To solve the problem, we need to use some properties of angles, particularly focusing on angle bisectors and relationships between angles in the triangle.

Given:

• $\angle QPR = 110^\circ$
• $\angle TQS = \frac{1}{5} \times \angle QTR$

Step 1: Identify Known Angles

First, let’s denote some of the angles with variables to make our work easier.

• Let $\angle PQR = x$.
• Since $QR$ is extended to $S$, $\angle PRS = 180^\circ - x$.
• Since $T$ is the intersection of the angle bisectors of $\angle PQR$ and $\angle PRS$, the properties of angle bisectors will be crucial.

Step 2: Use the Relationship Given in the Problem

We know that: $\angle TQS = \frac{1}{5} \times \angle QTR$ Since $T$ lies on the angle bisectors, $\angle TQS$ and $\angle QTR$ are related to the angles of the triangle. Specifically: $\angle TQS = \angle PQR/2 = \frac{x}{2}$ $\angle QTR = \angle PRS/2 = \frac{180^\circ - x}{2}$ Given $\angle TQS = \frac{1}{5} \times \angle QTR$: $\frac{x}{2} = \frac{1}{5} \times \frac{180^\circ - x}{2}$ Multiply both sides by 2 to eliminate the denominator: $x = \frac{1}{5} \times (180^\circ - x)$ Multiply through by 5: $5x = 180^\circ - x$ Add $x$ to both sides: $6x = 180^\circ$ Divide by 6: $x = 30^\circ$ So, $\angle PQR = 30^\circ$.

Step 3: Find $\angle TRS$

Now, $\angle PRS$ is: $\angle PRS = 180^\circ - \angle PQR = 180^\circ - 30^\circ = 150^\circ$

Since $T$ is the bisector: $\angle TRS = \frac{150^\circ}{2} = 75^\circ$

Thus, the value of $\angle TRS$ is $75^\circ$.

Would you like more details or have any questions?

Related Questions:

1. What is the sum of all angles in a triangle?
2. How do angle bisectors affect the angles in a triangle?
3. What is the relationship between the exterior angle and the interior opposite angles?
4. How does extending a side of a triangle affect the adjacent angle?
5. Can the sum of angles on a straight line be used to solve any triangle problem?

Tip:

Always double-check your work when dealing with angle bisectors, as small errors in identifying or calculating angles can lead to incorrect conclusions.