In `DeltaPQR,/_Qgt/_R` PS is the bisector of `/_P` and `PT_|_PQ`. If `/_SPT=28^(@)` and `/_R=23^(@)` then the measure of `/_Q` is

Options

A. `89^(@)`
B. `82^(@)`
C. `79^(@)`
D. `74^(@)`

Answer

To solve the problem step by step, we will analyze the given triangle \( \Delta PQR \) and apply the properties of angles and triangles.

### Step 1: Write down the given information
- \( \angle SPT = 28^\circ \)
- \( \angle R = 23^\circ \)
- \( PS \) is the bisector of \( \angle P \)
- \( PT \perp PQ \) (meaning \( PT \) is perpendicular to \( PQ \))
- \( \angle Q > \angle R \)

### Step 2: Understand the implications of the angle bisector
Since \( PS \) bisects \( \angle P \), we can denote:
- \( \angle QPS = \angle SPR = x \)

Thus, we have:
\[ \angle P = 2x \]

### Step 3: Analyze triangle \( PTS \)
In triangle \( PTS \), we know:
- \( \angle SPT = 28^\circ \)
- \( PT \perp PQ \) implies \( \angle PTS = 90^\circ \)

Using the triangle angle sum property:
\[ \angle PTS + \angle SPT + \angle TSP = 180^\circ \]
Substituting the known values:
\[ 90^\circ + 28^\circ + \angle TSP = 180^\circ \]
This simplifies to:
\[ \angle TSP = 180^\circ - 118^\circ = 62^\circ \]

### Step 4: Find \( \angle PSR \)
Since \( PS \) bisects \( \angle P \), we have:
\[ \angle PSR = 180^\circ - \angle TSP = 180^\circ - 62^\circ = 118^\circ \]

### Step 5: Analyze triangle \( PSR \)
In triangle \( PSR \):
\[ \angle SPR + \angle PSR + \angle R = 180^\circ \]
Substituting the known values:
\[ \angle SPR + 118^\circ + 23^\circ = 180^\circ \]
This simplifies to:
\[ \angle SPR + 141^\circ = 180^\circ \]
Thus:
\[ \angle SPR = 180^\circ - 141^\circ = 39^\circ \]

### Step 6: Relate \( \angle Q \) to the angles found
Since \( PS \) bisects \( \angle P \):
\[ \angle QPS = \angle SPR = 39^\circ \]
Now, we can find \( \angle Q \):
\[ \angle Q = 180^\circ - \angle QPS - \angle PSR \]
Substituting the known values:
\[ \angle Q = 180^\circ - 39^\circ - 62^\circ \]
This simplifies to:
\[ \angle Q = 180^\circ - 101^\circ = 79^\circ \]

### Final Answer
Thus, the measure of \( \angle Q \) is \( 79^\circ \).

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