- Which angle pairs are formed by a transversal crossing parallel lines? - The two marked angles lie inside the parallel lines on opposite sides of the transversal. - What is true about alternate interior angles?

1. The given \(62^\circ\) angle and \(\beta\) are alternate interior angles. 2. Alternate interior angles are congruent when the lines are parallel. 3. Therefore, \(\beta = 62^\circ\).

\(\beta = 62^\circ\)

- What angle relationship is formed when a transversal crosses two parallel lines? - Identify the alternate interior angle corresponding to \(\beta\).

1. Angles \(\beta\) and \(\varepsilon\) are alternate interior angles formed by a transversal crossing parallel lines. 2. Therefore, \(\varepsilon = \beta = 68^\circ\).

\(\varepsilon = 68^\circ\)

- What is the name of the two angles in matching positions? - Which theorem lets you conclude something about the lines from the angle measures?

1. The two marked angles occupy matching positions at the intersections, so they are corresponding angles. 2. They have equal measures, so the converse of the corresponding angles theorem gives \(g\parallel h\).

Yes. The equal \(74^\circ\) corresponding angles show that \(g\parallel h\).

- What must be true about corresponding angles if the lines are parallel? - Compare the two measured angles rather than how the lines might look.

1. Parallel lines cut by a transversal have congruent corresponding angles. 2. Here \(55^\circ\ne51^\circ\), so the corresponding angles are not congruent. 3. The lines cannot be parallel.

No. Their corresponding angles have different measures.

- What is the sum of same-side interior angles when the lines are parallel? - Use the given \(110^\circ\) angle with that relationship.

1. Same-side interior angles formed by a transversal of parallel lines are supplementary. 2. The other angle measures \(180^\circ - 110^\circ = 70^\circ\).

The other angle measures \(70^\circ\).

- First decide whether each named angle lies inside or outside the two parallel lines. - Then compare which side of transversal \(t\) each angle occupies. - Match those two observations to the four relationship names in the question.

1. Both angles lie between the parallel lines \(g\) and \(h\), so they are interior angles. 2. They lie on opposite sides of transversal \(t\). 3. Therefore, \(\angle BPQ\) and \(\angle CQP\) are alternate interior angles.

Alternate interior angles.

- Compare the corner occupied by each named angle at its intersection with the transversal. - Decide whether the two angles occupy matching corners or opposite corners. - Match that position pattern to the four relationship names in the question.

1. \(\angle BPQ\) and \(\angle DQE\) occupy the same relative position at the two intersections. 2. Therefore, they are corresponding angles.

Corresponding angles.

- First decide which named angles lie inside the strip between \(g\) and \(h\) and which lie outside it. - Then compare which side of transversal \(t\) each angle occupies. - Use the three-letter angle names to locate the vertex and the two rays.

1. \(\angle EPB\) and \(\angle CQF\) lie outside the parallel lines and on opposite sides of transversal \(t\), so they are alternate exterior angles. 2. \(\angle APQ\) and \(\angle CQP\) lie between the parallel lines and on the same side of transversal \(t\), so they are same-side interior angles.

a) \(\angle EPB\) and \(\angle CQF\) b) \(\angle APQ\) and \(\angle CQP\)

- In which task are the lines already known to be parallel? - In which task are the angle measures used to establish parallelism?

1. When \(g\parallel h\) is given, the corresponding angles theorem says the matching angles are congruent. 2. When corresponding angles are equal, the converse theorem establishes \(g\parallel h\). 3. Therefore task b) uses the converse, while a) uses the original theorem.

a) Corresponding angles are congruent. b) \(g\parallel h\). Task b) uses the converse theorem.

- What relationship do corresponding angles have when the lines are parallel? - What is the sum of a linear pair? - Use vertical and corresponding angle relationships to count all congruent angles.

1. Corresponding angles formed by a transversal of parallel lines are congruent. Therefore, each angle measures \(156^\circ \div 2 = 78^\circ\). 2. Angles in a linear pair are supplementary, so the adjacent angle measures \(180^\circ - 78^\circ = 102^\circ\). 3. Vertical, corresponding, and alternate angles create a set of four congruent acute angles and four congruent obtuse angles. Therefore, four of the eight angles measure \(78^\circ\).

a) \(78^\circ\) b) \(102^\circ\) c) Four angles

- Recall the angle-pair names used with parallel lines and a transversal. - What is the sum of a linear pair? - What happens to a linear pair when one angle is \(90^\circ\)?

1. Three other \(65^\circ\) angles are the vertical angle to \(\alpha\), the corresponding angle at the other intersection, and the vertical angle to that corresponding angle. The last of these is also alternate to \(\alpha\). 2. Since a linear pair totals \(180^\circ\), \(\beta = 180^\circ - 65^\circ = 115^\circ\). 3. If the transversal is perpendicular to the parallel lines, one angle measures \(90^\circ\). Its linear-pair angle also measures \(180^\circ - 90^\circ = 90^\circ\), and vertical and corresponding angles are congruent. Therefore, all eight angles measure \(90^\circ\), so the claim is true.

a) The vertical angle to \(\alpha\), its corresponding angle, and the vertical angle to that corresponding angle all measure \(65^\circ\). b) \(\beta = 115^\circ\) c) True. All eight angles would measure \(90^\circ\).

