Let's go over the three diagrams step by step, analyzing the given information and proving why angles $\angle 1 \cong \angle 2$ in each case:

Diagram 1 (Parallelogram)

• Statement: $\angle 1 \cong \angle 2$
• Reason:
  • $PQRS$ is a parallelogram, which implies opposite sides are parallel and equal.
  • Diagonal $PR$ bisects the parallelogram.
  • $\triangle PTS$ and $\triangle QTR$ share side $PT = QT$ and side $TS = QR$, as marked.
  • $\triangle PTS \cong \triangle QTR$ by SSS (Side-Side-Side) congruence, leading to corresponding angles $\angle 1 \cong \angle 2$ by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Diagram 2 (Isosceles Trapezoid)

• Statement: $\angle 1 \cong \angle 2$
• Reason:
  • $ABCD$ is an isosceles trapezoid, which means $AB = CD$ and $AD \parallel BC$.
  • $BE = DE$, and line segment $E$ is an altitude or bisector of the trapezoid.
  • $\triangle ABE \cong \triangle CDE$ by SAS (Side-Angle-Side) congruence (equal sides $AB = CD$, equal angles at the base $\angle ABE \cong \angle CDE$, and common side $BE = DE$).
  • Therefore, $\angle 1 \cong \angle 2$ by CPCTC.

Diagram 3 (Isosceles Triangle)

• Statement: $\angle 1 \cong \angle 2$
• Reason:
  • $\triangle KLM$ is an isosceles triangle where $KM = LM$.
  • Segment $KN = LN$, showing that point $N$ is the midpoint.
  • Triangles $KNL$ and $LNM$ share side $NL$ and have equal sides $KN = LN$.
  • $\triangle KNL \cong \triangle LNM$ by SSS (Side-Side-Side) congruence, which makes corresponding angles $\angle 1 \cong \angle 2$ by CPCTC.

Summary:

In all three cases, angles $\angle 1$ and $\angle 2$ are congruent due to the application of triangle congruence theorems (SSS or SAS), and then by applying CPCTC.

Would you like further clarifications on any step, or details on a specific diagram?

Here are 5 related questions to consider:

1. How does the definition of a parallelogram help establish congruent triangles in Diagram 1?
2. Why is the SAS congruence theorem applicable in Diagram 2 for the isosceles trapezoid?
3. What role does the midpoint or bisector play in the congruence proof of Diagram 3?
4. Can you find another pair of congruent angles or segments in any of the diagrams?
5. What are some other scenarios where CPCTC (Corresponding Parts of Congruent Triangles are Congruent) can be applied?

Tip:

When proving angle or triangle congruence, always check for shared sides, equal sides, and parallelism, as these often lead directly to congruent triangles through the major congruence theorems.