In the figure below, $${X Z}$$ is a diameter of circle $${A}$$. What is the arc measure of minor arc $${Y Z}$$ in degrees? 

![](http://admin.getprephub.com/media/editor/c8_20230403191452925358.png)

<div id="id_cont_display" class="markdown-body editormd-html-preview"><textarea>In the figure below, $${X Z}$$ is a diameter of circle $${A}$$. What is the arc measure of minor arc $${Y Z}$$ in degrees? 

![](http://admin.getprephub.com/media/editor/c8_20230403191452925358.png)</textarea>In the figure below, <math><semantics><mrow><mrow><mi>X</mi><mi>Z</mi></mrow></mrow><annotation encoding="application/x-tex">{X Z}</annotation></semantics></math>XZ is a diameter of circle <math><semantics><mrow><mrow><mi>A</mi></mrow></mrow><annotation encoding="application/x-tex">{A}</annotation></semantics></math>A. What is the arc measure of minor arc <math><semantics><mrow><mrow><mi>Y</mi><mi>Z</mi></mrow></mrow><annotation encoding="application/x-tex">{Y Z}</annotation></semantics></math>YZ in degrees? 
<img src="http://admin.getprephub.com/media/editor/c8_20230403191452925358.png" alt="">
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>Solution

$${X Z}$$ is a diameter of circle $${A}$$.

$$m&lt;X A Z=180^{\circ}$$

Or, $$m&lt;X A Y+m&lt;Y A Z=180^{\circ}$$

Or, $$11 w+5 w+4=180$$

Or, $$16w=180-4$$

Or, $$16 w=176$$

Or,$$ w = 11$$

The arc measure of minor $${arc}{Y Z}$$ in degrees $$=5 w+4=5 \times 11+4=59^{\circ}$$

</textarea>Solution
<math><semantics><mrow><mrow><mi>X</mi><mi>Z</mi></mrow></mrow><annotation encoding="application/x-tex">{X Z}</annotation></semantics></math>XZ is a diameter of circle <math><semantics><mrow><mrow><mi>A</mi></mrow></mrow><annotation encoding="application/x-tex">{A}</annotation></semantics></math>A.
<math><semantics><mrow><mi>m</mi><mo>&lt;</mo><mi>X</mi><mi>A</mi><mi>Z</mi><mo>=</mo><mn>1</mn><mn>8</mn><msup><mn>0</mn><mrow><mo>∘</mo></mrow></msup></mrow><annotation encoding="application/x-tex">m&lt;X A Z=180^{\circ}</annotation></semantics></math>m&lt;XAZ=180​∘​​
Or, <math><semantics><mrow><mi>m</mi><mo>&lt;</mo><mi>X</mi><mi>A</mi><mi>Y</mi><mo>+</mo><mi>m</mi><mo>&lt;</mo><mi>Y</mi><mi>A</mi><mi>Z</mi><mo>=</mo><mn>1</mn><mn>8</mn><msup><mn>0</mn><mrow><mo>∘</mo></mrow></msup></mrow><annotation encoding="application/x-tex">m&lt;X A Y+m&lt;Y A Z=180^{\circ}</annotation></semantics></math>m&lt;XAY+m&lt;YAZ=180​∘​​
Or, <math><semantics><mrow><mn>1</mn><mn>1</mn><mi>w</mi><mo>+</mo><mn>5</mn><mi>w</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>8</mn><mn>0</mn></mrow><annotation encoding="application/x-tex">11 w+5 w+4=180</annotation></semantics></math>11w+5w+4=180
Or, <math><semantics><mrow><mn>1</mn><mn>6</mn><mi>w</mi><mo>=</mo><mn>1</mn><mn>8</mn><mn>0</mn><mo>−</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">16w=180-4</annotation></semantics></math>16w=180−4
Or, <math><semantics><mrow><mn>1</mn><mn>6</mn><mi>w</mi><mo>=</mo><mn>1</mn><mn>7</mn><mn>6</mn></mrow><annotation encoding="application/x-tex">16 w=176</annotation></semantics></math>16w=176
Or,<math><semantics><mrow><mi>w</mi><mo>=</mo><mn>1</mn><mn>1</mn></mrow><annotation encoding="application/x-tex"> w = 11</annotation></semantics></math>w=11
The arc measure of minor <math><semantics><mrow><mrow><mi>a</mi><mi>r</mi><mi>c</mi></mrow><mrow><mi>Y</mi><mi>Z</mi></mrow></mrow><annotation encoding="application/x-tex">{arc}{Y Z}</annotation></semantics></math>arcYZ in degrees <math><semantics><mrow><mo>=</mo><mn>5</mn><mi>w</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>5</mn><mo>×</mo><mn>1</mn><mn>1</mn><mo>+</mo><mn>4</mn><mo>=</mo><mn>5</mn><msup><mn>9</mn><mrow><mo>∘</mo></mrow></msup></mrow><annotation encoding="application/x-tex">=5 w+4=5 \times 11+4=59^{\circ}</annotation></semantics></math>=5w+4=5×11+4=59​∘​​
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