- Which angle-pair name describes two angles between the lines on the same side of the transversal? - What angle sum would force the two lines to be parallel? - Do you need the theorem or its converse when parallelism is the conclusion?

1. The angles are same-side interior angles. 2. Their sum is \(70^\circ+110^\circ=180^\circ\). 3. By the converse of the same-side interior angles theorem, \(g\parallel h\).

Yes. The same-side interior angles are supplementary, so \(g\parallel h\).

- Which sides of the trapezoid are parallel? - What is the sum of the two interior angles on the same leg? - Use the two legs separately to find the unknown angles.

1. Consecutive interior angles along each leg of a trapezoid with parallel bases add to \(180^\circ\). 2. Along \(AD\), \(\delta=180^\circ-70^\circ=110^\circ\). 3. Along \(BC\), \(\beta=180^\circ-135^\circ=45^\circ\).

\(\beta=45^\circ\) and \(\delta=110^\circ\).

- What angle relationships are formed when a transversal cuts parallel lines? - Identify the angle corresponding to the given angle. - What is the sum of a linear pair?

1. The angle corresponding to the given \(55^\circ\) angle also measures \(55^\circ\). 2. That corresponding angle and \(\alpha\) form a linear pair. 3. Therefore, \(\alpha = 180^\circ - 55^\circ = 125^\circ\).

\(\alpha = 125^\circ\)

- Which pair of sides is parallel? - What angle relationships are created when a transversal crosses parallel lines? - Look for corresponding angles and same-side interior angles. - Compare each unknown angle with a given angle along the same transversal.

1. Because \(AB \parallel CD\), \(\alpha\) and the \(70^\circ\) angle are corresponding angles formed by transversal \(AD\). Therefore, \(\alpha = 70^\circ\). 2. The \(60^\circ\) angle and \(\beta\) are same-side interior angles formed by transversal \(BC\), so they are supplementary. Therefore, \(\beta = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 70^\circ\) and \(\beta = 120^\circ\)

- First find the angle next to the given obtuse angle. - What does a matching alternate interior angle pair tell you about the lines?

1. The angle adjacent to \(120^\circ\) is \(180^\circ-120^\circ=60^\circ\). 2. It equals the alternate interior angle at \(h\). 3. The converse of the alternate interior angles theorem gives \(g\parallel h\).

Yes. The alternate interior angles are both \(60^\circ\), so \(g\parallel h\).

- Recall the angle relationships formed when a transversal crosses parallel lines. - How are \(\angle A\) and \(\delta\) related when \(AB \parallel CD\)? - What is the sum of the two same-side interior angles along either leg?

1. Because \(AB \parallel CD\), the same-side interior angles along each leg of the trapezoid are supplementary. 2. Along leg \(AD\), \(\delta = 180^\circ - 75^\circ = 105^\circ\). 3. Along leg \(BC\), \(\gamma = 180^\circ - 55^\circ = 125^\circ\).

\(\gamma = 125^\circ\) and \(\delta = 105^\circ\)

- Identify the corresponding angles formed by the transversal. - Which angles are vertical angles? - Which adjacent angles form a linear pair? - Use the fact that the transversal is not perpendicular to the parallel lines.

1. Angles \(\alpha\) and \(\beta\) are corresponding angles, so \(\alpha = \beta\). 2. Angles \(\alpha\) and \(\gamma\) are vertical angles, so \(\alpha = \gamma\). 3. Angles \(\beta\) and \(\delta\) form a linear pair, so \(\beta + \delta = 180^\circ\). 4. Since \(\gamma = \alpha = \beta\), the equation \(\gamma = \delta\) could hold only if both angles measured \(90^\circ\). The transversal is not perpendicular to the parallel lines, so statement d) is false.

d) \(\gamma = \delta\) is false.

- What angle relationships occur when a transversal crosses parallel lines? - Which angles must have equal measures? - What are adjacent angles that form a straight line called? - Review corresponding, alternate interior, vertical, and supplementary angles.

1. Angle \(\alpha\) is alternate interior to the given \(115^\circ\) angle, so \(\alpha = 115^\circ\). 2. Angles \(\alpha\) and \(\beta\) form a linear pair, so \(\beta = 180^\circ - 115^\circ = 65^\circ\). 3. Angle \(\gamma\) forms a linear pair with the given \(115^\circ\) angle, so \(\gamma = 65^\circ\).

\(\alpha = 115^\circ\), \(\beta = 65^\circ\), and \(\gamma = 65^\circ\)

- What angle pairs are formed when a transversal crosses two parallel lines? - Review corresponding angles and linear pairs. - What is the sum of two angles that form a straight line?

1. Angle \(\beta\) corresponds to \(\alpha\). Corresponding angles formed by a transversal crossing parallel lines are congruent, so \(\beta = 65^\circ\). 2. Angles \(\beta\) and \(\gamma\) form a linear pair. Therefore, \(\gamma = 180^\circ - 65^\circ = 115^\circ\).

