Question

match the expression for the measure of the given angle
column a

1. $m\angle acd$
2. $m\angle jnk$
3. $m\angle uwv$
4. $m\angle swt$

column b
a. $m\angle hji=\frac{1}{2}(m\widehat{fg}+m\widehat{hi})$
b. $m\angle uwv=\frac{1}{2}(m\widehat{uv}+m\widehat{xy})$
c. $m\angle acd=\frac{1}{2}(m\widehat{ad}+m\widehat{be})$
d. $m\angle swt=\frac{1}{2}(m\widehat{st}+m\widehat{vu})$
e. $m\angle jnk=\frac{1}{2}(m\widehat{lm}+m\widehat{jk})$

Explanation:

Step1: Recall the theorem for the measure of an angle formed by two chords intersecting inside a circle

The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.

Step2: Analyze \(m\angle ACD\)

For \(m\angle ACD\), the intercepted arcs are \(\overset{\frown}{AD}\) and \(\overset{\frown}{BE}\). So \(m\angle ACD=\frac{1}{2}(m\overset{\frown}{AD}+m\overset{\frown}{BE})\), which matches option c.

Step3: Analyze \(m\angle JNK\)

For \(m\angle JNK\), the intercepted arcs are \(\overset{\frown}{LM}\) and \(\overset{\frown}{JK}\). So \(m\angle JNK = \frac{1}{2}(m\overset{\frown}{LM}+m\overset{\frown}{JK})\), which matches option e.

Step4: Analyze \(m\angle UWV\)

For \(m\angle UWV\), the intercepted arcs are \(\overset{\frown}{UV}\) and \(\overset{\frown}{XY}\). So \(m\angle UWV=\frac{1}{2}(m\overset{\frown}{UV}+m\overset{\frown}{XY})\), which matches option b.

Step5: Analyze \(m\angle SWT\)

For \(m\angle SWT\), the intercepted arcs are \(\overset{\frown}{ST}\) and \(\overset{\frown}{VU}\). So \(m\angle SWT=\frac{1}{2}(m\overset{\frown}{ST}+m\overset{\frown}{VU})\), which matches option d.

Answer:

1. c. \(m\angle ACD=\frac{1}{2}(m\overset{\frown}{AD}+m\overset{\frown}{BE})\)
2. e. \(m\angle JNK=\frac{1}{2}(m\overset{\frown}{LM}+m\overset{\frown}{JK})\)
3. b. \(m\angle UWV=\frac{1}{2}(m\overset{\frown}{UV}+m\overset{\frown}{XY})\)
4. d. \(m\angle SWT=\frac{1}{2}(m\overset{\frown}{ST}+m\overset{\frown}{VU})\)