- What relationship do consecutive angles in a rhombus have? - Use that relationship with the given \(74^\circ\) angle.

1. Consecutive angles in a rhombus are supplementary, so \(\alpha + \beta = 180^\circ\). 2. Substitute \(\alpha = 74^\circ\): \(74^\circ + \beta = 180^\circ\). 3. \(\beta = 106^\circ\).

\(\beta = 106^\circ\)

- What is the relationship between consecutive angles in a parallelogram? - Apply that relationship to \(\angle A\) and \(\angle B\).

1. Consecutive angles in a parallelogram are supplementary. 2. Therefore, \(\beta = 180^\circ - 55^\circ = 125^\circ\).

\(\beta = 125^\circ\)

- What total do complementary angles have? - Subtract the known angle from that total.

1. Complementary angles have a sum of \(90^\circ\). 2. The missing angle is \(90^\circ-28^\circ=62^\circ\).

\(62^\circ\)

- What total is formed by a linear pair? - Which vocabulary word describes two angles whose measures have that total?

1. A linear pair has a sum of \(180^\circ\). 2. \(\beta=180^\circ-117^\circ=63^\circ\). 3. Because their measures sum to \(180^\circ\), the angles are supplementary.

\(\beta=63^\circ\). The angles are supplementary.

- What parts must two adjacent angles share? - Look at the sides that are not shared. - How does a straight line help identify a linear pair?

1. The angles share vertex \(S\) and the dividing ray. 2. Their other sides are opposite rays on the same straight line. 3. They form a linear pair, so together they measure \(180^\circ\).

Yes. They share one side, and their other sides are opposite rays. Their measures add to \(180^\circ\).

- Which angles lie across the vertex from each other? - Which marked angle shares a side with \(\alpha\)? - Recall the relationship between vertical angles.

1. Angle \(\gamma\) is opposite \(\alpha\) across the intersection. 2. Opposite angles formed by two intersecting lines are vertical angles and have equal measures. 3. Angle \(\beta\) is next to \(\alpha\) and is supplementary, not necessarily equal.

\(\gamma\), because \(\alpha\) and \(\gamma\) are vertical angles.

- What angle measure makes one right angle? - How are angles named when their measures add to that total? - Subtract the known measure from the full right angle.

1. Two angles that together make a right angle are complementary. 2. A right angle measures \(90^\circ\), so the missing measure is \(90^\circ-38^\circ=52^\circ\).

A complement, measuring \(52^\circ\).

- Picture the given angle beside a straight angle and a full turn. - Which benchmark is smaller than the given angle, and which is larger?

1. A full turn is \(360^\circ\), so the angle is \(360^\circ-215^\circ=145^\circ\) short of a full turn. 2. A straight angle is \(180^\circ\), so the angle is \(215^\circ-180^\circ=35^\circ\) past a straight angle.

It is \(145^\circ\) short of a full turn and \(35^\circ\) past a straight angle.

- Subtract the given angle from \(360^\circ\). - A reflex angle is greater than \(180^\circ\) and less than \(360^\circ\).

1. A full circle measures \(360^\circ\). 2. The remaining angle is \(\alpha=360^\circ-75^\circ=285^\circ\). 3. Because \(180^\circ<285^\circ<360^\circ\), \(\alpha\) is a reflex angle.

\(\alpha=285^\circ\), a reflex angle

- Compare each marked angle with a right angle and a straight angle. - A complement adds to \(90^\circ\), while a supplement adds to \(180^\circ\). - A reflex angle is more than \(180^\circ\); how much greater is \(\gamma\)?

1. For a supplement to be acute, the given angle must be between \(90^\circ\) and \(180^\circ\). That is \(\beta\), with supplement \(180^\circ-135^\circ=45^\circ\). 2. A straight angle measures \(180^\circ\). Therefore \(\gamma\) exceeds it by \(220^\circ-180^\circ=40^\circ\). 3. Only \(\alpha\) is less than \(90^\circ\), so it can have a complement: \(90^\circ-40^\circ=50^\circ\).

a) \(\beta\) has an acute supplement of \(45^\circ\). b) \(40^\circ\). c) \(\alpha\) has a complement of \(50^\circ\).

- What symmetry property does an isosceles trapezoid have? - How are consecutive interior angles related when lines are parallel? - Work along one leg at a time.