\(\beta = 65^\circ\) and \(\gamma = 115^\circ\)

- Review vertical angles and linear pairs at an intersection. - Which angle relationships occur when a transversal crosses parallel lines? - What is the sum of a linear pair?

1. Angle \(\beta\) is vertical to the given \(54^\circ\) angle, so \(\beta = 54^\circ\). 2. Angle \(\gamma\) corresponds to the given \(54^\circ\) angle, so \(\gamma = 54^\circ\). 3. Angles \(\gamma\) and \(\delta\) form a linear pair. Therefore, \(\delta = 180^\circ - 54^\circ = 126^\circ\).

\(\beta = 54^\circ\), \(\gamma = 54^\circ\), and \(\delta = 126^\circ\)

- What angle pairs are formed when a transversal crosses parallel lines? - What is the sum of adjacent angles that form a straight angle? - Which unknown angles are related directly to the given angles?

1. The angles \(50^\circ\), \(\alpha\), and \(60^\circ\) form a straight angle along line \(g\). Therefore, \(\alpha = 180^\circ - 50^\circ - 60^\circ = 70^\circ\). 2. Angle \(\beta\) is alternate interior to the given \(50^\circ\) angle, so \(\beta = 50^\circ\). 3. Angle \(\gamma\) is supplementary to the angle corresponding to the given \(60^\circ\) angle. Therefore, \(\gamma = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 70^\circ\), \(\beta = 50^\circ\), and \(\gamma = 120^\circ\)

- What angle pairs are formed by two intersecting lines? - Which angles must be congruent when a transversal crosses parallel lines? - Are the marked angles opposite at one intersection or in matching positions at two intersections?

1. Angle \(\beta\) is vertical to \(\alpha\), so \(\beta = 68^\circ\). 2. Angle \(\gamma\) corresponds to \(\alpha\), so \(\gamma = 68^\circ\).

\(\beta = 68^\circ\) because it is vertical to \(\alpha\). \(\gamma = 68^\circ\) because it corresponds to \(\alpha\).

- Identify which marked angle forms a linear pair with \(\alpha\). - Identify the marked angle outside the parallel lines on the opposite side of the transversal from \(\alpha\). - Use the appropriate relationship for each pair rather than estimating from the drawing.

1. Angles \(\alpha\) and \(\beta\) form a linear pair, so \(m\angle\beta=180^\circ-132^\circ=48^\circ\). 2. Since \(a\parallel b\), \(\alpha\) and \(\delta\) are alternate exterior angles. Therefore, \(m\angle\delta=132^\circ\).

\(m\angle\beta=48^\circ\) \(m\angle\delta=132^\circ\)

- Look for corresponding or alternate interior angles formed by the parallel lines. - What is the sum of adjacent angles that form a straight line? - Which angles must be congruent because \(g \parallel h\)?

1. Angle \(\alpha\) is alternate interior to the given \(36^\circ\) angle, so \(\alpha = 36^\circ\). 2. Angle \(\gamma\) is alternate interior to the given \(68^\circ\) angle, so \(\gamma = 68^\circ\). 3. The angles \(68^\circ\), \(\beta\), and \(36^\circ\) form a straight angle on line \(g\). Therefore, \(\beta = 180^\circ - 68^\circ - 36^\circ = 76^\circ\).

\(\alpha = 36^\circ\), \(\beta = 76^\circ\), and \(\gamma = 68^\circ\)

- What relationship do corresponding angles have when the lines are parallel? - What is the sum of the measures in a linear pair?

1. The corresponding obtuse angle at line \(h\) also measures \(115^\circ\). 2. That angle and \(\beta\) form a linear pair, so \(\beta = 180^\circ - 115^\circ = 65^\circ\).

\(\beta = 65^\circ\)

- Which opposite sides of the parallelogram are parallel? - Use diagonal \(QS\) as a transversal for each pair. - Match the named angles by their vertices, not their positions on a sketch.

1. Opposite sides of a parallelogram are parallel: \(PQ\parallel RS\) and \(PS\parallel QR\). 2. Diagonal \(QS\) is a transversal. Alternate interior angles give \(\angle QSR=\angle PQS=35^\circ\). 3. The other pair gives \(\angle SQR=\angle PSQ=45^\circ\).

\(\angle QSR=35^\circ\) because \(PQ\parallel SR\), and \(\angle SQR=45^\circ\) because \(PS\parallel QR\). Each pair is alternate interior along diagonal \(QS\).

- Locate both three-letter angles in the diagram before writing an equation. - Decide whether their positional relationship makes the measures equal or supplementary when the lines are parallel. - Substitute your value of \(x\) into both expressions to check consistency.

1. From the diagram, \(\angle APQ\) and \(\angle PQD\) are alternate interior angles. 2. Since \(g\parallel h\), their measures are equal: \(4x+7=6x-29\). 3. Solving gives \(36=2x\), so \(x=18\). 4. Substitution gives \(4(18)+7=79\) and \(6(18)-29=79\). Each angle measures \(79^\circ\).

\(x=18\); both angles measure \(79^\circ\). The relationship is alternate interior angles.

- Use the three-letter names to locate both angles relative to the two parallel lines and the transversal. - Decide whether the pair is forced to be congruent or supplementary. - Check that your final angle measures satisfy the relationship you identified.

