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Add and subtract angle measures

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5314904
Rays \(g\) and \(h\) form a right angle. Ray \(s\) lies inside the right angle, and the angle between \(g\) and \(s\) measures \(35^\circ\). Find the angle \(\alpha\) between \(s\) and \(h\).
Figure for problem 531490

Hints

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

Solution

1. A right angle measures \(90^\circ\). 2. Ray \(s\) divides the right angle into angles of \(35^\circ\) and \(\alpha\). 3. Therefore, \(\alpha=90^\circ-35^\circ=55^\circ\).

Answer

\(\alpha=55^\circ\)
5330784
A laser pointer is aimed at an angle of \(40^\circ\) above a horizontal line. It is rotated another \(25^\circ\) counterclockwise. What angle \(\alpha\) does it now make with the horizontal line?
Figure for problem 533078

Hints

- Decide whether the rotation increases or decreases the angle. - Add the initial angle and the rotation.

Solution

1. The initial angle is \(40^\circ\). 2. The counterclockwise rotation increases the angle by \(25^\circ\). 3. Therefore, \(\alpha=40^\circ+25^\circ=65^\circ\).

Answer

\(\alpha=65^\circ\)
5330604
Angle \(\alpha\) measures \(36^\circ\). An adjacent angle \(\beta\) is three times as large as \(\alpha\). Find the total angle formed by the two outside rays.
Figure for problem 533060

Hints

- First find the angle that is three times \(36^\circ\). - Add the measures of adjacent angles.

Solution

1. Angle \(\beta\) measures \(3 \times 36^\circ=108^\circ\). 2. The adjacent angles combine to form a total angle of \(36^\circ+108^\circ=144^\circ\).

Answer

\(144^\circ\)
5330624
Angle \(\gamma\) measures \(125^\circ\). Adjacent angle \(\delta\) is one-fifth as large as \(\gamma\). Find the total angle formed by combining \(\gamma\) and \(\delta\).
Figure for problem 533062

Hints

- Find one-fifth of \(125^\circ\). - Add the measures of the adjacent angles.

Solution

1. Angle \(\delta\) measures \(125^\circ \div 5=25^\circ\). 2. The combined angle measures \(125^\circ+25^\circ=150^\circ\).

Answer

\(150^\circ\)
5330794
A crane boom makes an angle of \(155^\circ\), measured counterclockwise from a horizontal ray pointing right. The boom rotates \(60^\circ\) clockwise. What angle \(\beta\) does it then make with the horizontal ray?
Figure for problem 533079

Hints

- A clockwise rotation decreases the given counterclockwise angle. - Subtract the rotation from the initial angle.

Solution

1. A clockwise rotation decreases the counterclockwise angle. 2. Therefore, \(\beta=155^\circ-60^\circ=95^\circ\).

Answer

\(\beta=95^\circ\)
5331054
Ray \(c\) lies inside the angle formed by rays \(a\) and \(b\). The angle between \(a\) and \(c\) measures \(40^\circ\). The whole angle between \(a\) and \(b\) is \(25^\circ\) greater than the angle between \(a\) and \(c\). Find the angle between \(c\) and \(b\). Does the problem include any unnecessary information?
Figure for problem 533105

Hints

- Focus on how much greater the whole angle is than the first part. - Decide whether the exact \(40^\circ\) measure is needed to find the difference.

Solution

1. The whole angle measures \(40^\circ+25^\circ=65^\circ\). 2. The remaining angle measures \(65^\circ-40^\circ=25^\circ\). 3. The specific measure \(40^\circ\) is unnecessary because the problem directly states that the whole angle is \(25^\circ\) greater than the first part.

Answer

The angle between \(c\) and \(b\) is \(25^\circ\). The \(40^\circ\) measure is unnecessary.
5331064
Ray \(s_2\) divides the angle between rays \(s_1\) and \(s_3\) into two parts. The angle between \(s_1\) and \(s_2\) measures \(30^\circ\). The whole angle between \(s_1\) and \(s_3\) is twice as large as that angle. Find the angle between \(s_2\) and \(s_3\).
Figure for problem 533106

Hints

- Find twice \(30^\circ\). - Subtract the known part from the whole angle.

Solution

1. The whole angle measures \(2 \times 30^\circ=60^\circ\). 2. The remaining part measures \(60^\circ-30^\circ=30^\circ\).

Answer

\(30^\circ\)
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. Rays \(q\) and \(r\) lie inside it. The angle between \(p\) and \(r\) measures \(68^\circ\), and the angle between \(q\) and \(s\) measures \(55^\circ\). Find the angle \(\gamma\) between \(q\) and \(r\).
Figure for problem 533112

Hints

- First find the angle between \(p\) and \(q\). - Both ray positions can be measured from ray \(p\). - Subtract their angle measures.

Solution

1. The angle between \(p\) and \(q\) is \(90^\circ-55^\circ=35^\circ\). 2. Ray \(r\) is \(68^\circ\) from \(p\), while ray \(q\) is \(35^\circ\) from \(p\). 3. Therefore, \(\gamma=68^\circ-35^\circ=33^\circ\).

Answer

\(\gamma=33^\circ\)
5331134
A cake slice has a central angle of \(120^\circ\) and is divided by two straight cuts, \(a\) and \(b\). From the right edge to cut \(b\) is \(75^\circ\). From the left edge to cut \(a\) is \(85^\circ\). Find the angle \(\delta\) between cuts \(a\) and \(b\).
Figure for problem 533113

Hints

- Measure both cuts from the same edge. - Find the position of cut \(a\) from the right edge. - Subtract the two positions.

Solution

1. Measure all ray positions from the right edge, which is \(0^\circ\). The left edge is at \(120^\circ\). 2. Cut \(b\) is at \(75^\circ\). 3. Cut \(a\) is \(85^\circ\) from the left edge, so its position from the right edge is \(120^\circ-85^\circ=35^\circ\). 4. Therefore, \(\delta=75^\circ-35^\circ=40^\circ\).

Answer

\(\delta=40^\circ\)

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