Question

12 construct arguments write a two - column proof of the alternate exterior angles theorem. given: m || n prove: ∠1 ≅ ∠2

1. higher order thinking using what you know about angle pairs formed by parallel lines and a transversal, how are ∠1, ∠2, ∠3, and ∠4 related in the trapezoid? explain.

14 error analysis what error did tyler make? m∠1 = 72 by same - side exterior angles theorem

1. generalize in the diagram shown, if x + y = 180, label the remaining angles as x° or y°.
2. mathematical connections a transversal intersects two parallel lines. the measures of a pair of alternate interior angles are 5v and 2w. the measures of a pair of same - side exterior angles are 10w and 5v. what are the values of w and v?
3. identify a pair of angles for each type. see example 1 same - side interior

18 corresponding

1. alternate exterior
2. which angles are supplementary to ∠1? which are congruent to ∠1? see example 2
3. find each measure. see example 3 m∠1

22 x

1. opposite sides of a parallelogram are parallel. prove that opposite angles of a parallelogram are congruent. see example 4 given: abcd is a parallelogram. prove: ∠a ≅ ∠c, ∠b ≅ ∠d
2. for safety and strength, the supports of a set of scissor steps form parallelograms. what should the measure of ∠1 be to allow a dog to climb safely into its owner’s car? see example 5

Explanation:

Step1: Identify problem - angle relationships

We have multiple sub - problems related to angle relationships in parallel lines and geometric shapes.

Step2: Solve problem 12

Given \(m\parallel n\), we know that alternate exterior angles are congruent by the Alternate Exterior Angles Theorem. In a two - column proof:

Statements	Reasons
\(\angle1\) and \(\angle2\) are alternate exterior angles	Definition of alternate exterior angles
\(\angle1\cong\angle2\)	Alternate Exterior Angles Theorem

Step3: Solve problem 17

Same - side interior angles in the given figure are \(\angle3\) and \(\angle5\) (assuming standard labeling of angles formed by two lines and a transversal).

Step4: Solve problem 18

Corresponding angles are \(\angle1\) and \(\angle5\) (assuming standard labeling of angles formed by two lines and a transversal).

Step5: Solve problem 19

Alternate exterior angles are \(\angle1\) and \(\angle7\) (assuming standard labeling of angles formed by two lines and a transversal).

Step6: Solve problem 20

Angles supplementary to \(\angle1\) are \(\angle2\) (linear pair) and \(\angle4\) (if the lines are parallel and considering same - side interior or other supplementary relationships depending on the figure). Congruent to \(\angle1\) are angles that are corresponding or alternate exterior (e.g., if there are parallel lines) such as \(\angle5\) (corresponding) or \(\angle7\) (alternate exterior).

Step7: Solve problem 21

If the angle adjacent to \(\angle1\) is \(123^{\circ}\) and they are a linear pair, then \(m\angle1 = 180 - 123=57^{\circ}\).

Step8: Solve problem 22

If \((3x + 6)^{\circ}\) and \(123^{\circ}\) are corresponding or alternate interior/exterior angles (assuming parallel lines), then \(3x+6 = 123\).
Subtract 6 from both sides: \(3x=123 - 6=117\).
Divide both sides by 3: \(x = 39\).

Step9: Solve problem 23

Given parallelogram \(ABCD\):

Statements	Reasons
\(\angle A+\angle B = 180^{\circ}\) and \(\angle B+\angle C=180^{\circ}\)	Same - side interior angles of parallel lines are supplementary
\(\angle A+\angle B=\angle B+\angle C\)	Transitive property of equality
\(\angle A=\angle C\)	Subtraction property of equality

Similarly, we can prove \(\angle B=\angle D\).

Step10: Solve problem 24

If the supports form parallelograms, and we assume the angle adjacent to \(\angle1\) is \(99^{\circ}\), and they are a linear pair, then \(m\angle1=180 - 99 = 81^{\circ}\).

Answer:

1. Two - column proof shown above.
2. \(\angle3\) and \(\angle5\)
3. \(\angle1\) and \(\angle5\)
4. \(\angle1\) and \(\angle7\)
5. Supplementary: \(\angle2,\angle4\); Congruent: \(\angle5,\angle7\)
6. \(57^{\circ}\)
7. \(x = 39\)
8. Proof shown above.
9. \(81^{\circ}\)