1. Base angles of an isosceles trapezoid are congruent, so \(\beta = \alpha = 65^\circ\). 2. Consecutive interior angles along each leg are supplementary because \(AB \parallel CD\). 3. Therefore, \(\delta = 180^\circ - 65^\circ = 115^\circ\). 4. Likewise, \(\gamma = 180^\circ - 65^\circ = 115^\circ\).

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

- What is true about opposite angles in a parallelogram? - How are consecutive angles in a parallelogram related? - You may also use the quadrilateral angle sum.

1. Opposite angles of a parallelogram are congruent, so \(\delta = \beta = 120^\circ\). 2. Consecutive angles of a parallelogram are supplementary. 3. Therefore, \(\alpha = 180^\circ - 120^\circ = 60^\circ\). 4. Since opposite angles are congruent, \(\gamma = \alpha = 60^\circ\).

\(\alpha = 60^\circ\), \(\gamma = 60^\circ\), and \(\delta = 120^\circ\).

- Classify each angle by comparing it with a right angle and a straight angle. - A positive complement is possible only for an angle smaller than \(90^\circ\). - A positive supplement is possible only for an angle smaller than \(180^\circ\).

1. Angle \(\alpha\) is acute, \(\beta\) is obtuse, and \(\gamma\) is reflex. 2. Only an angle less than \(90^\circ\) can have a positive complement, so only \(\alpha\) can. 3. Any angle less than \(180^\circ\) can have a positive supplement, so \(\alpha\) and \(\beta\) can. 4. Since \(\gamma>180^\circ\), subtracting it from either \(90^\circ\) or \(180^\circ\) would give a negative value, so it has neither a positive complement nor a positive supplement.

\(\alpha\): acute; \(\beta\): obtuse; \(\gamma\): reflex a) \(\alpha\) only b) \(\alpha\) and \(\beta\) c) \(\gamma>180^\circ\), so neither \(90^\circ-\gamma\) nor \(180^\circ-\gamma\) is positive.

- How are consecutive interior angles in a parallelogram related? - What total must the angles at \(A\) and \(D\) have? - Write an equation containing \(\alpha\).

1. Consecutive interior angles of a parallelogram are supplementary because opposite sides are parallel. 2. The angles at \(A\) and \(D\) are consecutive, so \(72^\circ + \alpha = 180^\circ\). 3. Therefore, \(\alpha = 180^\circ - 72^\circ = 108^\circ\).

\(\alpha = 108^\circ\)

- A vertical and horizontal grid line form \(90^\circ\). - A square’s diagonal divides a right angle into two equal angles. - Pay attention to the direction in which the reflex angle is marked.

1. In a), the diagonal of a square divides a right angle in half, so \(\alpha=45^\circ\). 2. In b), the ray points along the northwest diagonal, so \(\beta=90^\circ+45^\circ=135^\circ\). 3. In c), the angle is measured counterclockwise from the right-pointing ray to the downward ray, which is three-fourths of a full turn: \(\gamma=270^\circ\). 4. Therefore, \(\alpha\) is acute, \(\beta\) is obtuse, and \(\gamma\) is reflex.

a) \(\alpha=45^\circ\), acute b) \(\beta=135^\circ\), obtuse c) \(\gamma=270^\circ\), reflex

- What angle measure forms a straight line? - How are adjacent angles at an intersection related? - What is true about opposite angles at an intersection?

1. Angle \(\beta\) forms a linear pair with the \(142^\circ\) angle, so \(\beta = 180^\circ - 142^\circ = 38^\circ\). 2. Angle \(\gamma\) is vertical to the \(142^\circ\) angle, so \(\gamma = 142^\circ\). 3. Angle \(\delta\) is vertical to \(\beta\), so \(\delta = 38^\circ\). 4. Opposite angles formed by two intersecting lines are called vertical angles.

\(\beta = 38^\circ\), \(\gamma = 142^\circ\), and \(\delta = 38^\circ\). The opposite angle pairs are vertical angles.

- What is the sum of two angles that form a linear pair? - Treat the two vertices independently; the quadrilateral angle sum is not needed.

1. Since \(\alpha\) and \(100^\circ\) form a linear pair, \(\alpha = 180^\circ - 100^\circ = 80^\circ\). 2. Since \(\delta\) and \(85^\circ\) form a linear pair, \(\delta = 180^\circ - 85^\circ = 95^\circ\).

