To find the missing angle at vertex $Z$ (angle $\angle YZX$), we need to use the properties of circles and arcs.
Step 1: Understand the relationship between arcs and the circle.
The total measure of the arcs around a circle is always $360^\circ$. The circle in the problem is divided into three arcs by points $X$, $Y$, and $Z$:
1. Arc $YZ$ (given as $110^\circ$)
2. Arc $XY$ (unknown)
3. Arc $ZX$ (given as $146^\circ$)
Step 2: Calculate the measure of the unknown arc $XY$.
We can find the measure of arc $XY$ by subtracting the known arcs from the total $360^\circ$.
$$ \text{Arc } XY = 360^\circ - (\text{Arc } YZ + \text{Arc } ZX) $$
$$ \text{Arc } XY = 360^\circ - (110^\circ + 146^\circ) $$
$$ \text{Arc } XY = 360^\circ - 256^\circ $$
$$ \text{Arc } XY = 104^\circ $$
So, the arc connecting points $X$ and $Y$ measures $104^\circ$.
Step 3: Calculate the inscribed angle.
The angle marked with a question mark is an
inscribed angle because its vertex ($Z$) is on the circle and its sides are chords. This angle subtends (opens up to) arc $XY$.
The rule for inscribed angles is:
$$ \text{Inscribed Angle} = \frac{1}{2} \times \text{Measure of Intercepted Arc} $$
Therefore:
$$ \text{Angle } Z = \frac{1}{2} \times \text{Arc } XY $$
$$ \text{Angle } Z = \frac{1}{2} \times 104^\circ $$
$$ \text{Angle } Z = 52^\circ $$
Final Answer:
A) 52°
Parent Tip: Review the logic above to help your child master the concept of inscribed angle worksheet.