1. From the diagram, \(\angle BPQ\) and \(\angle PQD\) are same-side interior angles. 2. Since \(p\parallel q\), they are supplementary: \((5x+8)+(3x+12)=180\). 3. Thus \(8x+20=180\), so \(x=20\). 4. The measures are \(5(20)+8=108^\circ\) and \(3(20)+12=72^\circ\).

\(x=20\); \(m\angle BPQ=108^\circ\) and \(m\angle PQD=72^\circ\). They are same-side interior angles.

- First decide whether each marked angle is inside or outside the parallel lines. - Are the marked angles on the same side or opposite sides of the transversal?

1. The angles lie between the parallel lines and on opposite sides of transversal \(s\), so they are alternate interior angles. 2. Alternate interior angles are congruent when the lines are parallel. Hence \(\alpha=65^\circ\).

Alternate interior angles; \(\alpha=65^\circ\).

- What sum do same-side interior angles have when two lines are parallel? - A crew wants to infer parallelism from measurements. Which direction of the angle rule is needed? - Compare the new angle sum with the sum required for parallel lines.

1. Same-side interior angles must sum to \(180^\circ\) for the lines to be parallel. 2. The original angles sum to \(71^\circ+109^\circ=180^\circ\). Their measures establish parallelism by the converse theorem. 3. The new sum is \(71^\circ+106^\circ=177^\circ\). If those are the actual measurements, the edges cannot be parallel.

a) Yes. The angles sum to \(180^\circ\), so the edges are parallel. b) If both measurements are correct, the edges are not parallel.

- Let the smaller angle be \(x\), and express the larger angle in terms of \(x\). - What is true about alternate interior angles when lines are parallel? - The two adjacent angles must continue to total \(180^\circ\).

1. Let \(x\) be the acute angle. The adjacent obtuse angle is \(2x\). 2. Since the angles form a linear pair, \(x + 2x = 180^\circ\). Thus \(3x = 180^\circ\), so \(x = 60^\circ\). The obtuse angle is \(120^\circ\). 3. Alternate interior angles formed by parallel lines are congruent. Therefore, the requested sum is \(60^\circ + 60^\circ = 120^\circ\). 4. The acute and obtuse angles remain a linear pair. If the acute angle decreases to \(50^\circ\), the obtuse angle becomes \(180^\circ - 50^\circ = 130^\circ\). It increases by \(10^\circ\).

a) The acute angle is \(60^\circ\), and the obtuse angle is \(120^\circ\). b) \(120^\circ\) c) The obtuse angle increases by \(10^\circ\), from \(120^\circ\) to \(130^\circ\).

- Describe the angle positions created by each transversal. - What is true about corresponding angles when two lines are parallel? - What is the sum of a linear pair?

1. The \(35^\circ\) angle and \(\gamma\) are corresponding angles formed by transversal \(s\). Since \(g \parallel h\), \(\gamma = 35^\circ\). 2. The \(120^\circ\) angle corresponds to a \(120^\circ\) angle at the other parallel line. That angle and \(\epsilon\) form a linear pair. 3. Therefore, \(\epsilon = 180^\circ - 120^\circ = 60^\circ\).

\(\gamma = 35^\circ\) and \(\epsilon = 60^\circ\).

- Use corresponding angle relationships on the parallel lines. - Identify the triangle formed by the three intersecting lines. - What is the sum of a triangle’s interior angles? - Use a linear pair to find the remaining angle.

1. The \(50^\circ\) angle and \(\gamma\) are corresponding angles. Since \(g \parallel h\), \(\gamma = 50^\circ\). 2. In the triangle formed by \(g\), \(s\), and \(t\), \(\alpha + \gamma + 75^\circ = 180^\circ\). 3. Substitute \(\gamma = 50^\circ\): \(\alpha + 50^\circ + 75^\circ = 180^\circ\), so \(\alpha = 55^\circ\). 4. At the other parallel line, the angle corresponding to \(\alpha\) also measures \(55^\circ\). It forms a linear pair with \(\beta\), so \(\beta = 180^\circ - 55^\circ = 125^\circ\).

\(\alpha = 55^\circ\), \(\beta = 125^\circ\), and \(\gamma = 50^\circ\).

- What does an angle bisector do? - Use corresponding or alternate interior angles formed by the parallel lines. - Which angle forms a linear pair with \(\beta\)?

1. Since \(w\) is an angle bisector, the two angles between \(g\) and \(t\) are congruent. The given half is \(28^\circ\), so \(\delta = 28^\circ\). 2. The full angle between \(g\) and \(t\) is \(28^\circ + 28^\circ = 56^\circ\). 3. The corresponding angle at line \(h\) also measures \(56^\circ\). It forms a linear pair with \(\beta\), so \(\beta = 180^\circ - 56^\circ = 124^\circ\). 4. The \(28^\circ\) angle formed by \(g\) and \(w\) is alternate interior to \(\gamma\). Therefore, \(\gamma = 28^\circ\).

\(\delta = 28^\circ\), \(\beta = 124^\circ\), and \(\gamma = 28^\circ\).