\(\alpha = 80^\circ\) and \(\delta = 95^\circ\)

- What is the total measure of a linear pair? - Each unknown angle is adjacent to a \(42^\circ\) angle on a straight line.

1. Each unknown angle and an adjacent \(42^\circ\) angle form a linear pair, so their measures sum to \(180^\circ\). 2. \(\alpha = 180^\circ - 42^\circ = 138^\circ\). 3. Likewise, \(\beta = 180^\circ - 42^\circ = 138^\circ\).

\(\alpha = 138^\circ\) and \(\beta = 138^\circ\)

- What stays the same when equal angles are part of two different straight angles? - Let both given angles measure \(x^\circ\). - How would you describe the unknown angle in either pair?

1. Each remaining angle makes a straight angle with one of the given congruent angles. 2. If each given angle measures \(x^\circ\), each remaining angle measures \(180^\circ-x^\circ\). 3. Both remaining angles therefore have the same measure and are congruent.

Both remaining angles are supplements of congruent angles, so each measures \(180^\circ-x^\circ\). They are congruent.

- Locate the inward-pointing vertex to identify the reflex interior angle. - Decide which marked angles are below \(90^\circ\) and which are below \(180^\circ\). - Connect those comparisons to the definitions of complement and supplement.

1. The inward vertex is \(Q\), so \(\phi_2\) is greater than \(180^\circ\) and is reflex. 2. Angles \(\phi_1\), \(\phi_3\), and \(\phi_4\) are acute, so each is less than \(90^\circ\). 3. Therefore, \(\phi_1\), \(\phi_3\), and \(\phi_4\) each have both a positive complement and a positive supplement. 4. Since \(\phi_2>180^\circ\), it has neither a positive complement nor a positive supplement.

a) \(\phi_2\) b) \(\phi_1,\phi_3,\phi_4\) c) \(\phi_1,\phi_3,\phi_4\) d) \(\phi_2>180^\circ\), so subtracting it from \(90^\circ\) or \(180^\circ\) cannot give a positive angle measure.

- Which angles are vertical when two lines intersect? - What is the sum of the measures in a linear pair?

1. Vertical angles are congruent, so the opposite angle also measures \(32^\circ\). 2. Each adjacent angle forms a linear pair with the given angle, so its measure is \(180^\circ - 32^\circ = 148^\circ\). 3. Therefore, both adjacent angles measure \(148^\circ\).

The vertical angle measures \(32^\circ\), and each adjacent angle measures \(148^\circ\).

- What is the total angle measure around a point? - First find one ninth of that total. - Then use the relationship between angles in a linear pair.

1. The four angles around the intersection total \(360^\circ\). 2. Find one ninth of the total: \(\alpha = \frac{1}{9} \cdot 360^\circ = 40^\circ\). 3. An adjacent angle forms a linear pair with \(\alpha\), so its measure is \(180^\circ - 40^\circ = 140^\circ\).

\(\alpha = 40^\circ\), and an adjacent angle measures \(140^\circ\).

- Separate the relationships “vertical” and “adjacent.” - Which pair is congruent, and which pair forms a straight angle? - Use the relationship before doing any subtraction.

1. Vertical angles are congruent, so the vertical angle also measures \(68^\circ\). 2. Adjacent angles at the intersection form linear pairs with the \(68^\circ\) angle. 3. Each adjacent angle measures \(180^\circ-68^\circ=112^\circ\). 4. Priya is incorrect: the vertical angle is congruent to the given angle, while an adjacent linear-pair angle is supplementary to it.

Priya is incorrect. The vertical angle is \(68^\circ\), and each adjacent angle is \(112^\circ\).

- Write the target total for a complementary pair and for a supplementary pair. - “Congruent angles” means their measures are equal.

1. A complementary angle has measure \(90^\circ-42^\circ=48^\circ\). 2. A supplementary angle has measure \(180^\circ-42^\circ=138^\circ\). 3. A congruent angle has the same measure, \(42^\circ\).

Complementary: \(48^\circ\) Supplementary: \(138^\circ\) Congruent: \(42^\circ\)

- Which total belongs to complementary angles? - Which total belongs to supplementary angles? - Check which of Nora's calculations actually matches her result.

