Geometry practice problems identifying congruent triangles using SSS, SAS, AAS, and HL theorems.

Here are the solutions for the practice problems. For each problem, we identify if the triangles are congruent, name the matching triangle, and state the reason (postulate).

22.
* Analysis: We have two right angles ($\angle B$ and $\angle N$), one pair of equal sides ($BC \cong NY$), and one pair of equal angles ($\angle C \cong \angle Y$). The side is between the two angles.
* Reason: ASA (Angle-Side-Angle)
* Answer: $\triangle ABC \cong \triangle PNY$ by ASA

23.
* Analysis: We have two pairs of equal sides ($AB \cong AD$ and $CB \cong CD$) and they share a common side in the middle ($AC$).
* Reason: SSS (Side-Side-Side)
* Answer: $\triangle ABC \cong \triangle ADC$ by SSS

24.
* Analysis: We have two pairs of parallel-looking sides marked equal ($AB \cong CD$ and $BC \cong DA$) and they share a diagonal ($AC$).
* Reason: SSS (Side-Side-Side)
* Answer: $\triangle ABC \cong \triangle CDA$ by SSS

25.
* Analysis: We have a right angle at $B$, a shared side ($CB$), and equal segments on the base ($AB \cong DB$). This is a Right Triangle with two legs known.
* Reason: SAS (Side-Angle-Side) or LL (Leg-Leg)
* Answer: $\triangle ABC \cong \triangle DBC$ by SAS

26.
* Analysis: All three sides of one triangle match all three sides of the other ($FN \cong BA$, $NW \cong AC$, $FW \cong BC$).
* Reason: SSS (Side-Side-Side)
* Answer: $\triangle FNW \cong \triangle BAC$ by SSS

27.
* Analysis: We have right angles, equal hypotenuses ($AB \cong DE$), and vertical angles at $B$ which are equal.
* Reason: AAS (Angle-Angle-Side)
* Answer: $\triangle ABC \cong \triangle EDB$ by AAS

28.
* Analysis: We have two pairs of equal sides ($BC \cong DC$ and $BA \cong DA$) and a shared side ($AC$).
* Reason: SSS (Side-Side-Side)
* Answer: $\triangle ABC \cong \triangle ADC$ by SSS

29.
* Analysis: We have two pairs of equal sides ($AD \cong BC$ and $CD \cong AB$) and a shared side ($AC$).
* Reason: SSS (Side-Side-Side)
* Answer: $\triangle ABC \cong \triangle CDA$ by SSS

30.
* Analysis: These are right triangles. They have equal hypotenuses ($AC \cong DF$) and one equal leg ($BC \cong EF$).
* Reason: HL (Hypotenuse-Leg)
* Answer: $\triangle ABC \cong \triangle DEF$ by HL

31.
* Analysis: We have two pairs of equal sides ($AB \cong CD$ and $BC \cong DA$) and a shared side ($AC$).
* Reason: SSS (Side-Side-Side)
* Answer: $\triangle ABC \cong \triangle CDA$ by SSS

32.
* Analysis: This is a rectangle cut by a diagonal. Opposite sides are equal ($AB \cong CD$ and $BC \cong DA$) and they share side $AC$. Also, the corners are right angles.
* Reason: SSS (Side-Side-Side) or SAS (Side-Angle-Side)
* Answer: $\triangle ABC \cong \triangle CDA$ by SSS

33.
* Analysis: We have two pairs of equal sides ($AB \cong DE$ and $BC \cong EC$) and vertical angles at $C$ ($\angle ACB \cong \angle DCE$). The angle is between the sides.
* Reason: SAS (Side-Angle-Side)
* Answer: $\triangle ABC \cong \triangle DEC$ by SAS

34.
* Analysis: We have two pairs of equal sides ($AC \cong CE$ and $BC \cong CD$) and vertical angles at $C$. The angle is between the sides.
* Reason: SAS (Side-Angle-Side)
* Answer: $\triangle ABC \cong \triangle EDC$ by SAS

35.
* Analysis: We have an isosceles triangle split down the middle. We have equal outer sides ($AB \cong DB$), a shared side ($BC$), and right angles at the bottom.
* Reason: HL (Hypotenuse-Leg)
* Answer: $\triangle ABC \cong \triangle DBC$ by HL

36.
* Analysis: We have two pairs of equal angles ($\angle A \cong \angle T$ and $\angle B \cong \angle P$) and the side between them is equal ($AB \cong TP$).
* Reason: ASA (Angle-Side-Angle)
* Answer: $\triangle ABC \cong \triangle TPR$ by ASA