Question

proving opposite angles of an inscribed quadrilateral are supplementary
an inscribed quadrilateral is a quadrilateral drawn inside a circle such that each vertex of the quadrilateral lies on the circle. the sides of the quadrilateral are chords of the circle.
quadrilateral quad is inscribed in circle o.
angles of an inscribed quadrilateral

1. quadrilateral quad is inscribed in circle o 1. given
2. ( mwidehat{uqd}=2cdot mangle a ) and ( mwidehat{uad}=2cdot mangle q ) 2. inscribed angle theorem
3. ( mwidehat{uqd}+mwidehat{uad}=360^{circ} ) 3. a circle is ( 360^{circ} ).
4. ( 2cdot mangle a + 2cdot mangle q=square^{circ} ) 4. substitution property
5. ( 2(mangle a + mangle q)=360^{circ} ) 5. (square) property
6. ( mangle a + mangle q=square^{circ} ) 6. division
7. ( angle q ) and ( angle a ) are supplementary angles. 7. definition of supplementary angles

because the sum of the measures of a quadrilateral is ( 360^{circ} ), ( angle u ) and ( angle d )

Explanation:

Step1: Substitute values

From step 3 \(m\overset{\frown}{UQD}+m\overset{\frown}{UAD} = 360^{\circ}\), and step 2 \(m\overset{\frown}{UQD}=2\cdot m\angle A\) and \(m\overset{\frown}{UAD}=2\cdot m\angle Q\). Substituting into step 4 gives \(2\cdot m\angle A+2\cdot m\angle Q = 360^{\circ}\).

Step2: Factor out

For \(2\cdot m\angle A+2\cdot m\angle Q\), using the distributive property \(a(b + c)=ab+ac\) (here \(a = 2\), \(b=m\angle A\), \(c = m\angle Q\)), we get \(2(m\angle A+m\angle Q)=360^{\circ}\).

Step3: Solve for \(m\angle A + m\angle Q\)

Dividing both sides of \(2(m\angle A+m\angle Q)=360^{\circ}\) by 2 (using the division property of equality: if \(a=b\), then \(\frac{a}{c}=\frac{b}{c}\), \(c
eq0\)), we have \(m\angle A+m\angle Q=\frac{360^{\circ}}{2}=180^{\circ}\).

Answer:

1. \(360\)
2. Distributive
3. \(180\)