Regents Exam Study Guide: Triangle Congruence Proofs

Core Methods of Triangle Congruence

To prove that two triangles are congruent ($\cong$), specific sets of corresponding parts must be shown to be equal in measure. The methods discussed in the Regents Exam materials include:

• SSS (Side-Side-Side): All three pairs of corresponding sides are congruent.
• SAS (Side-Angle-Side): Two pairs of corresponding sides and the angle included between them are congruent.
• ASA (Angle-Side-Angle): Two pairs of corresponding angles and the side included between them are congruent.
• AAS (Angle-Angle-Side): Two pairs of corresponding angles and a non-included side are congruent.
• HL (Hypotenuse-Leg): Specifically for right triangles, the hypotenuses and one pair of legs are congruent.

Proving Congruence in Specific Geometric Configurations

• Direct Identification of Method (Questions 1-3):

  • In $\triangle ABC$ and $\triangle DEF$, if $AB \cong DE$, $\angle A \cong \angle D$, and $\angle B \cong \angle E$, the triangles are congruent by ASA because the congruent side is between the two congruent angles.
  • If $AC$ bisects $\angle BAD$ and $\angle B \cong \angle D$, the triangles $\triangle ABC$ and $\triangle ADC$ are congruent by AAS. This relies on the reflexive property ($AC \cong AC$) and the definition of an angle bisector ($\angle BAC \cong \angle DAC$).
  • In $\triangle DAE$ and $\triangle BCE$, where $AB$ and $CD$ intersect at $E$, if $AE \cong CE$ and $\angle BCE \cong \angle DAE$, then $\triangle DAE \cong \triangle BCE$ by ASA. The third required part is the pair of vertical angles: $\angle AED \cong \angle CEB$.

• Isosceles Triangles and Fractional Segments (Question 4):

  • In $\triangle ABC$, if $AB \cong AC$, then the base angles $\angle ABC$ and $\angle ACB$ are congruent.
  • If $BD = \frac{1}{3} BA$ and $CE = \frac{1}{3} CA$, then $BD \cong CE$.
  • Using the shared side $BC \cong CB$ (reflexive property), $\triangle EBC$ can be proved congruent to $\triangle DCB$ by SAS.

• Parallelogram Properties in Proofs (Questions 7-9, 18):

  • In parallelogram $ABCD$, opposite sides are parallel and congruent ($AD \parallel BC$, $AB \parallel CD$). Diagonals bisect each other.
  • For $\triangle EGC$ and $\triangle FGA$ in a parallelogram where $DE \cong BF$, congruence is proven via AAS. This uses vertical angles ($\angle EGC \cong \angle FGA$) and alternate interior angles formed by the parallel sides.
  • In a parallelogram with diagonal $DB$ and segment $EF$ bisecting $DB$ at $M$, $\triangle EMB \sim \triangle FMD$ can be proven by both ASA and AAS because the parallel lines provide multiple pairs of congruent alternate interior angles.
  • In quadrilateral $ABCD$, if $AB \parallel CD$ and $\angle ABC \cong \angle CDA$, then $\triangle ABC \cong \triangle CDA$ by AAS using the diagonal $AC$ as a shared side and alternate interior angles $\angle BAC \cong \angle DCA$.

Identifying Missing Information for Proofs

To complete a proof, students must identify what specific side or angle is missing to satisfy a given method:

• For SAS Congruence (Questions 11, 12, 19, 24):

  • In $\triangle AGE$ and $\triangle OLD$, given $\angle GAE \cong \angle LOD$ and $AE \cong OD$, the side $AG \cong OL$ is required.
  • In $\triangle ABE$ and $\triangle CBD$, if $DB \cong BE$, matching the included angle requires either further side data like $AD \cong CE$ or specific segment sums to establish $AB \cong CB$.
  • If given $\angle A \cong \angle EDF$ and $AB \cong DE$, the pair $BC \cong EF$ would NOT prove congruence by SAS because it would create an SSA (Side-Side-Angle) situation, which is invalid.
  • In $\triangle PQR$ and $\triangle LMN$ with $PQ \cong LM$, the additional statement $QR \cong MN$ and $\angle Q \cong \angle M$ is sufficient for SAS.

