Question

name xicely hqu0aa 2 - 1 additional practice parallel lines use the figure for exercises 1 - 4. identify all pairs of each type of angle. 1. corresponding angles 2. same - side interior angles 3. alternate interior angles 4. alternate exterior angles use the figure for exercises 5 and 6. 5. which angles are supplementary to the given angle? 6. which angles are congruent to the given angle? 7. complete the two - column proof. given: x || y prove: ∠3 ≅ ∠5 statements 1) x || y 2) m∠3 + m∠8 = 180° 3) m∠5 + m∠8 = 180° 4) 5) 6) reasons 1) 2) 3) 4) transitive property of equality 5) subtraction property of equality 6) definition of congruence 8. in the figure, eh || bi and bi || ci. a. what is m∠1? explain. b. what is m∠3? explain.

Explanation:

Step1: Recall angle - pair definitions

Corresponding angles are in the same relative position with respect to the parallel lines and the transversal.
Same - side interior angles are between the parallel lines and on the same side of the transversal.
Alternate interior angles are between the parallel lines and on opposite sides of the transversal.
Alternate exterior angles are outside the parallel lines and on opposite sides of the transversal.

Step2: For supplementary angles

Two angles are supplementary if their sum is 180°. For congruent angles, they have the same measure.

Step3: For two - column proof

1) Given \(x\parallel y\), the reason is "Given".
2) \(m\angle3 + m\angle8=180^{\circ}\) because same - side interior angles of parallel lines are supplementary.
3) \(m\angle5 + m\angle8 = 180^{\circ}\) because they are also same - side interior angles of parallel lines.
4) Since \(m\angle3 + m\angle8=180^{\circ}\) and \(m\angle5 + m\angle8 = 180^{\circ}\), by the Transitive Property of Equality, \(m\angle3=m\angle5\).
5) Subtracting \(m\angle8\) from both sides of \(m\angle3 + m\angle8=m\angle5 + m\angle8\) (using the Subtraction Property of Equality), we get \(m\angle3=m\angle5\).
6) By the Definition of congruence (\(\angle A\cong\angle B\) if and only if \(m\angle A = m\angle B\)), \(\angle3\cong\angle5\).

Step4: For angle - measure problems

Use properties of parallel lines and angle - relationships such as vertical angles (which are congruent) and corresponding angles (which are congruent when lines are parallel) to find the measures of \(\angle1\) and \(\angle3\).

1. **Corresponding angles**:
- Without seeing the specific figure, in general, if we have two parallel lines \(l_1\) and \(l_2\) cut by a transversal \(t\), pairs of corresponding angles are \(\angle1\) and \(\angle5\), \(\angle2\) and \(\angle6\), \(\angle3\) and \(\angle7\), \(\angle4\) and \(\angle8\).
2. **Same - side interior angles**:
- Pairs of same - side interior angles are \(\angle3\) and \(\angle5\), \(\angle4\) and \(\angle6\).
3. **Alternate interior angles**:
- Pairs of alternate interior angles are \(\angle3\) and \(\angle6\), \(\angle4\) and \(\angle5\).
4. **Alternate exterior angles**:
- Pairs of alternate exterior angles are \(\angle1\) and \(\angle7\), \(\angle2\) and \(\angle8\).
5. **Supplementary angles**:
- If the given angle is \(\angle1\), its supplementary angles are those that form a linear - pair with it or are same - side interior angles with it (depending on the figure). For example, if \(\angle1\) and \(\angle2\) form a linear - pair, \(\angle2\) is supplementary to \(\angle1\). Also, if there are parallel lines and \(\angle1\) is related to another angle as a same - side interior angle, that angle is supplementary to \(\angle1\).
6. **Congruent angles**:
- If the given angle is \(\angle1\), its congruent angles are corresponding angles (if there are parallel lines), vertical angles. For example, if \(\angle1\) has a vertical angle \(\angle3\), \(\angle1\cong\angle3\), and if there are parallel lines and \(\angle1\) has a corresponding angle \(\angle5\), \(\angle1\cong\angle5\).
7. **Two - column proof**:
- Statements:
- 1) \(x\parallel y\) (Given)
- 2) \(m\angle3 + m\angle8=180^{\circ}\) (Same - side interior angles of parallel lines are supplementary)
- 3) \(m\angle5 + m\angle8 = 180^{\circ}\) (Same - side interior angles of parallel lines are supplementary)
- 4) \(m\angle3=m\angle5\) (Transitive Property of Equality)
- 5) \(\angle3\cong\angle5\) (Definition of congruence)
- Reasons:
- 1) Given
- 2) Same - side interior angles of parallel lines are supplementary
- 3) Same - side interior angles of parallel lines are supplementary
- 4) Transitive Property of Equality
- 5) Definition of congruence
8. **a. Measure of \(\angle1\)**:
- Without seeing the figure details, if \(EH\parallel BI\) and \(BI\parallel CJ\), use angle - relationships such as corresponding angles, vertical angles. If there is a known angle related to \(\angle1\) by one of these relationships, we can find its measure. For example, if \(\angle1\) is a vertical angle to an angle of measure \(50^{\circ}\), then \(m\angle1 = 50^{\circ}\). If \(\angle1\) is a corresponding angle to an angle of measure \(50^{\circ}\) formed by parallel lines, then \(m\angle1=50^{\circ}\).
8. **b. Measure of \(\angle3\)**:
- Similar to part a, use properties of parallel lines and angle - relationships. If \(\angle3\) is related to a known - measure angle by vertical - angle or corresponding - angle relationships, we can find its measure. For example, if \(\angle3\) is a corresponding angle to an angle of measure \(60^{\circ}\) formed by parallel lines, then \(m\angle3 = 60^{\circ}\).

Answer:

1. Corresponding angles: Vary by figure (e.g., \(\angle1\) and \(\angle5\), \(\angle2\) and \(\angle6\), \(\angle3\) and \(\angle7\), \(\angle4\) and \(\angle8\) in a standard parallel - lines - transversal setup).
2. Same - side interior angles: Vary by figure (e.g., \(\angle3\) and \(\angle5\), \(\angle4\) and \(\angle6\) in a standard parallel - lines - transversal setup).
3. Alternate interior angles: Vary by figure (e.g., \(\angle3\) and \(\angle6\), \(\angle4\) and \(\angle5\) in a standard parallel - lines - transversal setup).
4. Alternate exterior angles: Vary by figure (e.g., \(\angle1\) and \(\angle7\), \(\angle2\) and \(\angle8\) in a standard parallel - lines - transversal setup).
5. Supplementary angles: Vary by figure and given angle.
6. Congruent angles: Vary by figure and given angle.
7. Statements: 1) \(x\parallel y\); 2) \(m\angle3 + m\angle8=180^{\circ}\); 3) \(m\angle5 + m\angle8 = 180^{\circ}\); 4) \(m\angle3=m\angle5\); 5) \(\angle3\cong\angle5\). Reasons: 1) Given; 2) Same - side interior angles of parallel lines are supplementary; 3) Same - side interior angles of parallel lines are supplementary; 4) Transitive Property of Equality; 5) Definition of congruence.
8. a. \(m\angle1\): Vary by figure, use angle - relationships. b. \(m\angle3\): Vary by figure, use angle - relationships.