle Theorem
What is the value of angle 'a' in the triangle with exterior angles of 152° and 85°?
a = 152° + 85° - 180° = 152° + 85° - 180° = 57°.
p.9
Types of Quadrilaterals
What is a defining feature of a kite?
There are two pairs of sides of equal length and diagonals intersect at right angles.
p.21
Regular Polygons and Their Angles
What is the size of an interior angle of an 82-sided regular polygon?
Each interior angle is approximately 178.29°.
p.37
Regular Polygons and Their Angles
What is the size of each exterior angle of a regular polygon with 26 sides?
7° (to the nearest degree).
p.30
Solving for Unknown Angles
What is the task in problem 4?
Find the size of ∠ ABC in the quadrilateral.
p.9
Types of Quadrilaterals
What do quadrilaterals with parallel sides have in common regarding angles?
They include two pairs of co-interior angles.
p.11
Solving for Unknown Angles
How do you find the value of pronumerals in quadrilaterals?
By using the angle sum property and solving for unknown angles.
p.11
Solving for Unknown Angles
What is the value of 'a' in a quadrilateral with angles 130°, a°, b°, and 95°?
a = 360° - (130° + 95° + b°).
p.3
Solving for Unknown Angles
In a triangle with angles a, 70°, and 30°, how do you find the value of a?
a = 180° - (70° + 30°) = 80°.
p.11
Solving for Unknown Angles
If a quadrilateral has angles a°, 110°, 70°, and b°, how can you find 'a'?
Use the equation a + 110° + 70° + b = 360°.
What is the relationship between the angles in a quadrilateral?
The sum of the interior angles is always 360 degrees.
p.21
Regular Polygons and Their Angles
What is the size of an interior angle of a regular dodecagon?
Each interior angle is 150°.
p.31
Euler's Rule for Polyhedra
What is Euler's formula for polyhedra?
E = F + V - 2, where E is edges, F is faces, and V is vertices.
p.12
Types of Quadrilaterals
Is a rhombus a type of parallelogram?
Yes, a rhombus is a type of parallelogram.
p.35
Euler's Rule for Polyhedra
What type of polyhedron has 6 faces?
A triangular prism or a cube.
p.4
Classification of Triangles
What type of triangle is described as right and scalene?
A triangle with one right angle and all sides of different lengths.
p.22
Applications of Geometry in Design
What are the requirements for Spinning Wheel A?
Must include three obtuse angles.
p.16
Properties of Polygons
How does the number of interior angles relate to the number of sides in a polygon?
The number of interior angles equals the number of sides.
p.25
Euler's Rule for Polyhedra
What is a pyramid?
A polyhedron with a base face and triangular faces meeting at a vertex called the apex.
p.34
Solving for Unknown Angles
How do you find the size of angle ∠AOB using the dashed construction line?
By using the angles 45° and 50° to calculate the remaining angle.
p.26
Properties of Polygons
Which of the following solids are polyhedra? A) Cube B) Pyramid C) Cone D) Sphere E) Cylinder F) Rectangular prism G) Tetrahedron H) Hexahedron
A) Cube, B) Pyramid, F) Rectangular prism, G) Tetrahedron, H) Hexahedron.
What is the angle sum of a quadrilateral?
The angle sum of a quadrilateral is 360°.
p.28
Euler's Rule for Polyhedra
If a polyhedron has 18 edges and 9 vertices, how many faces does it have?
Using Euler's rule: F = E + 2 - V, so F = 18 + 2 - 9 = 11.
p.35
Properties of Polygons
How do you find the value of 'a' and 'b' in the given quadrilaterals?
Use the properties of angles in quadrilaterals, where the sum of interior angles equals 360°.
p.30
Types of Quadrilaterals
What is the challenge in problem 1b?
Move 3 matchsticks to form 3 squares.
p.35
Properties of Polygons
What is the angle sum of a nonagon?
The angle sum of a nonagon is 1260°.
p.28
Classification of Triangles
Can a hexahedron be a pyramid?
True, a hexahedron can be a pyramid.
p.28
Classification of Triangles
Are there solids with 0 vertices?
False, all solids have at least one vertex.
p.29
Convex vs Non-Convex Shapes
How can you test if a solid is non-convex?
