Question

key postulates: protractor postulate. angle addition postulate. guided practice: use the figure shown for exercises 11–14. 11. what is ( mangle ptr )? 12. what is ( mangle ptq )? independent practice: 13. what is ( mangle qts )? 14. understand luis said that ( mangle qtr = 80^circ ). explain luis’s error.

Explanation:

To solve these angle - measurement problems, we will use the protractor postulate and the angle addition postulate. The protractor postulate allows us to measure angles using a protractor, and the angle addition postulate states that if a point \(B\) lies in the interior of \(\angle AOC\), then \(m\angle AOB + m\angle BOC=m\angle AOC\).

Exercise 11: Find \(m\angle PTR\)

Step 1: Identify the angle on the protractor

Looking at the protractor, the ray \(TP\) is along the \(0^{\circ}\) mark (or the initial side), and the ray \(TR\) is at \(60^{\circ}\) (by reading the protractor scale).
So, by the protractor postulate, \(m\angle PTR = 60^{\circ}\)

Exercise 12: Find \(m\angle PTQ\)

Step 1: Identify the angle on the protractor

The ray \(TP\) is along the \(0^{\circ}\) mark, and the ray \(TQ\) is at \(30^{\circ}\) (by reading the protractor scale).
So, by the protractor postulate, \(m\angle PTQ=30^{\circ}\)

Exercise 13: Find \(m\angle QTS\)

Step 1: Use the angle addition postulate

We know that \(m\angle PTS\) (a straight angle) is \(180^{\circ}\), and \(m\angle PTQ = 30^{\circ}\)
Let \(m\angle QTS=x\). By the angle addition postulate, \(m\angle PTQ + m\angle QTS=m\angle PTS\)
Substitute the known values: \(30^{\circ}+x = 180^{\circ}\)

Step 2: Solve for \(x\)

\(x=m\angle QTS=180^{\circ}- 30^{\circ}=150^{\circ}\)

Exercise 14: Explain Luis's error

Step 1: Analyze the angle \(\angle QTR\)

First, we know that \(m\angle PTR = 60^{\circ}\) and \(m\angle PTQ = 30^{\circ}\)
By the angle addition postulate, \(m\angle QTR=m\angle PTR - m\angle PTQ\) (since \(TQ\) is in the interior of \(\angle PTR\))

Step 2: Calculate the correct measure

\(m\angle QTR = 60^{\circ}-30^{\circ}=30^{\circ}\)
Luis probably used the wrong scale on the protractor (the outer scale instead of the inner scale or vice - versa) or misapplied the angle addition postulate. He might have thought that the measure of \(\angle QTR\) was \(80^{\circ}\) by incorrectly reading the protractor or by miscalculating the difference or sum of the related angles.

Answer:

s:

1. \(m\angle PTR=\boldsymbol{60^{\circ}}\)
2. \(m\angle PTQ=\boldsymbol{30^{\circ}}\)
3. \(m\angle QTS=\boldsymbol{150^{\circ}}\)
4. Luis likely misread the protractor scale or misapplied the angle addition postulate. The correct measure of \(\angle QTR\) is \(30^{\circ}\) (calculated as \(m\angle PTR - m\angle PTQ=60^{\circ}-30^{\circ}\)), not \(80^{\circ}\).