- Use corresponding and alternate interior angle relationships. - Find the interior angle at \(C\) by using a linear pair. - What is the sum of the interior angles of triangle \(ACE\)?

1. The \(60^\circ\) angle and \(\beta\) are alternate interior angles formed by transversal \(s\). Therefore, \(\beta = 60^\circ\). 2. At \(C\), the angle corresponding to the given \(130^\circ\) angle measures \(130^\circ\). The interior angle of triangle \(ACE\) at \(C\) is its linear-pair angle: \(180^\circ - 130^\circ = 50^\circ\). 3. In triangle \(ACE\), \(\alpha = 180^\circ - 60^\circ - 50^\circ = 70^\circ\).

\(\alpha = 70^\circ\) and \(\beta = 60^\circ\).

- Which corresponding or alternate angles can you identify? - What total do adjacent angles on a straight line have? - You may also identify the triangle formed by the two transversals and one parallel line.

1. By the corresponding and alternate angle relationships for parallel lines, the two angles adjacent to \(\alpha\) along line \(h\) measure \(60^\circ\) and \(50^\circ\). 2. These three angles form a straight angle, so \(60^\circ + \alpha + 50^\circ = 180^\circ\). 3. Therefore, \(\alpha = 180^\circ - 60^\circ - 50^\circ = 70^\circ\).

\(\alpha = 70^\circ\)

- What total do the three angles below line \(g\) form? - Use alternate interior angle relationships between \(g\) and \(h\). - Which known angle forms a linear pair with \(\gamma\)?

1. The three angles below line \(g\) form a straight angle. Therefore, \(\alpha = 180^\circ - 70^\circ - 50^\circ = 60^\circ\). 2. The \(70^\circ\) angle and \(\beta\) are alternate interior angles, so \(\beta = 70^\circ\). 3. The angle alternate interior to the given \(50^\circ\) angle measures \(50^\circ\). It forms a linear pair with \(\gamma\), so \(\gamma = 180^\circ - 50^\circ = 130^\circ\).

\(\alpha = 60^\circ\), \(\beta = 70^\circ\), and \(\gamma = 130^\circ\).

- Identify alternate interior and corresponding angles. - Which unknown angle is congruent to the given \(55^\circ\) angle? - Which unknown angle forms a linear pair with a \(115^\circ\) angle?

1. The angle \(\gamma\) is alternate interior to the given \(55^\circ\) angle. Since \(g \parallel h\), \(\gamma = 55^\circ\). 2. The angle corresponding to the given \(115^\circ\) angle forms a linear pair with \(\delta\). 3. Therefore, \(\delta = 180^\circ - 115^\circ = 65^\circ\).

\(\gamma = 55^\circ\) and \(\delta = 65^\circ\).

- What relationship do vertical angles have? - What is the sum of a linear pair? - How are corresponding angles related when lines are parallel?

1. The given \(55^\circ\) angle and \(\alpha\) are vertical angles, so \(\alpha = 55^\circ\). 2. The angle corresponding to \(\alpha\) at the upper intersection also measures \(55^\circ\). It forms a linear pair with \(\beta\), so \(\beta = 180^\circ - 55^\circ = 125^\circ\). 3. The angles \(\beta\) and \(\gamma\) are vertical angles, so \(\gamma = 125^\circ\).

\(\alpha = 55^\circ\), \(\beta = 125^\circ\), and \(\gamma = 125^\circ\).

- Classify each pair by whether the angles are interior or exterior and by which side of the transversal they occupy. - For parallel lines, decide which named relationships force congruence and which force supplementarity. - Do not use the apparent angle sizes in the drawing as measurements; justify each statement from its position.

1. \(\angle EPB\) and \(\angle PQD\) are corresponding angles, so a) is true because \(g\parallel h\). 2. \(\angle APQ\) and \(\angle PQD\) are alternate interior angles, so b) is true. 3. \(\angle BPQ\) and \(\angle PQD\) are same-side interior angles, so their measures are supplementary. Thus c) is true. 4. \(\angle EPB\) and \(\angle DQF\) are same-side exterior angles. For parallel lines they are supplementary, not generally congruent. Thus d) is false.

d) is false. \(\angle EPB\) and \(\angle DQF\) are same-side exterior angles, so their measures are supplementary rather than necessarily equal.

- Which three angles form a straight angle where the transversals meet \(g\)? - Look for corresponding angles at the lower line. - How can a linear pair give each remaining angle?

1. At the intersection on \(g\), three adjacent angles form a straight angle: \(60^\circ+\alpha+45^\circ=180^\circ\). Thus \(\alpha=75^\circ\). 2. The angle next to \(\beta\) corresponds to the given \(60^\circ\) angle. They form a linear pair, so \(\beta=180^\circ-60^\circ=120^\circ\). 3. The angle next to \(\gamma\) corresponds to the given \(45^\circ\) angle. They form a linear pair, so \(\gamma=180^\circ-45^\circ=135^\circ\).

The straight angle gives \(\alpha=180^\circ-60^\circ-45^\circ=75^\circ\). Corresponding angles and linear pairs give \(\beta=180^\circ-60^\circ=120^\circ\) and \(\gamma=180^\circ-45^\circ=135^\circ\).

