Congruence - Geometry

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a degree angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"A truck is traveling down a hill"

From this statement, it cannot be assumed that the hill is a straight line nor can it be assumed that the hill goes on forever. Therefore, the truck and the hill will never be parallel. Also, for these same reasons it is known that the truck will never be perpendicular to the hill. The relation of the truck's tires to the hill will never be parallel since they constantly touch.

Thus, the correct answer choice is,

"The body of the truck is not perpendicular to the hill."

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches."

The question is asking to define the relationship between the two chains that hold the swing to the swing set. Since the two chains are exactly inches apart from one another and attached to the pole which is horizontal from the swing and the swing seat itself is inches, it is concluded that the two chains are parallel to one another.

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches."

The question is asking to define the relationship between one of the chains and the horizontal bar it is attached to. Since the swing will hang directly down from the two chains and the bar is horizontal to ground it can be assumed that the chain and the bar form a angle and thus, they are perpendicular to one another.

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches."

The question is asking to define the relationship between the horizontal bar and the swing seat. Since the two chains are exactly inches apart from one another and of equal length and attached to the pole which is horizontal from the swing and the swing seat itself is inches, it is concluded that the seat and the horizontal bar are parallel to one another.

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

Knowing these characteristics, solve for the central angle of one slice of pizza.

Therefore, the correct answer is

"The central angle of one pizza slice is degrees."

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

The circumference of a circle is the length around the circle and the radius is the length from the center of the circle to any point on the circle's edge.

For this particular question, calculate the circumference and then calculate the arc length of each slice pizza slice.

Since there are 8 equal slices, divide the circumference by 8.

Therefore, the correct answer is

"The arc length of one slice of pizza is inches."

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

The area of a circle is found by using the formula .

For this particular problem first calculate the area of the clock.

Now, since the clock reads 4:50, the distance between the hour and minute hands is of the total clock. From here, calculate the area between the two hands.

Therefore, the correct answer is

The area between the hour and minute hand is .

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

Also recall that a straight line measures 180 degrees.

Looking at the given clock, it is seen that a straight line can be created by connecting the 12 and 6 on the clock. Since the clock reads 11:35 the angle between the hour and minute hand is greater than 180 degrees because the hour hand is behind the 12 and the minute hand is behind the 6 on the clock.

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

Also recall that a straight line measures 180 degrees.

Looking at the given clock, it is seen that a straight line can be created by connecting the 12 and 6 on the clock. Since the clock reads 3:05 the angle between the hour and minute hand is less than 180 degrees. Further more, it is less than 45 degrees because the hour hand and minute hand are both between one quadrant of the circle.

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

Knowing these characteristics, solve for the central angle of one slice of pizza.

Therefore, the correct answer is

"The central angle of one pizza slice is 72 degrees."

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a degree angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"A truck is traveling up a hill"

From this statement, it cannot be assumed that the hill is a straight line nor can it be assumed that the hill goes on forever. Therefore, the truck and the hill will never be parallel. Also, for these same reasons it is known that the truck will never be perpendicular to the hill. The relation of the truck's tires to the hill will never be parallel since they constantly touch.

Thus, the correct answer choice is,

"The body of the truck is not perpendicular to the hill."

First, recall the definitions of the terms in the possible answer choices.

Collinear: Represents points that all fall on the same line.

Equidistance: Represents points that are the same length away from one another.

Parallel: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular: In a plane, perpendicular lines are lines that intersect by creating a degree angle. This also means they have opposite sign, reciprocal slopes.

Therefore, the correct answer is collinear.

Both squares and rhombi are quadrilaterals and parallelograms. Quadrilaterals are four sided figures and parallelograms are figures that have opposite sides that are parallel.

Squares by definition contain four angles; rhombi on the other hand have two sets of opposite congruent angles. Therefore, for a square to to altered into a rhombus, the interior angles must be altered.

Recall that a rigid motion preserves the distance from points within a shape in the plane. Therefore, if has had a rigid motion applied to it to result in that means that all corresponding sides and angles of these two triangle are congruent.

Therefore, the statement "both corresponding angles and sides are congruent" is the correct solution to this particular question.

Recall that a rigid motion preserves the distance from points within a shape in the plane. Since the statement clearly states that the two triangles are congruent, that means that the distances between the points within the shape are preserved. Also, since the corresponding points between the two triangles are separated by five spaces this is describing a translation which when combined with the preserved shape, describes a rigid motion. Therefore, the statement is true.

Recall that a rigid motion preserves the distance from points within a shape in the plane. It appears that the rigid motion was a rotation. The third angle can be identified by subtracting the two known angles from 180 degrees.

Triangle A:

Triangle C:

Since the side between two angles on triangle C is congruent to the side between those same two angles on triangle A, the two triangles are congruent.

To determine whether this statement is true or false, first recall what it means to be "congruent". Congruent means equal or in more mathematical terms, both corresponding angles and sides of the two triangles are congruent.

The statement says that is an equilateral triangle which means all sides and angles are the same. After the triangle goes through a transformation, it becomes an isosceles triangle. To be isosceles means that two of the sides and angles of the triangle are the same. Since, not all of the angles are the same in it is not congruent to .

Therefore, the statement, " is congruent to ." is false.

To determine which theorem cannot prove triangle congruency, first recall what it means to be "congruent". Congruent means equal or in more mathematical terms, both corresponding angles and sides of the two triangles are congruent.

There are five theorems used in geometry to prove whether two triangles are congruent.

1. Side, Side, Side

2. Side, Angle, Side

3. Angle, Side, Angle

4. Angle, Angle, Side

5, Hypotenuse and One Leg from Right Triangle

Of the answers below, Angle, Angle, Angle (AAA) is not among one of the theorems to prove triangle congruency. The reason AAA does not prove triangle congruency because two triangles can have the same angles but have different side lengths thus, the triangles would not be congruent.

For this particular problem there are two geometric theorems that can prove that the triangles are similar.

The geometric theorems that could be used:

1. Angle, Side, Angle (ASA)

2. Angle, Angle, Side (AAS)

For triangle C the AAS is the most evident to use. Since ASA is not an option in the answer selections, AAS is the correct answer.

For this particular problem there are two geometric theorems that can prove that the triangles are similar.

The geometric theorems that could be used:

1. Angle, Side, Angle (ASA)

2. Angle, Angle, Side (AAS)

For triangle C the AAS is the most evident to use. Since AAS is not an option in the answer selections, ASA is the correct answer.