Apply properties of quadrilaterals inscribed in a circle to determine the mAD if mQU=55° and mAU=112° , while the m∠UAD=100° .

Question

(1 point)
mAD=
°

Answer 1

To determine $m\mathrm{\angle }ADU$ in an inscribed quadrilateral $QUAD$, we can use the property that opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) are supplementary. This means that the sum of the measures of opposite angles is equal to 180 degrees.

We know the following angles:

• $m\mathrm{\angle }QUA={55}^{\circ }$
• $m\mathrm{\angle }AUQ={112}^{\circ }$
• $m\mathrm{\angle }UAD={100}^{\circ }$

Since $QUAD$ is a cyclic quadrilateral, we can apply the following relationships:

1. $m\mathrm{\angle }UAD+m\mathrm{\angle }QUA={180}^{\circ }$
2. $m\mathrm{\angle }AUQ+m\mathrm{\angle }ADU={180}^{\circ }$

From the first equation: $$m\mathrm{\angle }UAD+m\mathrm{\angle }QUA={100}^{\circ }+{55}^{\circ }={155}^{\circ }$$ Since $m\mathrm{\angle }UAD+m\mathrm{\angle }QUA$ does not equal ${180}^{\circ }$, we confirm that these angles are not opposite but adjacent. We need to check the opposite angle $ADU$.

To find $m\mathrm{\angle }ADU$: Using the second equation: $$m\mathrm{\angle }AUQ+m\mathrm{\angle }ADU={180}^{\circ }$$ $${112}^{\circ }+m\mathrm{\angle }ADU={180}^{\circ }$$ Now, solving for $m\mathrm{\angle }ADU$: $$m\mathrm{\angle }ADU={180}^{\circ }-{112}^{\circ }={68}^{\circ }$$

Thus, the measure $m\mathrm{\angle }ADU$ is: $$\boxed{{\displaystyle {68}^{\circ }}}$$