- What can you calculate from the two angles of triangle \(BSC\)? - Which angle relationship would hold if the two lines were parallel? - Does the angle you found satisfy that relationship?

1. The triangle angle sum gives \(\angle SBC=180^\circ-90^\circ-58^\circ=32^\circ\). 2. The given alternate interior angle at \(g\) is \(35^\circ\), which differs from \(32^\circ\). 3. If \(g\parallel h\), alternate interior angles would be equal. Therefore, \(g\) and \(h\) cannot be parallel.

No. The triangle gives \(\angle SBC=32^\circ\), but its alternate interior angle is \(35^\circ\). Parallel lines would require equal angles.

- Transfer the given angle from the upper parallel line to the lower one. - Use the triangle between the two parallel lines. - What is the sum of its three interior angles?

1. Because \(g\parallel h\), the angle at \(A\) in triangle \(ASB\) equals the given \(50^\circ\) angle at \(S\). 2. The triangle above \(g\) has known interior angles \(50^\circ\) and \(80^\circ\). 3. Thus \(\beta=180^\circ-50^\circ-80^\circ=50^\circ\).

\(\beta=50^\circ\).

- What angle relationships occur when a transversal crosses two parallel lines? - What is true about adjacent angles that form a straight line? - Review corresponding, alternate interior, and vertical angles. - Can you first find the angle next to the given angle? - How are the angles at the upper intersection related to those at the lower intersection?

1. The angle supplementary to the given \(115^\circ\) angle measures \(180^\circ - 115^\circ = 65^\circ\). 2. Angle \(\alpha\) corresponds to this \(65^\circ\) angle, so \(\alpha = 65^\circ\). 3. Angles \(\alpha\) and \(\beta\) are vertical angles, so \(\beta = \alpha = 65^\circ\).

\(\alpha = 65^\circ\) and \(\beta = 65^\circ\)

- What angle relationships occur when a transversal crosses two parallel lines? - Review corresponding and alternate interior angles. - What is the sum of the interior angles of a triangle? - Which shown angle is supplementary to the interior angle at \(B\)?

1. Because \(g \parallel h\), \(\alpha\) is alternate interior to the given \(50^\circ\) angle. Therefore, \(\alpha = 50^\circ\). 2. The \(130^\circ\) angle at \(D\) corresponds to the exterior angle at \(B\). The interior angle of triangle \(ABC\) at \(B\) is supplementary to that angle, so it measures \(180^\circ - 130^\circ = 50^\circ\). 3. The interior angles of triangle \(ABC\) total \(180^\circ\). Therefore, \(\gamma = 180^\circ - 50^\circ - 50^\circ = 80^\circ\).

\(\alpha = 50^\circ\) and \(\gamma = 80^\circ\)

- What angle relationships occur when other lines cross two parallel lines? - Review alternate interior and corresponding angles. - How could an auxiliary line through \(P\), parallel to \(g\) and \(h\), help? - Can you identify a triangle whose angle sum will help?

1. Because \(g \parallel h\), \(\beta\) is alternate interior to the given \(40^\circ\) angle at \(A\). Therefore, \(\beta = 40^\circ\). 2. The angle at \(B\) that is exterior to triangle \(ABP\) corresponds to the given \(125^\circ\) angle at \(D\). The interior angle at \(B\) is its supplement, so it measures \(180^\circ - 125^\circ = 55^\circ\). 3. In triangle \(ABP\), \(\alpha = 180^\circ - 40^\circ - 55^\circ = 85^\circ\).

\(\alpha = 85^\circ\) and \(\beta = 40^\circ\)

- Angles that form a straight angle have a sum of \(180^\circ\). - What angle relationships occur when a transversal crosses parallel lines? Think about corresponding and alternate interior angles. - Can you identify a triangle in the figure and use its angle sum? - What is true about vertical angles?

1. Angles \(\alpha\), \(55^\circ\), and \(60^\circ\) are adjacent along line \(g\) and form a straight angle. Therefore, \(\alpha = 180^\circ - 55^\circ - 60^\circ = 65^\circ\). 2. Because \(g \parallel h\), \(\delta\) is alternate interior to \(\alpha\). Therefore, \(\delta = 65^\circ\). 3. Angle \(\varepsilon\) corresponds to the given \(60^\circ\) angle. Therefore, \(\varepsilon = 60^\circ\).

\(\alpha = 65^\circ\), \(\delta = 65^\circ\), and \(\varepsilon = 60^\circ\)

- Which angle at the upper line matches the lower-left marked angle? - Look for the triangle bounded by the lower parallel and the two transversals. - At the upper-right intersection, find the angle next to \(\gamma\) first.

1. Corresponding angles give \(\beta=50^\circ\). 2. In triangle \(ABC\), the other two base angles are \(50^\circ\) and \(60^\circ\), so \(\alpha=180^\circ-50^\circ-60^\circ=70^\circ\). 3. The angle at \(E\) next to \(\gamma\) corresponds to the \(60^\circ\) interior angle at \(B\). 4. The pair at \(E\) is a linear pair, so \(\gamma=180^\circ-60^\circ=120^\circ\).

