Here are the step-by-step solutions for each problem on the worksheet.
1. Find $k$
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Reasoning: The large angle is split into two smaller angles. To find the total measure ($k$), you add the two parts together.
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Calculation: $30^\circ + 42^\circ = 72^\circ$.
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Result: $k = 72$
2. Find $r$
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Reasoning: The square symbol in the corner indicates a right angle, which always measures $90^\circ$. The angle $r$ and the $58^\circ$ angle combine to make this $90^\circ$ angle. We subtract the known part from the total to find the missing part.
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Calculation: $90 - 58 = 32$.
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Result: $r = 32$
3. Find $t$
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Reasoning: Angles on a straight line add up to $180^\circ$. The angle $t$ and the $143^\circ$ angle form a linear pair (a straight line). We subtract the known angle from 180.
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Calculation: $180 - 143 = 37$.
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Result: $t = 37$
4. Find $a$
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Reasoning: When two lines cross, opposite angles (vertical angles) are equal. The angle labeled $(5a)^\circ$ is directly across from the $140^\circ$ angle. Therefore, they are equal.
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Calculation:
$$5a = 140$$
Divide both sides by 5:
$$a = 140 / 5$$
$$a = 28$$
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Result: $a = 28$
5. Find $d$
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Reasoning: The square symbol indicates a right angle ($90^\circ$). The two angles inside it, $(5d)^\circ$ and $25^\circ$, add up to $90^\circ$.
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Calculation:
$$5d + 25 = 90$$
Subtract 25 from both sides:
$$5d = 65$$
Divide by 5:
$$d = 13$$
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Result: $d = 13$
6. Find $w$
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Reasoning: Angles on a straight line add up to $180^\circ$. The angle labeled $135^\circ$ and the angle labeled $(45 + 3w)^\circ$ sit next to each other on a straight line.
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Calculation:
$$(45 + 3w) + 135 = 180$$
Combine the numbers ($45 + 135 = 180$):
$$180 + 3w = 180$$
Subtract 180 from both sides:
$$3w = 0$$
Divide by 3:
$$w = 0$$
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Result: $w = 0$
Final Answer:
1. k = 72
2. r = 32
3. t = 37
4. a = 28
5. d = 13
6. w = 0
Parent Tip: Review the logic above to help your child master the concept of geometry angle relationships worksheet answer key.
