Math / Geometry

QuestionS
8. Given: $\overline{\mathrm{DE}} \cong \overline{\mathrm{HG}}, \mathrm{DE} / / \mathrm{GH}$
Prove: $\overline{\mathrm{DF}} \cong \overline{\mathrm{HF}}$ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. Line segments $\overline{\mathrm{DE}}$ and $\overline{\mathrm{HG}}$ are congruent.
2. Line segments $\mathrm{DE}$ and $\mathrm{GH}$ are parallel.
3. $\mathrm{F}$ is the intersection point of lines $\mathrm{DG}$ and $\mathrm{EH}$ .

STEP 2

1. Identify corresponding angles.
2. Use the properties of parallel lines.
3. Apply the Side-Angle-Side (SAS) Congruence Theorem.
4. Conclude congruence of $\overline{\mathrm{DF}}$ and $\overline{\mathrm{HF}}$ .

STEP 3

Identify corresponding angles formed by the intersecting lines and parallel lines:
- Since $\mathrm{DE} \parallel \mathrm{GH}$ and $\mathrm{DF}$ is a transversal, $\angle \mathrm{EDF} \cong \angle \mathrm{GHF}$ (Alternate Interior Angles).

STEP 4

Use the properties of parallel lines to establish angle congruence:
- $\angle \mathrm{DEF} \cong \angle \mathrm{HGF}$ because they are corresponding angles.

STEP 5

Apply the Side-Angle-Side (SAS) Congruence Theorem:
- We have $\overline{\mathrm{DE}} \cong \overline{\mathrm{HG}}$ , $\angle \mathrm{EDF} \cong \angle \mathrm{GHF}$ , and $\angle \mathrm{DEF} \cong \angle \mathrm{HGF}$ . - By SAS, $\triangle \mathrm{DEF} \cong \triangle \mathrm{HGF}$ .

STEP 6

Conclude congruence of $\overline{\mathrm{DF}}$ and $\overline{\mathrm{HF}}$ :
- Since $\triangle \mathrm{DEF} \cong \triangle \mathrm{HGF}$ , corresponding parts of congruent triangles are congruent (CPCTC), so $\overline{\mathrm{DF}} \cong \overline{\mathrm{HF}}$ .
The proof is complete, and $\overline{\mathrm{DF}} \cong \overline{\mathrm{HF}}$ .

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Studdy Solution

STEP 1

1. Line segments $\overline{\mathrm{DE}}$ and $\overline{\mathrm{HG}}$ are congruent.
2. Line segments $\mathrm{DE}$ and $\mathrm{GH}$ are parallel.
3. $\mathrm{F}$ is the intersection point of lines $\mathrm{DG}$ and $\mathrm{EH}$ .

STEP 2

1. Identify corresponding angles.
2. Use the properties of parallel lines.
3. Apply the Side-Angle-Side (SAS) Congruence Theorem.
4. Conclude congruence of $\overline{\mathrm{DF}}$ and $\overline{\mathrm{HF}}$ .

STEP 3

Identify corresponding angles formed by the intersecting lines and parallel lines:
- Since $\mathrm{DE} \parallel \mathrm{GH}$ and $\mathrm{DF}$ is a transversal, $\angle \mathrm{EDF} \cong \angle \mathrm{GHF}$ (Alternate Interior Angles).

STEP 4

Use the properties of parallel lines to establish angle congruence:
- $\angle \mathrm{DEF} \cong \angle \mathrm{HGF}$ because they are corresponding angles.

STEP 5

Apply the Side-Angle-Side (SAS) Congruence Theorem:
- We have $\overline{\mathrm{DE}} \cong \overline{\mathrm{HG}}$ , $\angle \mathrm{EDF} \cong \angle \mathrm{GHF}$ , and $\angle \mathrm{DEF} \cong \angle \mathrm{HGF}$ . - By SAS, $\triangle \mathrm{DEF} \cong \triangle \mathrm{HGF}$ .

STEP 6

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Studdy Solution

STEP 1

1. Line segments DE‾\overline{\mathrm{DE}}DE and HG‾\overline{\mathrm{HG}}HG are congruent.2. Line segments DE\mathrm{DE}DE and GH\mathrm{GH}GH are parallel.3. F\mathrm{F}F is the intersection point of lines DG\mathrm{DG}DG and EH\mathrm{EH}EH.

STEP 2

1. Identify corresponding angles.2. Use the properties of parallel lines.3. Apply the Side-Angle-Side (SAS) Congruence Theorem.4. Conclude congruence of DF‾\overline{\mathrm{DF}}DF and HF‾\overline{\mathrm{HF}}HF.

STEP 3

Identify corresponding angles formed by the intersecting lines and parallel lines:- Since DE∥GH\mathrm{DE} \parallel \mathrm{GH}DE∥GH and DF\mathrm{DF}DF is a transversal, ∠EDF≅∠GHF\angle \mathrm{EDF} \cong \angle \mathrm{GHF}∠EDF≅∠GHF (Alternate Interior Angles).

STEP 4

Use the properties of parallel lines to establish angle congruence:- ∠DEF≅∠HGF\angle \mathrm{DEF} \cong \angle \mathrm{HGF}∠DEF≅∠HGF because they are corresponding angles.

STEP 5

STEP 6

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1. Line segments $\overline{\mathrm{DE}}$ and $\overline{\mathrm{HG}}$ are congruent.
2. Line segments $\mathrm{DE}$ and $\mathrm{GH}$ are parallel.
3. $\mathrm{F}$ is the intersection point of lines $\mathrm{DG}$ and $\mathrm{EH}$ .

1. Identify corresponding angles.
2. Use the properties of parallel lines.
3. Apply the Side-Angle-Side (SAS) Congruence Theorem.
4. Conclude congruence of $\overline{\mathrm{DF}}$ and $\overline{\mathrm{HF}}$ .

Identify corresponding angles formed by the intersecting lines and parallel lines:
- Since $\mathrm{DE} \parallel \mathrm{GH}$ and $\mathrm{DF}$ is a transversal, $\angle \mathrm{EDF} \cong \angle \mathrm{GHF}$ (Alternate Interior Angles).

Use the properties of parallel lines to establish angle congruence:
- $\angle \mathrm{DEF} \cong \angle \mathrm{HGF}$ because they are corresponding angles.