\(\beta=50^\circ\) by corresponding angles. Triangle \(ABC\) gives \(\alpha=180^\circ-50^\circ-60^\circ=70^\circ\). The angle beside \(\gamma\) corresponds to \(60^\circ\), so their linear pair gives \(\gamma=120^\circ\).

- Start with the large triangle \(ABC\). What is the sum of its interior angles? - Because \(DE \parallel BC\), which corresponding or alternate interior angles can you identify? - How are \(\beta\) and \(\angle ADE\) related along line \(AB\)? - What is the sum of a linear pair?

1. In triangle \(ABC\), \(\alpha = 180^\circ - 50^\circ - 60^\circ = 70^\circ\). 2. Because \(DE \parallel BC\), \(\angle ADE\) corresponds to \(\angle ABC\). Therefore, \(\angle ADE = 50^\circ\). 3. Angles \(\beta\) and \(\angle ADE\) form a linear pair along line \(AB\). Thus, \(\beta = 180^\circ - 50^\circ = 130^\circ\).

\(\alpha = 70^\circ\) and \(\beta = 130^\circ\)

- Which lines are parallel? - What angle relationships occur when a transversal crosses parallel lines? - Can you identify a triangle in the diagram and use its angle sum? - Which angles form linear pairs? - Begin with the angles directly related to the given measures.

1. The angle corresponding to the given \(70^\circ\) angle is supplementary to \(\beta\). Therefore, \(\beta = 180^\circ - 70^\circ = 110^\circ\). 2. The two transversals and line \(g\) form a triangle with base angles \(70^\circ\) and \(60^\circ\). Thus, \(\alpha = 180^\circ - 70^\circ - 60^\circ = 50^\circ\). 3. The angle corresponding to the given \(60^\circ\) angle is supplementary to \(\gamma\). Therefore, \(\gamma = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 50^\circ\), \(\beta = 110^\circ\), and \(\gamma = 120^\circ\)

- Transfer the marked angle from \(h\) to \(g\). - Use the triangle formed between the two lines and the intersecting transversals. - What is the relationship between the angle at \(S\) and \(\beta\)?

1. Alternate interior angles give \(\gamma=45^\circ\). 2. In triangle \(SBD\), the angles at \(B\) and \(D\) are \(30^\circ\) and \(45^\circ\). 3. Hence \(\angle BSD=180^\circ-30^\circ-45^\circ=105^\circ\). 4. Angle \(\beta\) is vertical to \(\angle BSD\), so \(\beta=105^\circ\).

\(\beta=105^\circ\), \(\gamma=45^\circ\).

- Find an angle pair along transversal \(t\). - Use the parallel lines before considering the triangle. - Apply the triangle angle sum to the two known angles.

1. Angle \(\alpha\) and the given \(135^\circ\) angle are same-side interior angles. 2. Therefore, \(\alpha=180^\circ-135^\circ=45^\circ\). 3. The angles of triangle \(PQR\) are \(40^\circ\), \(45^\circ\), and \(\beta\). Thus \(\beta=95^\circ\).

\(\alpha=45^\circ\), \(\beta=95^\circ\).

- Use the triangle angle sum. - How are an interior angle and its adjacent exterior angle related? - Look for corresponding angles formed by the parallel lines.

1. The two transversals and line \(g\) form a triangle. Its two base angles are \(65^\circ\) and \(40^\circ\), so its interior angle at their intersection is \(180^\circ-65^\circ-40^\circ=75^\circ\). 2. The marked angle \(\beta\) is vertical to that \(75^\circ\) triangle angle, so \(\beta=75^\circ\). 3. Angles \(\alpha\) and \(\beta\) form a linear pair, so \(\alpha=180^\circ-75^\circ=105^\circ\). 4. At the lower intersection with transversal \(t\), the angle adjacent to the given \(40^\circ\) angle measures \(140^\circ\). That angle corresponds to \(\gamma\) because \(g\parallel h\). Therefore, \(\gamma=140^\circ\).

\(\alpha=105^\circ\), \(\beta=75^\circ\), and \(\gamma=140^\circ\)

- Start with the linear pair containing the marked \(135^\circ\) angle. - Use the parallel lines to transfer the acute angle made by line \(s\). - Then use the right-angle mark where the two diagonal lines meet.

1. The angle that forms a linear pair with the marked \(135^\circ\) angle measures \(180^\circ-135^\circ=45^\circ\). 2. Because \(a\parallel b\), line \(s\) forms the same \(45^\circ\) acute angle with line \(b\). 3. The right-angle mark shows that \(s\perp t\). Therefore, line \(t\) also forms a \(45^\circ\) acute angle with line \(b\). 4. Angle \(\alpha\) is supplementary to that acute angle, so \(m\angle\alpha=180^\circ-45^\circ=135^\circ\).

\(m\angle\alpha=135^\circ\)

- Compare the angle positions in a) and b). - The two transversals and \(g\) form a triangle below \(g\). - Find \(\gamma\) before checking the two statements containing it.

