Congruence - Geometry

For two triangles to be congruent they must have equal corresponding angles and sides. There are five geometric theorems that can be used to prove whether triangles are congruent or not. Since, these two triangles have two defined angles and the side between the angles are defined as well, the Angle, Side, Angle geometric theorem can be used.

Looking at the triangles the corresponding angles are not equal therefore, the triangles are not congruent.

For two triangles to be congruent they must have equal corresponding angles and sides. There are five geometric theorems that can be used to prove whether triangles are congruent or not. Since, these two triangles have two defined angles and the side between the angles are defined as well, the Angle, Side, Angle geometric theorem can be used.

Looking at the triangles the corresponding angles are not equal and the correspond side is also not equal therefore, the triangles are not congruent.

For two triangles to be congruent they must have equal corresponding angles and sides. There are five geometric theorems that can be used to prove whether triangles are congruent or not. Since, these two triangles have all three sides defined, the Side, Side, Side geometric theorem can be used.

Looking at the triangles the corresponding sides are not equal therefore, the triangles are not congruent.

This situation describes a real life application of geometry where the coordinate grid can be seen as the piece of paper. Since Jenny is tracing the rectangle after the paper is folded it can be seen that the fold is an axis and the new rectangle is congruent to the original one. Therefore, the rigid motion describing this situation is a reflection.

Recall that a rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types.

These basic type of rigid motions include the following:

1. Rotation
2. Reflection
3. Translation
4. Glide Reflection

Therefore, of the answer selections, "Expansion" is the term that is NOT a rigid motion.

Recall that a rigid motion is that that preserves the distances between points within the object while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types. The piece of paper is the object in this situation. Since the paper is cut in half, it does not preserve the shape of the object and thus, is not a rigid motion.

Therefore, the answer is "No".

Recall that a rigid motion is that that preserves the distances between points within the object while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types. The water balloons are the objects in this situation. Since the they are filled with more water to increase their circumferences, it does not preserve the shape of the object and thus, is not a rigid motion.

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides. Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle). So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles. By SAS Theorem, these right triangles are congruent. When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

When we look at this figure we see that we have two pairs of congruent corresponding angles, . The Angle-Angle Theorem (AA) states that if two angles of one triangle are congruent to two angles of another triangle, then these triangles are similar.

For this particular problem there are two geometric theorems that could potentially be used to prove that the triangles are similar.

The geometric theorems that could be used:

1. Hypotenuse and One Leg of a Right Triangle (HL)

2. Side, Side, Side (SSS)

To use SSS or HL, the Pythagorean theorem will need to be used to calculate the missing side.

For these triangles the HL is the most evident to use and since SSS is not an option in the answer selections, HL is the correct answer.

For this particular problem there are two geometric theorems that could potentially be used to prove that the triangles are similar.

The geometric theorems that could be used:

1. Hypotenuse and One Leg of a Right Triangle (HL)

2. Side, Side, Side (SSS)

To use SSS or HL, the Pythagorean theorem will need to be used to calculate the missing side.

For these triangles the HL is the most evident to use but is not an option in the answer selections, therefore, SSS is the correct answer.

The statement, "The __________ geometric theorem deals with proving congruency among right triangles; specifically when the length of one leg and the length of the hypotenuse are known." is describing the geometric theorem known as the Hypotenuse and One Leg theorem. When abbreviated this is seen as, HL.

Therefore, the missing term is, Hypotenuse and One Leg (HL)

The statement, "The __________ geometric theorem can be used to identify whether triangles that each have three known side lengths are congruent." is describing the geometric theorem known as the Side, Side, Side theorem. When abbreviated this is seen as, SSS.

Therefore, the missing term is, Side, Side, Side (SSS).

The statement, "When two triangles have two known angles and a known side length that is in between the angles, the geometric theorem that can be used to prove congruency is known as __________. " is describing the geometric theorem known as the Angle, Side, Angle theorem. When abbreviated this is seen as, ASA.

Therefore, the missing term is, Angle, Side, Angle (ASA).

Rigid motion follows these criteria because the motion is rigid meaning that everything that is moving stays the same except for the location of the entire figure. There are three common types of rigid motion; translation, reflection, and rotation.

A parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

A parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

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 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.