Rigid Motions
Moving an object like an ice cube melting or filling a balloon with gas does not qualify as this type of motion because the object's size and shape change.
What is a rigid motion?
This term describes a line that divides a geometric figure into two matching halves that are mirror images of each other, such as reflecting a regular decagon across a segment connecting opposite vertices.
What is a line of symmetry (or axis of symmetry)?
Unlike rigid motions, this transformation changes the size of a figure but preserves its shape and angle measures, making the figures similar.
What is a dilation?
When a transversal intersects two lines, angles that add up to 180 degrees and lie on a straight line together are known by this geometric term.
What is a linear pair (or supplementary angles)?
If triangle ABC is congruent to triangle PQR, this angle in triangle PQR must be congruent to angle C.
What is angle R?
This type of transformation moves every point of a figure the same distance and in the same direction, such as moving furniture across a room.
What is a translation?
This type of symmetry exists when a figure can be rotated less than 360 degrees around its center point and still look exactly like the original figure.
What is rotational symmetry?
These two conditions justifies why two shapes are similar to each other.
What is "corresponding angles are congruent and all sides are proportional"?
This geometric reason justifies why opposite angles are congruent when two lines intersect.
What is "vertical angles are congruent"?
If triangle ABC is congruent to triangle RQP, this line segment in the second triangle is strictly congruent to side CB.
What is segment PQ?
These three types of transformations always preserve the side lengths and angle measures of a polygon, producing a final image congruent to the original.
What are translations, rotations, and reflections? (Rigid motions)
This term describes a figure that looks identical after a 180 degree rotation about its center point.
What is point symmetry?
If a dilation scale factor is greater than 0 but less than 1, the resulting image undergoes this specific structural change compared to its pre-image.
What is a reduction (or it gets smaller)?
In a geometric proof, if you know that lines are parallel, this specific type of angle pairs can be immediately declared congruent because they lie in the same relative position at each intersection.
What are corresponding angles?
If triangle JKL is congruent to triangle MNO and triangle MNO is congruent to triangle PQR, this statement correctly identifies the congruence relationship for triangle LKJ.
What is "triangle LKJ is congruent to triangle RQP?
A line segment has a length of 10 units. If it undergoes a translation up 5 units and left 5 units, this will be the length of the resulting line segment.
What is 10 units? (Translations preserve length)
This is the total number of distinct lines of symmetry that can be drawn through a regular octagonal figure.
What is 8?
In a scale drawing where a segment of 20 units corresponds to a segment of 15 units on a similar figure, this is the simplified scale factor of the reduction.
What is 3/4 (or 0.75)?
When parallel lines are cut by a transversal, the two angles situated between the parallel lines on opposite sides of the transversal are congruent because of this theorem.
What is the Alternate Interior Angles Theorem?
To prove two triangles are congruent using the Side-Angle-Side (SAS) postulate, the congruent angle must be located in this position relative to the two congruent sides.
What is the included angle? (The angle between the two sides)
If line segment AB is reflected across a line p, and the perpendicular distance from point A to line p is 4 cm, this is the perpendicular distance between line p and the reflected point A'
What is 4 centimeters?
This is the smallest positive angle of rotation about its center that will map a regular octagon onto itself.
What is 45 degrees?
This similarity criterion can be used to prove two triangles are similar.
What is Angle-Angle (AA) similarity?
If Line A is parallel to Line B, Line C is parallel to Line D, and Line D is perpendicular to Line B, then Line C must have this relationship to Line A.
What is perpendicular?
If you are trying to prove triangle congruence and you have two pairs of congruent angles, you must find a congruent side in this position to use the Angle-Side-Angle (ASA) postulate.
What is the included side? (The side between the two angles)