Triangle congruence (SSS, SAS, ASA, AAS, HL) and proofs Knowledge-Based Practice

Questions

1. 1.

   Segments $\overline{AC}$ and $\overline{BD}$ intersect at point $E$ such that $\overline{AE} \cong \overline{CE}$ and $\overline{BE} \cong \overline{DE}$. Which congruence criterion and supporting geometric property justify the conclusion that $\triangle ABE \cong \triangle CDE$?

   • A) SSS criterion, using the Reflexive Property of Congruence
   • B) ASA criterion, using the Alternate Interior Angles Theorem
   • C) SAS criterion, using the Vertical Angles Congruence Theorem
   • D) AAS criterion, using the Definition of a Midpoint
2. 2.

   Which of the following sets of corresponding congruent parts between two triangles is insufficient to prove that the triangles are congruent?

   • A) Two pairs of angles and the included pair of sides (ASA)
   • B) Two pairs of sides and a non-included pair of angles (SSA)
   • C) Three pairs of corresponding sides (SSS)
   • D) Two pairs of angles and a non-included pair of sides (AAS)
3. 3.

   In right-angled triangles $ABC$ and $DEF$, $m\angle B = m\angle E = 90^\circ$. Which pair of congruent corresponding parts is sufficient to prove that $\triangle ABC \cong \triangle DEF$ by the Hypotenuse-Leg (HL) theorem?

   • A) $\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$
   • B) $\overline{AC} \cong \overline{DF}$ and $\overline{AB} \cong \overline{DE}$
   • C) $\angle A \cong \angle D$ and $\overline{AB} \cong \overline{DE}$
   • D) $\angle A \cong \angle D$ and $\angle C \cong \angle F$
4. 4.

   In $\triangle ABC$ and $\triangle DEF$, it is given that $\angle A \cong \angle D$, $\angle B \cong \angle E$, and $\overline{BC} \cong \overline{EF}$. Which congruence theorem or postulate directly proves that $\triangle ABC \cong \triangle DEF$?

   • A) Angle-Angle-Side (AAS)
   • B) Angle-Side-Angle (ASA)
   • C) Side-Angle-Side (SAS)
   • D) Hypotenuse-Leg (HL)
5. 5.

   In $\triangle PQR$ and $\triangle STU$, it is given that $\overline{PQ} \cong \overline{ST}$ and $\angle Q \cong \angle T$. Which additional congruence statement is sufficient to prove that $\triangle PQR \cong \triangle STU$ by the Side-Angle-Side (SAS) criterion?

   • A) $\overline{PR} \cong \overline{SU}$
   • B) $\angle P \cong \angle S$
   • C) $\overline{QR} \cong \overline{TU}$
   • D) $\angle R \cong \angle U$

Answers and step-by-step solutions appear after you finish the test.