Add and Subtract Angle Measures - 4th Grade Math

1. In the figure shown, angle $PQS$ has been divided into two parts by ray $QR$. The measure of angle $PQR$ is $67°$, and the measure of the entire angle $PQS$ is $115°$. What is the measure of angle $RQS$?
   Answer: $48°$ Since ray QR divides angle PQS into two parts (angles PQR and RQS), we have: angle PQR + angle RQS = angle PQS. Therefore: 67° + angle RQS = 115°, so angle RQS = 115° - 67° = 48°. Choice A adds all given measures, choice C repeats the given angle PQR, and choice D repeats the whole angle PQS.
2. A surveyor measures a large angle at a construction site. The angle measures $137°$ and is composed of four smaller, non-overlapping angles. Three of these angles measure $28°$, $35°$, and $41°$. What is the measure of the fourth angle?
   Answer: $33°$ The four angles must add to 137°: 28° + 35° + 41° + fourth angle = 137°. First, 28° + 35° + 41° = 104°. So the fourth angle = 137° - 104° = 33°. Choice B gives the sum of the three known angles (104°), choice C adds all given measurements (137° + 28° + 35° + 41°), and choice D gives an incorrect calculation.
3. Look at the diagram. Ray $BC$ divides angle $ABD$ into two smaller angles. If angle $ABC$ measures $35°$ and angle $CBD$ measures $42°$, what is the measure of angle $ABD$?
   Answer: $77°$ Since ray BC divides angle ABD into two non-overlapping parts (angles ABC and CBD), the measure of the whole angle ABD equals the sum of its parts: 35° + 42° = 77°. Choice B represents the difference (42° - 35°), choice C only considers one part, and choice D represents 2 × 35°.
4. Angle $XYZ$ is divided by two rays, $YA$ and $YB$, creating three separate angles. If angle $XYA = 29°$, angle $AYB = 41°$, and angle $BYZ = 35°$, what is the measure of angle $XYZ$?
   Answer: $105°$ Since the two rays split angle XYZ into three smaller angles, the total measure is the sum of all three: $29° + 41° + 35° = 105°$. Choice A comes from adding only two of the three smaller angles. Choice C comes from a small addition slip when combining the three angle measures. Choice D also comes from adding only two of the three angles, using a different pair. Choice B correctly adds all three angles together for the total measure.
5. A clock face shows that the hour hand and minute hand form different angles with the 12 o'clock position. The hour hand makes a $90°$ angle with 12 o'clock, and the minute hand makes a $54°$ angle with 12 o'clock. Both hands are on the same side of 12 o'clock. What is the angle between the two hands?
   Answer: $36°$ Since both hands are on the same side of 12 o'clock, the angle between them is the difference between their individual angles from 12 o'clock: 90° - 54° = 36°. Choice A adds the angles (90° + 54°), while choices C and D repeat one of the given angles.
6. Tommy draws an angle that measures $156°$. He then draws a ray inside this angle that creates two smaller angles. One of the smaller angles measures $89°$. If Tommy wants the other smaller angle to measure $75°$, by how many degrees is he off from his target?
   Answer: $8°\text{ too small}$ Choice A is correct because the two smaller angles must add up to 156°, so the second angle measures 156° minus 89°, which is 67°, and that is 8° less than the 75° target. Choice B is incorrect because 67° is smaller than 75°, not larger. Choice C is incorrect because the difference between 67° and 75° is 8°, not 2°. Choice D is incorrect for the same reason and also states the wrong direction.
7. Look at the angle diagram. A straight angle measuring $180°$ has been divided into three non-overlapping angles. Two of these angles measure $52°$ and $73°$. What equation could be used to find $x$, the measure of the third angle?
   Answer: $52 + 73 + x = 180$ Since the three angles are non-overlapping parts of a straight angle (180°), their measures must add up to 180°. The correct equation is 52° + 73° + x = 180°. Choice B subtracts x instead of adding, choice C uses 90° instead of 180°, and choice D incorrectly isolates x on the left side.
8. Look at the diagram. Angle $ABC$ measures $84°$. Ray $BD$ is drawn so that angle $ABD$ measures $49°$. Ray $BD$ is inside angle $ABC$. What is the measure of angle $DBC$?
   Answer: $35°$ Since ray BD is inside angle ABC, it divides the angle into two parts: angle ABD and angle DBC. Therefore: angle ABD + angle DBC = angle ABC, so 49° + angle DBC = 84°. This gives angle DBC = 84° - 49° = 35°. Choice A adds the angles (84° + 49°), while choices C and D repeat given angle measures.
9. In the figure, ray $FH$ divides angle $EFG$ into two parts. The measure of angle $EFH$ is three times the measure of angle $HFG$. If angle $HFG$ measures $32°$, what is the measure of angle $EFG$?
   Answer: $128°$ Angle EFH = 3 × 32° = 96°. Since angle EFG is made up of angles EFH and HFG, the total is 96° + 32° = 128°. Choice A gives only angle EFH (96°), choice C gives 2 × 32°, and choice D repeats the given angle HFG.

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