1. Nora used the total for supplementary angles, \(180^\circ\), instead of the total for complementary angles, \(90^\circ\). 2. The complement is \(90^\circ-63^\circ=27^\circ\). 3. The supplement is \(180^\circ-63^\circ=117^\circ\).

Nora used the supplementary total instead of the complementary total. The complement is \(27^\circ\), and the supplement is \(117^\circ\).

- Which angle is next to both angles you want to compare? - What sum do two angles in a linear pair have? - What can you conclude when the same angle is subtracted from two equal sums?

1. Angles \(APB\) and \(BPC\) form a linear pair, so \(m\angle APB+m\angle BPC=180^\circ\). 2. Angles \(BPC\) and \(CPD\) also form a linear pair, so \(m\angle BPC+m\angle CPD=180^\circ\). 3. Subtracting the same \(m\angle BPC\) from both sums gives \(m\angle APB=m\angle CPD\).

Both angles are \(180^\circ-m\angle BPC\), so \(m\angle APB=m\angle CPD\).

- Can you trust exact measurements from a drawing that gives no angles? - Could nearly parallel-looking lines meet if extended? - What would make your answer a proven fact rather than a guess from the picture?

1. The appearance of two lines does not prove they are parallel. 2. Corresponding angles are guaranteed equal when a transversal crosses parallel lines. 3. The needed information is that the two lines are parallel, given or shown by reliable markings.

No. The drawing alone is not enough; the two lines must be known to be parallel.

- What total do the original two angles have? - What happens to each measure when its angle is bisected? - Can you halve the combined total instead of working with each angle separately?

1. The other angle measures \(180^\circ-74^\circ=106^\circ\). 2. Bisecting gives adjacent halves of \(37^\circ\) and \(53^\circ\). 3. They add to \(90^\circ\). In general, half of two angles whose sum is \(180^\circ\) always totals \(90^\circ\).

The halves are \(37^\circ\) and \(53^\circ\), totaling \(90^\circ\). Halving supplementary angles makes complementary halves.

- What information does the question give about the whole angle \(AOB\)? - Does sharing a side always make two angles supplementary? - What total does each of the two relationship names describe?

1. The angles are adjacent because they share vertex \(O\) and side \(OC\). 2. Their outer sides \(OA\) and \(OB\) make a right angle, not a straight angle. 3. Their sum is \(90^\circ\), so they are complementary, not supplementary.

Being adjacent does not make angles supplementary. These two angles form a right angle, so they are complementary.

- Which two adjacent angles together make a straight angle? - Find the whole angle that line \(k\) divides. - What does an angle bisector do to the whole angle?

1. The given \(60^\circ\) angle and the angle from \(h\) to the left-hand ray of \(g\) make a straight angle. 2. That larger angle measures \(180^\circ-60^\circ=120^\circ\). 3. Line \(k\) divides it into two equal angles, so \(\alpha=\beta=120^\circ\div2=60^\circ\).

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

- What relationship connects an interior angle and its adjacent exterior angle? - What does the exterior angle theorem state? - Test the equations with a triangle whose angles have three different measures.

1. By the exterior angle theorem, an exterior angle of a triangle equals the sum of the two remote interior angles. Therefore, \(\delta = \alpha + \beta\), so statement a) is true. 2. Angles \(\gamma\) and \(\delta\) form a linear pair, so \(\delta + \gamma = 180^\circ\). Equivalently, \(\delta = 180^\circ - \gamma\). Therefore, statements b) and c) are true. 3. The equation \(\delta = \beta + \gamma\) would require \(\alpha = \gamma\), which is not true for every triangle. Therefore, statement d) is not always true.

d) \(\delta = \beta + \gamma\) is not always true.

- What is the sum of the interior angles of a triangle? - After finding the interior angle at \(C\), how can you use its linear pair? - Which theorem relates an exterior angle directly to the two remote interior angles?

1. Method 1: The triangle angle sum gives the interior angle at \(C\): \(180^\circ - 55^\circ - 65^\circ = 60^\circ\). This angle and \(\alpha\) form a linear pair, so \(m\angle \alpha = 180^\circ - 60^\circ = 120^\circ\). 2. Method 2: By the exterior angle theorem, an exterior angle equals the sum of the two remote interior angles. Therefore, \(m\angle \alpha = 55^\circ + 65^\circ = 120^\circ\).

