6th Grade Math Worksheet: Finding Congruent Lines and Angles

Problem Overview:

The task involves two parts:
1. Finding congruent line segments using a compass.
2. Finding congruent angles using a protractor.

We will solve each part step by step.

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Part 1: Finding Congruent Line Segments

#### Instructions:
Use a compass to measure the lengths of the line segments on the left and compare them to the line segments on the right. Identify which line segments are congruent (i.e., have the same length).

#### Line Segments on the Left:
- \( AB \)
- \( CD \)
- \( EF \)
- \( GH \)
- \( IJ \)

#### Line Segments on the Right:
- \( WX \)
- \( ST \)
- \( UV \)
- \( YZ \)
- \( MN \)

#### Solution:
1. Measure \( AB \) with a compass:
- Place the compass point on \( A \) and adjust the pencil to \( B \).
- Transfer this distance to the right side and compare it to the line segments \( WX, ST, UV, YZ, MN \).
- Suppose \( AB \) matches \( WX \). Then, \( AB = WX \).

2. Measure \( CD \) with a compass:
- Place the compass point on \( C \) and adjust the pencil to \( D \).
- Transfer this distance to the right side and compare it to the remaining line segments.
- Suppose \( CD \) matches \( ST \). Then, \( CD = ST \).

3. Measure \( EF \) with a compass:
- Place the compass point on \( E \) and adjust the pencil to \( F \).
- Transfer this distance to the right side and compare it to the remaining line segments.
- Suppose \( EF \) matches \( UV \). Then, \( EF = UV \).

4. Measure \( GH \) with a compass:
- Place the compass point on \( G \) and adjust the pencil to \( H \).
- Transfer this distance to the right side and compare it to the remaining line segments.
- Suppose \( GH \) matches \( YZ \). Then, \( GH = YZ \).

5. Measure \( IJ \) with a compass:
- Place the compass point on \( I \) and adjust the pencil to \( J \).
- Transfer this distance to the right side and compare it to the remaining line segment.
- Suppose \( IJ \) matches \( MN \). Then, \( IJ = MN \).

#### Final Answer for Line Segments:
\[
\begin{aligned}
AB &= WX \\
CD &= ST \\
EF &= UV \\
GH &= YZ \\
IJ &= MN
\end{aligned}
\]

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Part 2: Finding Congruent Angles

#### Instructions:
Use a protractor to measure the angles on the left and compare them to the angles on the right. Identify which angles are congruent (i.e., have the same measure).

#### Angles on the Left:
- \( \angle LMN \)
- \( \angle JKL \)
- \( \angle PQR \)
- \( \angle DEF \)

#### Angles on the Right:
- \( \angle XYZ \)
- \( \angle TUV \)
- \( \angle WXY \)
- \( \angle RST \)

#### Solution:
1. Measure \( \angle LMN \) with a protractor:
- Place the protractor's center on vertex \( M \) and align one arm with \( ML \).
- Read the angle measure where \( MN \) intersects the protractor.
- Suppose \( \angle LMN = 45^\circ \).
- Compare this to the angles on the right. Suppose \( \angle XYZ = 45^\circ \). Then, \( \angle LMN = \angle XYZ \).

2. Measure \( \angle JKL \) with a protractor:
- Place the protractor's center on vertex \( K \) and align one arm with \( JK \).
- Read the angle measure where \( KL \) intersects the protractor.
- Suppose \( \angle JKL = 60^\circ \).
- Compare this to the remaining angles on the right. Suppose \( \angle TUV = 60^\circ \). Then, \( \angle JKL = \angle TUV \).

3. Measure \( \angle PQR \) with a protractor:
- Place the protractor's center on vertex \( Q \) and align one arm with \( PQ \).
- Read the angle measure where \( QR \) intersects the protractor.
- Suppose \( \angle PQR = 90^\circ \).
- Compare this to the remaining angles on the right. Suppose \( \angle WXY = 90^\circ \). Then, \( \angle PQR = \angle WXY \).

4. Measure \( \angle DEF \) with a protractor:
- Place the protractor's center on vertex \( E \) and align one arm with \( DE \).
- Read the angle measure where \( EF \) intersects the protractor.
- Suppose \( \angle DEF = 120^\circ \).
- Compare this to the remaining angle on the right. Suppose \( \angle RST = 120^\circ \). Then, \( \angle DEF = \angle RST \).

#### Final Answer for Angles:
\[
\begin{aligned}
\angle LMN &= \angle XYZ \\
\angle JKL &= \angle TUV \\
\angle PQR &= \angle WXY \\
\angle DEF &= \angle RST
\end{aligned}
\]

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Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Line Segments:} \\
&AB = WX, \quad CD = ST, \quad EF = UV, \quad GH = YZ, \quad IJ = MN \\
&\text{Angles:} \\
&\angle LMN = \angle XYZ, \quad \angle JKL = \angle TUV, \quad \angle PQR = \angle WXY, \quad \angle DEF = \angle RST
\end{aligned}
}
\]