1. Statements a) and b) are true by alternate interior angle relationships. 2. The triangle below \(g\) has interior angles \(60^\circ\), \(40^\circ\), and \(\gamma\). Therefore, \(\gamma=180^\circ-60^\circ-40^\circ=80^\circ\), so d) is true. 3. Since \(60^\circ+40^\circ=100^\circ\ne80^\circ\), c) is false.

c) \(\gamma=\alpha_1+\alpha_2\) is false: \(\gamma=180^\circ-60^\circ-40^\circ=80^\circ\), but \(\alpha_1+\alpha_2=100^\circ\).

- Think of \(AB\) as a line crossing both \(AD\) and \(BC\). - How are the two named angles positioned along that line? - What two pairs of parallel sides must a parallelogram have?

1. Both pairs are corresponding angles formed by transversal \(AB\) with lines \(AD\) and \(BC\). 2. In a), both are \(105^\circ\). The converse of the corresponding angles theorem gives \(AD\parallel BC\). Together with \(AB\parallel CD\), the quadrilateral is a parallelogram. 3. In b), the corresponding angles differ, so \(AD\not\parallel BC\). Hence the quadrilateral is not a parallelogram.

a) The equal corresponding angles give \(AD\parallel BC\). With \(AB\parallel CD\), \(ABCD\) is a parallelogram. b) Unequal corresponding angles mean \(AD\not\parallel BC\), so \(ABCD\) is not a parallelogram.

- A linear pair has a sum of \(180^\circ\). - Use corresponding angles formed by the parallel lines.

1. Angle \(\alpha\) and the given \(115^\circ\) angle form a linear pair, so \(\alpha = 180^\circ - 115^\circ = 65^\circ\). 2. Because \(p \parallel q\), \(\gamma\) corresponds to the given \(48^\circ\) angle, so \(\gamma = 48^\circ\). 3. The two transversals and line \(p\) form a triangle. Therefore, \(\beta = 180^\circ - 65^\circ - 48^\circ = 67^\circ\).

\(\alpha = 65^\circ\), \(\beta = 67^\circ\), and \(\gamma = 48^\circ\)

- How much of a straight angle is taken up by the two exterior pieces? - What can you say about the angle adjacent to them? - When does a pair of alternate interior angles establish parallel lines?

1. The exterior angle and \(\theta\) form a linear pair. The exterior angle has measure \(90^\circ+56^\circ=146^\circ\), so \(\theta=180^\circ-146^\circ=34^\circ\). 2. The alternate interior angles at the two lines both measure \(34^\circ\). 3. By the converse of the alternate interior angles theorem, \(g\parallel h\).

\(\theta=34^\circ\). Yes, \(g\parallel h\) because the alternate interior angles match.

- Which pair of rays at \(C\) are opposite rays along the parallel line? - Transfer the two given angles to the base \(AB\). - Find the last angle from the triangle angle sum.

1. Because \(g\parallel h\), alternate interior angles give \(\angle A=50^\circ\) and \(\angle B=60^\circ\). 2. The triangle angle sum gives \(\angle C=180^\circ-50^\circ-60^\circ=70^\circ\).

\(\angle A=50^\circ\) and \(\angle B=60^\circ\) by alternate interior angles. Then \(\angle C=180^\circ-50^\circ-60^\circ=70^\circ\) by the triangle angle sum.

- Which opposite sides of a parallelogram are parallel? - Use diagonal \(AC\) as a transversal to identify alternate interior angles. - Then use the triangle angle sum in triangle \(ABC\).

1. Since opposite sides of a parallelogram are parallel, alternate interior angles give \(\angle ACB = \angle CAD = 40^\circ\) and \(\angle ACD = \angle BAC = 25^\circ\). 2. In triangle \(ABC\), \(\beta = 180^\circ - 25^\circ - 40^\circ = 115^\circ\).

a) \(\angle ACB = 40^\circ\) and \(\angle ACD = 25^\circ\) b) \(\beta = 115^\circ\)

- Consider a line through the vertex of \(\alpha\) parallel to \(g\) and \(h\). - How does that line split \(\alpha\) into two smaller angles? - Which alternate interior angle relationships can you use?

1. Consider an auxiliary line through the vertex of \(\alpha\) parallel to \(g\) and \(h\). 2. This line divides \(\alpha\) into two angles, so \(\alpha = \alpha_1 + \alpha_2\). 3. By alternate interior angle relationships, \(\alpha_1 = 35^\circ\) and \(\alpha_2 = 40^\circ\). 4. Therefore, \(\alpha = 35^\circ + 40^\circ = 75^\circ\).

\(\alpha = 75^\circ\)

- Consider an auxiliary line through vertex \(V\) parallel to \(g\) and \(h\). - Which corresponding, alternate interior, or supplementary angle relationships can you use? - How does the auxiliary line divide the unknown angle into two smaller angles?

1. Consider an auxiliary line through vertex \(V\) parallel to \(g\) and \(h\). This divides \(\gamma\) into two smaller angles. 2. The lower part is alternate interior to the given \(50^\circ\) angle, so it measures \(50^\circ\). 3. The supplement of the given \(120^\circ\) angle is \(60^\circ\). The upper part is alternate interior to this \(60^\circ\) angle. 4. Therefore, \(\gamma = 50^\circ + 60^\circ = 110^\circ\).

\(\gamma = 110^\circ\)