• For ASA and AAS Congruence (Questions 10, 13, 14, 25):

  • In $\triangle BAT$ and $\triangle FLU$, given $\angle B \cong \angle F$ and $BA \cong FL$, the angle $\angle A \cong \angle L$ is needed for ASA.
  • If $AE$ bisects $BD$ at $C$ ($BC \cong DC$) and $\angle ABC \cong \angle EDC$, the angle $\angle BCA \cong \angle DCE$ (vertical angles) is needed for ASA.
  • To prove $\triangle KAN \cong \triangle KSC$ by AAS when $AN \cong SC$, the condition $AN \parallel SC$ is sufficient as it provides the necessary alternate interior angles.

Invalid Methods and Counter-Examples

• SSA (Side-Side-Angle): This sequence of parts does not guarantee congruence. It is frequently cited as a method that cannot be used (Questions 19, 22, 23).
• AAA (Angle-Angle-Angle): This proves that triangles are similar ($\sim$), meaning they have the same shape, but it does not prove they are the same size (congruent).
• Non-Congruent Similar Triangles (Questions 16, 17):
  • Parallel lines ($AB \parallel DE$) intersected by segments $BD$ and $AE$ indicate $\triangle ABC \sim \triangle EDC$ (similarity via AA). However, $\triangle ABC \cong \triangle EDC$ is not always true unless a specific pair of sides is also proved congruent.
  • In $\triangle ABD$ and $\triangle CBE$, shared angles or given interior angles ($\angle ADB \cong \angle CEB$) may prove similarity ($\triangle ADB \sim \triangle CBE$), but congruence cannot be proven without side lengths.

Right Triangle Congruence

• Conditions for Right Triangles (Question 21):
  • Two right triangles must be congruent if their corresponding legs are congruent (SAS, where the included angle is the $90^{\circ}$ angle).
  • Congruent hypotenuses alone or congruent acute angles alone are insufficient.
• Specific Example (Question 27):
  • $\triangle ABC$ has sides $AB = 5$, $AC = 12$, and $m\angle A = 90^{\circ}$. By the Pythagorean Theorem ($a^2 + b^2 = c^2$), finding $BC$: $\sqrt{5^2 + 12^2} = 13$.
  • $\triangle DEF$ has $m\angle D = 90^{\circ}$, $DF = 12$, and $EF = 13$. Solving for leg $DE$: $\sqrt{13^2 - 12^2} = 5$.
  • Both triangles are 5-12-13 right triangles. They are congruent by SSS (and thus also similar, as all congruent triangles are similar).

Vertical Angles and Bisectors

• Vertical Angles: When two lines intersect, the opposite angles are always congruent ($\angle BCA \cong \angle DCE$). This is a frequent "hidden" piece of information in proofs involving intersecting segments (Questions 3, 5, 7, 13).
• Segment Bisectors: If a segment is bisected at a point, it is divided into two congruent segments. For example, if $HK$ bisects $IL$ at $J$, then $IJ \cong LJ$ (Question 5).

Questions & Discussion

Question 26: In $\triangle BAT$ and $\triangle CRE$, $\angle A \cong \angle R$ and $BA \cong CR$. Write one additional statement that could be used to prove that the two triangles are congruent and state the method.

• Response 1: $\angle B \cong \angle C$ using the ASA method.
• Response 2: $\angle T \cong \angle E$ using the AAS method.
• Response 3: $AT \cong RE$ using the SAS method.

Question 6: In a diagram where $CA \perp AB$, $ED \perp DF$, $ED \parallel AB$, $CE \cong BF$, $AB \cong ED$, and $m\angle CAB = m\angle FDE = 90$. Which statement would not be used to prove $\triangle ABC \cong \triangle DEF$?

• Response: SSS cannot be used because there is no statement or proof provided that the third sides ($AC$ and $DF$) or the hypotenuses ($BC$ and $EF$) are congruent before the triangles are proved congruent.