\(m\angle \alpha = 120^\circ\)

- Recall what must be true about consecutive angles in every parallelogram. - Recall what must be true about opposite angles in every parallelogram. - For the exterior-angle case, first determine the adjacent interior angle.

1. In a parallelogram, consecutive interior angles are supplementary. For a), \(68^\circ+110^\circ=178^\circ\), so these measurements cannot occur in a parallelogram. 2. In a parallelogram, opposite angles are congruent. For b), opposite angles of \(80^\circ\) are consistent with a parallelogram; the other pair could each be \(100^\circ\). Thus at least one parallelogram can have these measurements. 3. For c), the interior angle adjacent to the \(105^\circ\) exterior angle is \(180^\circ-105^\circ=75^\circ\). 4. Its opposite interior angle is given as \(73^\circ\). Since \(75^\circ\ne73^\circ\), these measurements cannot occur in a parallelogram.

a) No; \(68^\circ+110^\circ\ne180^\circ\). b) Yes; a parallelogram can have opposite angles of \(80^\circ\) and the other opposite pair of \(100^\circ\). c) No; the interior angle next to the \(105^\circ\) exterior angle is \(75^\circ\), not equal to its opposite \(73^\circ\) angle.

- What is the sum of the angles in a linear pair? - Express both \(\alpha\) and \(\alpha'\) in terms of \(\beta\). - What follows if two angles have the same measure?

1. Since \(\alpha\) and \(\beta\) form a linear pair, \(\alpha + \beta = 180^\circ\), so \(\alpha = 180^\circ - \beta\). 2. Since \(\alpha'\) and \(\beta\) also form a linear pair, \(\alpha' + \beta = 180^\circ\), so \(\alpha' = 180^\circ - \beta\). 3. Both angles equal \(180^\circ - \beta\), so \(\alpha = \alpha'\). This proves that vertical angles are congruent.

Both \(\alpha\) and \(\alpha'\) equal \(180^\circ - \beta\), so \(\alpha = \alpha'\).

- What is the sum of adjacent angles in a parallelogram? - What does an angle bisector do? - Use the angle sum of triangle \(ABP\).

1. Adjacent angles in a parallelogram are supplementary, so \(\alpha + \beta = 180^{\circ}\). 2. The angle bisectors create angles of \(\frac{\alpha}{2}\) and \(\frac{\beta}{2}\) in triangle \(ABP\). 3. Thus \(\frac{\alpha}{2} + \frac{\beta}{2} = \frac{180^{\circ}}{2} = 90^{\circ}\). 4. By the triangle angle-sum theorem, \(\angle APB = 180^{\circ} - 90^{\circ} = 90^{\circ}\).

\(\angle APB = 90^{\circ}\)

- Let the shared middle angle between \(b\) and \(c\) have measure \(x^\circ\). - Express each of the two target angles as the remainder of a right angle. - What follows if both expressions are the same?

1. Let the angle between rays \(b\) and \(c\) have measure \(x^\circ\). 2. Since the angle from \(a\) to \(c\) is a right angle, the angle between \(a\) and \(b\) measures \(90^\circ - x^\circ\). 3. Since the angle from \(b\) to \(d\) is also a right angle, the angle between \(c\) and \(d\) measures \(90^\circ - x^\circ\). 4. The two angles have the same measure, so they are congruent.

Both angles are what remains after subtracting the shared angle between \(b\) and \(c\) from \(90^\circ\), so they are congruent.

- First find the measure of the supplementary angle. - Bisect each angle. - Add the two half-angles that form the requested angle.

1. Since \(\alpha\) and \(\beta\) are supplementary, \(\beta = 180^{\circ} - 124^{\circ} = 56^{\circ}\). 2. Their half-angles are \(124^{\circ} \div 2 = 62^{\circ}\) and \(56^{\circ} \div 2 = 28^{\circ}\). 3. The angle between the bisectors is \(\delta = 62^{\circ} + 28^{\circ} = 90^{\circ}\).

The angle between the two angle bisectors is \(90^{\circ}\).

- What angle measure is required for two rays to form a straight line? - What does each bisector do to a \(60^\circ\) angle? - Include the angle supplementary to \(60^\circ\) between the two bisected angles.

