Triangle Congruence and Geometry Proofs

Practice flashcards covering geometry definitions, theorems, and triangle congruence proofs (SSS, SAS, ASA) from the lecture notes.

What is the definition of an Angle Bisector according to the lecture?

It is a ray or segment that divides an angle into two congruent angles, for example: if $AD$ bisects $\angle BAC$, then $\angle 1 \cong \angle 2$.

According to the Vertical Angles Congruent Theorem, what can be concluded when two lines form an 'X' shape?

The vertical angles formed are congruent, for example: $\angle 1 \cong \angle 4$.

Define the Right Angles Congruent Postulate.

If two angles (such as $\angle AXD$ and $\angle AXC$) are both right angles, then they are congruent ($\angle AXD \cong \angle AXC$).

What is the Definition of Perpendicular Lines in the context of angle measure?

If $AX \perp BC$, then $\angle AXD$ is a right angle ($90^\circ$).

What is the Definition of a Midpoint?

If a point $M$ is the midpoint of segment $PQ$, then $PM \cong MQ$.

What is the definition of a Segment Bisector?

A segment or line that divides another segment into two congruent parts; for example, if $AC$ bisects $BD$, then $BE \cong DE$.

In a geometry proof, what is the 'Reflexive Property'?

A property stating that a segment or angle is congruent to itself, such as $AD \cong AD$ or $RZ \cong RZ$.

What is the Side-Side-Side (SSS) Theorem?

If 3 sides of one triangle are congruent to the 3 sides of another triangle, then the triangles are congruent.

What is the Side-Angle-Side (SAS) Theorem?

If 2 sides and the included angle of one triangle are congruent to the 2 sides and the included angle of another triangle, then the triangles are congruent.

What is the Angle-Side-Angle (ASA) Theorem?

If 2 angles and the included side of one triangle are congruent to the 2 angles and the included side of another triangle, then the triangles are congruent.

In the proof where $Z$ is the midpoint of $TS$, what is the second statement and reason?

Statement: $TZ \cong ZS$; Reason: Defn of Midpt.

If $DB$ bisects $AE$, what congruent segments are identified by the Definition of Segment Bisector?

$AC \cong CE$.

In the practice proof for $\Delta ABD \cong \Delta ABX$, what reason allows the statement $AX \cong AX$?

Reflexive Property.

If $B$ is the midpoint of $AC$, what is the resulting congruent statement?

$AB \cong BC$.

If $GE$ bisects $\angle DEF$, what two angles are congruent?

$\angle 3 \cong \angle 4$.

In a proof where $AX \perp BC$, how is the statement '$\angle 2 \cong \angle 4$' justified?

By the Right Angles Congruent Postulate (R. L. S. Postulate).

According to the SAS theorem description, what is specific about the position of the angle?

The angle must be the 'included angle' between the two congruent sides.

If $RU$ bisects $QT$, what is the resulting segment statement?

$QS \cong ST$.

In the congruence statement for SSS using triangles $ABC$ and $XYZ$, what are the three side requirements?

$BC \cong ZY$, $AB \cong XY$, and $AC \cong XZ$.

Why is the triangle pair in practice problem 3 (Page 3) marked as 'Not Congruent'?

The angle has to be the included angle for SAS, and in this diagram, it is not.