Question

1. write a flow proof.

given: ∠e ≅ ∠h
∠hfg ≅ ∠egf
prove: △egf ≅ △hfg

1. write a two - column proof.

given: ∠k ≅ ∠m
(overline{kl} cong overline{ml})
prove: △jkl ≅ △pml

Explanation:

Problem 7: Flow Proof for \(\triangle EGF \cong \triangle HFG\)

Step 1: Identify Common Side

The side \( \overline{FG} \) is common to both \( \triangle EGF \) and \( \triangle HFG \), so \( \overline{FG} \cong \overline{FG} \) (Reflexive Property of Congruence).

Step 2: List Given Angles

We are given \( \angle E \cong \angle H \) and \( \angle HFG \cong \angle EGF \).

Step 3: Apply AAS Congruence

In \( \triangle EGF \) and \( \triangle HFG \):

• \( \angle E \cong \angle H \) (Given)
• \( \angle EGF \cong \angle HFG \) (Given)
• \( \overline{FG} \cong \overline{FG} \) (Reflexive Property)

By the Angle - Angle - Side (AAS) Congruence Theorem, \( \triangle EGF \cong \triangle HFG \).

Flow Proof Structure:

\[

$$\begin{align*} \angle E \cong \angle H &\quad \text{(Given)} \\ \angle HFG \cong \angle EGF &\quad \text{(Given)} \\ \overline{FG} \cong \overline{FG} &\quad \text{(Reflexive Property)} \\ \implies \triangle EGF \cong \triangle HFG &\quad \text{(AAS Congruence Theorem)} \end{align*}$$

\]

Problem 8: Two - Column Proof for \(\triangle JKL \cong \triangle PML\)

Step 1: Identify Vertical Angles

\( \angle JLK \) and \( \angle PML \) are vertical angles. By the Vertical Angles Theorem, \( \angle JLK \cong \angle PML \).

Step 2: List Given Information

We are given \( \angle K \cong \angle M \) and \( \overline{KL} \cong \overline{ML} \).

Step 3: Apply ASA Congruence

In \( \triangle JKL \) and \( \triangle PML \):

• \( \angle K \cong \angle M \) (Given)
• \( \overline{KL} \cong \overline{ML} \) (Given)
• \( \angle JLK \cong \angle PML \) (Vertical Angles Theorem)

By the Angle - Side - Angle (ASA) Congruence Theorem, \( \triangle JKL \cong \triangle PML \).

Two - Column Proof:

Statements	Reasons
\( \overline{KL} \cong \overline{ML} \)	Given
\( \angle JLK \cong \angle PML \)	Vertical Angles Theorem
\( \triangle JKL \cong \triangle PML \)	ASA Congruence Theorem

Problem 7 Answer:

By AAS congruence (using given angles and reflexive side), \( \triangle EGF \cong \triangle HFG \)

Problem 8 Answer:

By ASA congruence (using given angle, given side, and vertical angles), \( \triangle JKL \cong \triangle PML \)

Answer:

Step 1: Identify Vertical Angles

\( \angle JLK \) and \( \angle PML \) are vertical angles. By the Vertical Angles Theorem, \( \angle JLK \cong \angle PML \).

Step 2: List Given Information

We are given \( \angle K \cong \angle M \) and \( \overline{KL} \cong \overline{ML} \).

Step 3: Apply ASA Congruence

In \( \triangle JKL \) and \( \triangle PML \):

• \( \angle K \cong \angle M \) (Given)
• \( \overline{KL} \cong \overline{ML} \) (Given)
• \( \angle JLK \cong \angle PML \) (Vertical Angles Theorem)

By the Angle - Side - Angle (ASA) Congruence Theorem, \( \triangle JKL \cong \triangle PML \).

Two - Column Proof:

Statements	Reasons
\( \overline{KL} \cong \overline{ML} \)	Given
\( \angle JLK \cong \angle PML \)	Vertical Angles Theorem
\( \triangle JKL \cong \triangle PML \)	ASA Congruence Theorem

Problem 7 Answer:

By AAS congruence (using given angles and reflexive side), \( \triangle EGF \cong \triangle HFG \)

Problem 8 Answer:

By ASA congruence (using given angle, given side, and vertical angles), \( \triangle JKL \cong \triangle PML \)