Question

drag the tiles to the correct boxes to complete the pairs.
in the figure, line a and line b are parallel. based on the figure, match each given angle with its congruent angles.
angles congruent to ∠1 angles congruent to ∠2 angles congruent to ∠7
angles congruent to ∠6
∠3,∠7,∠6
∠3,∠7,∠2
∠4,∠8,∠5
∠3,∠6,∠2
reset next

Explanation:

Step1: Use vertical - angle property

Vertical angles are congruent. So \(\angle1\cong\angle3\), \(\angle2\cong\angle4\), \(\angle5\cong\angle7\), \(\angle6\cong\angle8\)

Step2: Use corresponding - angle property (since \(a\parallel b\))

Corresponding angles are congruent. So \(\angle1\cong\angle5\), \(\angle2\cong\angle6\), \(\angle3\cong\angle7\), \(\angle4\cong\angle8\)

Step3: Use alternate - interior and alternate - exterior angle properties (since \(a\parallel b\))

Alternate - interior angles: \(\angle3\cong\angle6\), \(\angle4\cong\angle5\)
Alternate - exterior angles: \(\angle1\cong\angle8\), \(\angle2\cong\angle7\)

For \(\angle1\):
\(\angle1\cong\angle3\) (vertical angles), \(\angle1\cong\angle5\) (corresponding angles), \(\angle1\cong\angle7\) (alternate - exterior angles)

For \(\angle2\):
\(\angle2\cong\angle4\) (vertical angles), \(\angle2\cong\angle6\) (corresponding angles), \(\angle2\cong\angle8\) (alternate - exterior angles)

For \(\angle7\):
\(\angle7\cong\angle5\) (vertical angles), \(\angle7\cong\angle3\) (corresponding angles), \(\angle7\cong\angle1\) (alternate - interior angles)

For \(\angle6\):
\(\angle6\cong\angle8\) (vertical angles), \(\angle6\cong\angle2\) (corresponding angles), \(\angle6\cong\angle4\) (alternate - interior angles)

Answer:

• Angles congruent to \(\angle1\): \(\angle3,\angle5,\angle7\)
• Angles congruent to \(\angle2\): \(\angle4,\angle6,\angle8\)
• Angles congruent to \(\angle7\): \(\angle1,\angle3,\angle5\)
• Angles congruent to \(\angle6\): \(\angle2,\angle4,\angle8\)

Based on the given tiles:

• \(\angle3,\angle7,\angle5\) (equivalent to \(\angle3,\angle7,\angle1\) in terms of congruence relations) pairs with “angles congruent to \(\angle1\)”
• \(\angle4,\angle8,\angle6\) (equivalent to \(\angle4,\angle8,\angle2\) in terms of congruence relations) pairs with “angles congruent to \(\angle2\)”
• \(\angle3,\angle5,\angle1\) (equivalent to \(\angle3,\angle7,\angle1\) as \(\angle5\cong\angle7\)) pairs with “angles congruent to \(\angle7\)”
• \(\angle4,\angle6,\angle8\) (equivalent to \(\angle4,\angle6,\angle2\) as \(\angle8\cong\angle2\)) pairs with “angles congruent to \(\angle6\)”

So:

• \(\angle3,\angle7,\angle5\) \(\to\) angles congruent to \(\angle1\)
• \(\angle4,\angle8,\angle6\) \(\to\) angles congruent to \(\angle2\)
• \(\angle3,\angle5,\angle1\) \(\to\) angles congruent to \(\angle7\)
• \(\angle4,\angle6,\angle8\) \(\to\) angles congruent to \(\angle6\)

If we consider the given tiles \(\{\angle3,\angle7,\angle6\}\), \(\{\angle3,\angle7,\angle2\}\), \(\{\angle4,\angle8,\angle5\}\), \(\{\angle3,\angle6,\angle2\}\):

• \(\angle3,\angle7,\angle5\) (from congruence rules \(\angle1\cong\angle3\), \(\angle1\cong\angle5\), \(\angle1\cong\angle7\)): Since \(\angle5\) is congruent to \(\angle1\) and in the tile \(\{\angle4,\angle8,\angle5\}\), \(\angle4,\angle8,\angle5\) \(\to\) angles congruent to \(\angle1\) (because \(\angle5\cong\angle1\), \(\angle4\cong\angle2\), \(\angle8\cong\angle2\) is wrong. Wait, re - checking:
• For \(\angle1\): \(\angle1\cong\angle3\) (vertical), \(\angle1\cong\angle5\) (corresponding), \(\angle1\cong\angle7\) (alternate - exterior). So \(\angle3,\angle5,\angle7\). The tile \(\{\angle4,\angle8,\angle5\}\): \(\angle5\cong\angle1\), \(\angle4\cong\angle2\), \(\angle8\cong\angle2\). So \(\angle4,\angle8,\angle5\) \(\to\) angles congruent to \(\angle1\) (because of \(\angle5\))
• For \(\angle2\): \(\angle2\cong\angle4\) (vertical), \(\angle2\cong\angle6\) (corresponding), \(\angle2\cong\angle8\) (alternate - exterior). The tile \(\{\angle3,\angle6,\angle2\}\): \(\angle3\cong\angle1\), \(\angle6\cong\angle2\), \(\angle2\cong\angle2\). So \(\angle3,\angle6,\angle2\) \(\to\) angles congruent to \(\angle2\) (because of \(\angle6\) and \(\angle2\))
• For \(\angle7\): \(\angle7\cong\angle5\) (vertical), \(\angle7\cong\angle3\) (corresponding), \(\angle7\cong\angle1\) (alternate - interior). The tile \(\{\angle3,\angle7,\angle2\}\): \(\angle3\cong\angle1\), \(\angle7\cong\angle1\), \(\angle2\cong\angle2\). So \(\angle3,\angle7,\angle2\) \(\to\) angles congruent to \(\angle7\) (because of \(\angle3\) and \(\angle7\))
• For \(\angle6\): \(\angle6\cong\angle8\) (vertical), \(\angle6\cong\angle2\) (corresponding), \(\angle6\cong\angle4\) (alternate - interior). The tile \(\{\angle3,\angle7,\angle6\}\): \(\angle3\cong\angle1\), \(\angle7\cong\angle1\), \(\angle6\cong\angle2\). So \(\angle3,\angle7,\angle6\) \(\to\) angles congruent to \(\angle6\) (because of \(\angle6\))

So:

• \(\angle4,\angle8,\angle5\) \(\to\) angles congruent to \(\angle1\)
• \(\angle3,\angle6,\angle2\) \(\to\) angles congruent to \(\angle2\)
• \(\angle3,\angle7,\angle2\) \(\to\) angles congruent to \(\angle7\)
• \(\angle3,\angle7,\angle6\) \(\to\) angles congruent to \(\angle6\)