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Add and subtract angle measures

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5314904
The diagram shows a right angle divided by ray \(s\). Find the angle \(\alpha\).
Figure for problem 531490

Hints

- Read the known part of the right angle from the diagram. - The two smaller angles add to \(90^\circ\). - Subtract the known part from the whole right angle.

Solution

1. A right angle measures \(90^\circ\). 2. The diagram shows one part measuring \(35^\circ\), so the two parts satisfy \(35^\circ+\alpha=90^\circ\). 3. Therefore, \(\alpha=90^\circ-35^\circ=55^\circ\).

Answer

\(\alpha=55^\circ\)
5330784
The diagram shows a laser pointer rotated counterclockwise from its first position. What angle \(\alpha\) does the pointer now make with the horizontal ray?
Figure for problem 533078

Hints

- Read the initial angle and the added rotation from the diagram. - A counterclockwise rotation here increases the angle from the horizontal ray. - Combine the two adjacent angle measures.

Solution

1. The diagram shows an initial angle of \(40^\circ\) and an additional counterclockwise rotation of \(25^\circ\). 2. The rotation increases the angle, so \(\alpha=40^\circ+25^\circ=65^\circ\).

Answer

\(\alpha=65^\circ\)
5543504
The two outer rays form a right angle. Find the missing angle shown in the diagram.
Figure for problem 554350

Hints

- Use the total measure of a right angle. - The two adjacent parts together make the whole angle. - Think about which operation recovers a missing part from a known whole and known part.

Solution

1. A right angle measures \(90^\circ\). 2. The known part measures \(32^\circ\). 3. The missing part is \(90^\circ - 32^\circ = 58^\circ\).

Answer

\(58^\circ\)
5330604
The diagram shows adjacent angles \(\alpha\) and \(\beta\). Angle \(\beta\) is three times as large as \(\alpha\). Find the total angle formed by the two outside rays.
Figure for problem 533060

Hints

- Read the measure of \(\alpha\) from the diagram. - Use the stated multiplicative relationship to find \(\beta\). - Add the adjacent angle measures to find the outside angle.

Solution

1. The diagram shows \(\alpha=36^\circ\). 2. Angle \(\beta\) measures \(3 \times 36^\circ=108^\circ\). 3. Because the angles are adjacent, the total angle is \(36^\circ+108^\circ=144^\circ\).

Answer

\(144^\circ\)
5330624
The diagram shows adjacent angles \(\gamma\) and \(\delta\). Angle \(\delta\) is one-fifth as large as \(\gamma\). Find the total angle formed by combining them.
Figure for problem 533062

Hints

- Read the measure of \(\gamma\) from the diagram. - Find one-fifth of that angle measure. - Add the two adjacent angle measures.

Solution

1. The diagram shows \(\gamma=125^\circ\). 2. One-fifth of \(125^\circ\) is \(25^\circ\), so \(\delta=25^\circ\). 3. The combined angle measures \(125^\circ+25^\circ=150^\circ\).

Answer

\(150^\circ\)
5330794
The diagram shows a crane boom rotating \(60^\circ\) clockwise from its initial position. What angle \(\beta\) does it then make with the horizontal ray?
Figure for problem 533079

Hints

- Read the initial angle from the diagram. - A clockwise rotation decreases the counterclockwise angle shown. - Subtract the rotation from the initial angle.

Solution

1. The diagram shows the initial angle as \(155^\circ\). 2. A \(60^\circ\) clockwise rotation decreases that counterclockwise angle. 3. Therefore, \(\beta=155^\circ-60^\circ=95^\circ\).

Answer

\(\beta=95^\circ\)
5331054
Use the diagram to find the missing angle between rays \(c\) and \(b\).
Figure for problem 533105

Hints

- Identify the measure of the whole angle and the measure of the known part in the diagram. - The two adjacent smaller angles combine to make the whole angle. - Check that your missing measure combines with the known part to match the whole.

Solution

1. The whole angle from ray \(a\) to ray \(b\) is \(65^\circ\). 2. The part from ray \(a\) to ray \(c\) is \(40^\circ\). 3. By angle addition, the missing angle is \(65^\circ-40^\circ=25^\circ\).

Answer

\(25^\circ\)
5331064
The diagram shows an angle divided by ray \(s_2\). The whole angle from \(s_1\) to \(s_3\) is twice as large as the marked known part. Find the angle between \(s_2\) and \(s_3\).
Figure for problem 533106

Hints

- Read the known part from the diagram. - Find twice that measure to get the whole angle. - Subtract the known part from the whole.

Solution

1. The diagram shows the known part as \(30^\circ\). 2. The whole angle is \(2 \times 30^\circ=60^\circ\). 3. The remaining part is \(60^\circ-30^\circ=30^\circ\).

