Question

1. name all the alternate exterior angles:
2. name all the linear pairs:
3. which of the above types of angles are cong
4. which of the above type of angles are suppl
5. complete the following proof:

give: a || b
prove: m∠1 + m∠8 = 180°

1. complete the following proof:

give: a || b
prove: m∠3 ≅ m∠6

Explanation:

Step1: Analyze the given (Question 10)

We know \( a \parallel b \), and we need to prove \( m\angle1 + m\angle8 = 180^\circ \). First, recall the properties of parallel lines and transversals. When two parallel lines are cut by a transversal, consecutive interior angles are supplementary, and also, vertical angles, corresponding angles, alternate interior angles have specific relationships. Also, linear pairs are supplementary.

Step2: Identify angle relationships

1. \( \angle1 \) and \( \angle2 \) form a linear pair, so \( m\angle1 + m\angle2 = 180^\circ \) (linear pair postulate).
2. Since \( a \parallel b \), \( \angle2 \) and \( \angle8 \) are alternate interior angles. By the alternate interior angles theorem, \( m\angle2 = m\angle8 \).
3. Substitute \( m\angle2 \) with \( m\angle8 \) in the equation \( m\angle1 + m\angle2 = 180^\circ \). We get \( m\angle1 + m\angle8 = 180^\circ \), which is what we needed to prove.

(For Question 11, similar approach: Given \( a \parallel b \), prove \( m\angle3 \cong m\angle6 \))

Step1: Analyze the given (Question 11)

We know \( a \parallel b \), need to prove \( m\angle3 \cong m\angle6 \). Recall alternate interior angles theorem.

Step2: Identify angle relationships

When two parallel lines \( a \) and \( b \) are cut by a transversal, \( \angle3 \) and \( \angle6 \) are alternate interior angles. By the alternate interior angles theorem, if two parallel lines are cut by a transversal, then alternate interior angles are congruent. So \( m\angle3 \cong m\angle6 \).

Answer:

Question 10 Proof:

1. \( a \parallel b \) (Given)
2. \( \angle1 \) and \( \angle2 \) are a linear pair (Definition of linear pair)
3. \( m\angle1 + m\angle2 = 180^\circ \) (Linear Pair Postulate)
4. \( \angle2 \cong \angle8 \) (Alternate Interior Angles Theorem, since \( a \parallel b \))
5. \( m\angle2 = m\angle8 \) (Definition of congruent angles)
6. Substitute \( m\angle2 \) with \( m\angle8 \) in step 3: \( m\angle1 + m\angle8 = 180^\circ \) (Substitution Property)

Question 11 Proof:

1. \( a \parallel b \) (Given)
2. \( \angle3 \) and \( \angle6 \) are alternate interior angles (Definition of alternate interior angles, as \( a \) and \( b \) are parallel and cut by a transversal)
3. \( m\angle3 \cong m\angle6 \) (Alternate Interior Angles Theorem)