Join two vertices or two faces with a line segment that passes outside the solid.
If angle a is x°, what can be said about the sum of angles a and c?
The sum of angles a and c will always equal 90°.
p.8
Types of Quadrilaterals
What is a kite?
A quadrilateral with two adjacent pairs of equal sides.
p.16
Properties of Polygons
What is a regular polygon?
A polygon with all sides and angles equal.
p.7
Applications of Geometry in Design
What is a method builders use to check if a wall is rectangular?
Measuring the lengths of the two diagonals.
p.12
Types of Quadrilaterals
Is a parallelogram a type of square?
No, a parallelogram is not a type of square.
p.35
Regular Polygons and Their Angles
What is the size of an interior angle of a regular dodecagon?
The size of an interior angle of a regular dodecagon is 150°.
p.4
Classification of Triangles
What type of triangle is described as acute and isosceles?
A triangle with all angles less than 90° and two sides of equal length.
p.7
Types of Quadrilaterals
What properties can be used to classify quadrilaterals?
Pairs of sides of equal length, parallel sides, and lengths of diagonals.
p.20
Solving for Unknown Angles
How do you find the value of 'x' in angle diagrams?
By applying the angle sum properties and solving for 'x'.
p.12
Types of Quadrilaterals
Is a square a type of rhombus?
Yes, a square is a type of rhombus.
p.28
Classification of Triangles
Is a tetrahedron considered a pyramid?
True, a tetrahedron is a type of pyramid.
p.3
Solving for Unknown Angles
In a triangle with angles a, b, and 24°, if b is 116°, how do you find a?
a = 180° - (116° + 24°) = 40°.
p.17
Properties of Polygons
What is a regular polygon?
A polygon with sides of equal length and equal interior angles.
p.3
Solving for Unknown Angles
What is the value of angle a in a triangle with angles 71° and 54°?
a = 180° - (71° + 54°) = 55°.
p.7
Types of Quadrilaterals
What quadrilateral has 2 pairs of parallel sides?
Parallelogram or rectangle.
p.12
Types of Quadrilaterals
Is a rectangle a type of square?
No, a rectangle is not a type of square.
p.35
Properties of Polygons
What is the angle sum of a heptagon?
The angle sum of a heptagon is 900°.
p.22
Applications of Geometry in Design
What is the angle at the center of the spinning wheel for each sector if it has 24 sectors?
15° (360° divided by 24).
p.12
Convex vs Non-Convex Shapes
Can you draw a non-convex quadrilateral with two or more interior reflex angles?
Yes, it is possible to draw such a quadrilateral.
p.22
Applications of Geometry in Design
What is the probability of landing on a prime number if each prime number has a sector angle of 30°?
The probability will depend on the number of prime numbers; if there are 9 primes, the probability is 30° * 9 / 360° = 0.075 or 7.5%.
p.28
Classification of Triangles
Will all pyramids have an odd number of faces?
False, pyramids can have an even number of faces depending on the base shape.
p.12
Types of Quadrilaterals
Is a square a type of rectangle?
Yes, a square is a type of rectangle.
p.28
Euler's Rule for Polyhedra
How many vertices does a polyhedron with 34 faces and 60 edges have?
Using Euler's rule: V = E + 2 - F, so V = 60 + 2 - 34 = 28.
p.25
Euler's Rule for Polyhedra
What is a heptahedron?
A polyhedron with 7 faces.
p.12
Types of Quadrilaterals
Is a rectangle a type of parallelogram?
Yes, a rectangle is a type of parallelogram.
p.28
Classification of Triangles
Are a cube and a rectangular prism both hexahedrons?
True, both are classified as hexahedrons.
p.22
Applications of Geometry in Design
What will be the sector angle for an odd number if each odd number is twice as likely to be landed on compared to each even number?
The sector angle for an odd number will be 20° and for an even number will be 10°.
p.22
Applications of Geometry in Design
What will be the sector angle for each composite number if each prime number has a sector angle of 30°?
The sector angle for each composite number will be 15°.
p.30
Solving for Unknown Angles
What is the task in problem 6?
Find the value of a + b + c + d + e in the star.
p.22
Applications of Geometry in Design
What are the requirements for Spinning Wheel D?
Must include one right angle and one obtuse angle.
