Here are the step-by-step solutions for the three problems shown in your quiz.
Problem 1
Goal: Find the measure of angle $K$ ($m\angle K$).
1.
Identify the parts: You have an inscribed angle, $\angle JKL$, which opens up to intercept arc $JL$.
2.
Find the known value: The problem gives the measure of the intercepted arc $JL$ as $40^\circ$.
3.
Apply the rule: The Inscribed Angle Theorem states that the measure of an inscribed angle is exactly half the measure of its intercepted arc.
* Formula: $m\angle K = \frac{1}{2} \times m(\text{arc } JL)$
4.
Calculate:
* $m\angle K = \frac{1}{2} \times 40^\circ$
* $m\angle K = 20^\circ$
Correct Option: A (20 degrees)
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Problem 2
Goal: Find the measure of arc $DF$ ($m\widehat{DF}$).
1.
Identify the parts: You have an inscribed angle, $\angle DEF$, with a measure of $72^\circ$. This angle intercepts arc $DF$.
2.
Apply the rule: Since the angle is half the arc, the arc must be twice the angle.
* Formula: $m(\text{arc } DF) = 2 \times m\angle E$
3.
Calculate:
* $m(\text{arc } DF) = 2 \times 72^\circ$
* $m(\text{arc } DF) = 144^\circ$
Correct Option: B (144 degrees)
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Problem 3
Goal: Find the measure of arc $ST$ ($m\widehat{ST}$).
1.
Identify the parts: You have an inscribed angle, $\angle TSU$, measuring $61^\circ$. The side $SU$ passes through the center point $P$, making it a diameter.
2.
Use the diameter property: Because $SU$ is a diameter, it cuts the circle in half. This means the entire arc from $S$ to $U$ (arc $STU$) measures $180^\circ$.
3.
Find the other arc first: The angle $\angle TSU$ intercepts arc $TU$.
* $m(\text{arc } TU) = 2 \times m\angle TSU$
* $m(\text{arc } TU) = 2 \times 61^\circ = 122^\circ$
4.
Calculate arc $ST$: Arc $ST$ and arc $TU$ make up the semi-circle ($180^\circ$). So, you subtract arc $TU$ from $180^\circ$.
* $m(\text{arc } ST) = 180^\circ - 122^\circ$
* $m(\text{arc } ST) = 58^\circ$
Correct Option: A (58 degrees)
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Final Answer:
1. A
2. B
3. A
Parent Tip: Review the logic above to help your child master the concept of inscribed polygons worksheet.
