Question

statements

1. quadrilateral abcd is inscribed in circle z.
2. mabc + madc = 360°

mdab + mdcb = 360°

1. m∠a = 1/2mdcb

m∠b = 1/2madc
m∠c = 1/2mbad
m∠d = 1/2mcba

1. m∠a + m∠c = 1/2mdcb+ 1/2mbad

reasons
given
arc addition postulate

central angle theorem
division property of equality
inscribed angle theorem
substitution property of equality

Explanation:

Step1: Identify given information

Quadrilateral \(ABCD\) is inscribed in circle \(Z\) (Given).

Step2: Recall arc - related property

By arc addition postulate, \(m\overset{\frown}{ABC}+m\overset{\frown}{ADC} = 360^{\circ}\) and \(m\overset{\frown}{DAB}+m\overset{\frown}{DCB}=360^{\circ}\).

Step3: Apply inscribed - angle theorem

The inscribed - angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. So \(m\angle A=\frac{1}{2}m\overset{\frown}{DCB}\), \(m\angle B=\frac{1}{2}m\overset{\frown}{ADC}\), \(m\angle C=\frac{1}{2}m\overset{\frown}{BAD}\), \(m\angle D=\frac{1}{2}m\overset{\frown}{CBA}\).

Step4: Use substitution property

Substitute the expressions for \(m\angle A\) and \(m\angle C\) into \(m\angle A + m\angle C\). We get \(m\angle A + m\angle C=\frac{1}{2}m\overset{\frown}{DCB}+\frac{1}{2}m\overset{\frown}{BAD}\).

The reason for step 4 is the substitution property of equality.

Answer:

Substitution property of equality