1. The opposite vertical angle also measures \(60^\circ\). 2. Each angle bisector divides a \(60^\circ\) angle into two \(30^\circ\) angles. 3. An angle adjacent to a \(60^\circ\) angle measures \(180^\circ - 60^\circ = 120^\circ\). 4. The angle from \(w_1\) to \(w_2\) measures \(30^\circ + 120^\circ + 30^\circ = 180^\circ\). 5. Therefore, \(w_1\) and \(w_2\) are opposite rays and lie on the same line.

The angle between \(w_1\) and \(w_2\) is \(30^\circ + 120^\circ + 30^\circ = 180^\circ\), so the rays lie on one straight line.

- How can repeated copies of the marked angle make a larger angle? - Which targets are whole-number multiples of that angle? - For the smallest target, compare a nearby multiple with a straight angle.

1. Three copies of the marked \(35^\circ\) angle make \(3\cdot35^\circ=105^\circ\). 2. Four copies make \(4\cdot35^\circ=140^\circ\). 3. Five copies make \(5\cdot35^\circ=175^\circ\). Its supplement in a straight angle is \(180^\circ-175^\circ=5^\circ\).

a) Use three copies of the marked angle. b) Use four copies of the marked angle. c) Use the supplement of five copies of the marked angle.

- What is the total of all four angles around the point? - Replace the two angles that are equal to others in that total. - What do you know about the outer rays of neighboring angles that total \(180^\circ\)?

1. Angles around a point total \(360^\circ\), so \(\alpha+\beta+\gamma+\delta=360^\circ\). 2. The given equalities make this \(2(\alpha+\beta)=360^\circ\), so \(\alpha+\beta=180^\circ\). 3. Rays \(a\) and \(c\), the outside rays of these two neighboring angles, must be opposite rays. 4. Also, \(\beta+\gamma=\beta+\alpha=180^\circ\), so rays \(b\) and \(d\) must be opposite rays. The four rays form two intersecting straight lines.

\(2(\alpha+\beta)=360^\circ\), so \(\alpha+\beta=180^\circ\). Rays \(a\) and \(c\) are opposite, as are rays \(b\) and \(d\). Thus, they form two intersecting lines.

- Use the complementary relationship first. - Then use the supplementary relationship that shares \(\beta\). - Compare the first and last angle only after both are known.

1. Since \(\alpha\) and \(\beta\) are complementary, \(\beta=90^\circ-34^\circ=56^\circ\). 2. Since \(\beta\) and \(\gamma\) are supplementary, \(\gamma=180^\circ-56^\circ=124^\circ\). 3. The difference is \(124^\circ-34^\circ=90^\circ\), so \(\gamma\) is \(90^\circ\) greater than \(\alpha\).

\(\beta=56^\circ\), \(\gamma=124^\circ\), and \(\gamma=\alpha+90^\circ\).

- Which angles are opposite each other when two lines intersect? - Which groups of adjacent angles form a straight angle? - Start with the two given measures and use vertical angles to find their opposites. - Check that all six angles sum to \(360^\circ\).

1. Claim 1 is true because \(\alpha\), \(\beta\), and \(\gamma\) form a straight angle. 2. Claim 2 is false. Angle \(\delta\) is vertical to \(\alpha\), so \(\delta = 60^\circ\), while \(\beta = 65^\circ\). 3. Claim 3 is true because \(\gamma\) and \(\varphi\) are vertical angles. 4. Claim 4 is true. Vertical angles give \(\delta = \alpha = 60^\circ\) and \(\varepsilon = \beta = 65^\circ\), so \(60^\circ + 65^\circ + 60^\circ + 65^\circ = 250^\circ\). 5. Since \(\alpha + \beta + \gamma = 180^\circ\), \(\gamma = 180^\circ - 60^\circ - 65^\circ = 55^\circ\). The vertical angle pairs then give \(\delta = 60^\circ\), \(\varepsilon = 65^\circ\), and \(\varphi = 55^\circ\).

a) Claims 1, 3, and 4 are true. Claim 2 is false. b) \(\alpha = 60^\circ\), \(\beta = 65^\circ\), \(\gamma = 55^\circ\), \(\delta = 60^\circ\), \(\varepsilon = 65^\circ\), and \(\varphi = 55^\circ\).