Question

a is the incenter of $\triangle pqr$. find each measure.
#1 $m\angle aru$
#2 $au$
(there is a triangle diagram with right angles at t, u, k, and angle 40° at r, length 20 at t to a.)

Explanation:

Step1: Find $\angle ARU$

The incenter is the intersection of angle bisectors, so $RA$ bisects $\angle PRQ$. Given $\angle PRK = 40^\circ$, $\angle PRQ = 40^\circ$, so $\angle ARU = \frac{1}{2}\angle PRQ = 20^\circ$.

Step2: Find length $AU$

The incenter is equidistant from all sides of the triangle. Given the distance from $A$ to side $PQ$ (i.e., $AT$) is 20, so $AU = AT = 20$.

Answer:

1. $m\angle ARU = 20^\circ$
2. $AU = 20$