Answer

\(30^\circ\)
5543514
The two outer rays form a straight angle. Find the missing angle shown in the diagram.
Figure for problem 554351

Hints

- Recall the total degree measure of a straight angle. - The two adjacent angles fill that whole straight angle. - Subtract the known part from the whole.

Solution

1. A straight angle measures \(180^\circ\). 2. One part measures \(147^\circ\). 3. The missing part is \(180^\circ - 147^\circ = 33^\circ\).

Answer

\(33^\circ\)
5543524
The whole angle from the first ray to the last ray measures \(120^\circ\). The diagram shows three adjacent parts. Find the missing part.
Figure for problem 554352

Hints

- Read the two known adjacent-angle measures from the diagram. - Combine the known parts before comparing them with the whole angle. - The three parts must add to the stated whole-angle measure.

Solution

1. The two known parts measure \(25^\circ\) and \(40^\circ\), for a total of \(65^\circ\). 2. The missing part is \(120^\circ - 65^\circ = 55^\circ\).

Answer

\(55^\circ\)
5330894
Four rays \(a\), \(b\), \(c\), and \(d\) share endpoint \(O\). The matching marks show that the three angles between neighboring rays are congruent. Which additional pair of angles must also be congruent? A) The angle between \(a\) and \(b\), and the angle between \(a\) and \(d\) B) The angle between \(a\) and \(c\), and the angle between \(b\) and \(d\) C) The angle between \(b\) and \(c\), and the angle between \(a\) and \(d\)
Figure for problem 533089

Hints

- Break each larger angle into neighboring smaller angles. - Compare how many congruent parts each angle contains.

Solution

1. The three smaller adjacent angles are congruent. 2. The angle between \(a\) and \(c\) is the sum of two adjacent congruent angles. 3. The angle between \(b\) and \(d\) is also the sum of two adjacent congruent angles. 4. Therefore, those two larger angles are congruent, so choice B is correct.

Answer

B) The angle between \(a\) and \(c\) is congruent to the angle between \(b\) and \(d\).
5331124
Rays \(p\) and \(s\) form a right angle, with rays \(q\) and \(r\) inside it as shown. Find the angle \(\gamma\) between \(q\) and \(r\).
Figure for problem 533112

Hints

- Read the two marked measures from the diagram. - Use the right angle to find the position of ray \(q\) from ray \(p\). - Compare the positions of rays \(q\) and \(r\).

Solution

1. The diagram shows \(55^\circ\) between \(q\) and \(s\), so the angle from \(p\) to \(q\) is \(90^\circ-55^\circ=35^\circ\). 2. The diagram shows \(68^\circ\) from \(p\) to \(r\). 3. Therefore, \(\gamma=68^\circ-35^\circ=33^\circ\).

Answer

\(\gamma=33^\circ\)
5331134
The diagram shows a cake slice with a central angle of \(120^\circ\), divided by two straight cuts \(a\) and \(b\). Find the angle \(\delta\) between the cuts.
Figure for problem 533113

Hints

- Read the two edge-to-cut measures from the diagram. - Express both cut positions from the same edge. - Subtract the two positions to find the angle between the cuts.

Solution

1. Use the right edge as \(0^\circ\). The diagram shows cut \(b\) at \(75^\circ\). 2. The diagram shows that cut \(a\) is \(85^\circ\) from the left edge. Since the whole slice is \(120^\circ\), cut \(a\) is at \(120^\circ-85^\circ=35^\circ\) from the right edge. 3. Therefore, \(\delta=75^\circ-35^\circ=40^\circ\).

Answer

\(\delta=40^\circ\)
5543534
The outer rays form a right angle. The larger of the two adjacent angles is twice the smaller angle. Find both angle measures.
Figure for problem 554353

Hints

- Represent the smaller angle as one equal part. - How many of those equal parts would the larger angle contain? - The parts together must fill one right angle.

Solution

1. Think of the smaller angle as one equal part. Then the larger angle is two equal parts. 2. Together the right angle contains three equal parts, so each part is \(90^\circ \div 3 = 30^\circ\). 3. The smaller angle is \(30^\circ\), and the larger angle is \(2 \times 30^\circ = 60^\circ\).

Answer

Smaller angle: \(30^\circ\) Larger angle: \(60^\circ\)
5543544
Ava says the missing angle in the diagram is \(22^\circ\) because she subtracted the known angle from \(90^\circ\). Explain her error and find the correct missing angle.
Figure for problem 554354

Hints

- Identify the type of angle formed by the two outer rays. - Decide whether its total measure is the measure of a right angle or a straight angle. - The known and missing adjacent parts must add to that whole.

Solution

1. The outer rays form a straight angle, not a right angle, so the whole is \(180^\circ\). 2. The known part is \(68^\circ\). 3. The missing angle is \(180^\circ - 68^\circ = 112^\circ\).

Answer

Ava used \(90^\circ\) for the whole angle, but the whole is a straight angle of \(180^\circ\). The missing angle is \(112^\circ\).

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