Consider ∆PQR.

Label the midpoint of PQ as point S, the midpoint of QR as point T, and the midpoint of RP as point U.

Next, draw PT, QU, and RS.

Which statements are true?

m∠P + m∠Q + m∠R = 180°

PT, QU, and RS intersect at the same point.

QU ≅ RS

The length of QU is half the length of RP.

The sum of the lengths of QU and RS is equal to the length of PT.

m∠Q = m∠R

Let's analyze each of the given statements one by one with respect to triangle \( \Delta PQR \), its midpoints, and the lines \( PT \), \( QU \), and \( RS \).

### 1. \( m\angle P + m\angle Q + m\angle R = 180^\circ \)
- **True**: This is the sum of the interior angles of any triangle. In any triangle, the sum of the interior angles is always \( 180^\circ \).

### 2. PT, QU, and RS intersect at the same point.
- **True**: The lines drawn from the vertices of a triangle to the midpoints of the opposite sides (called **medians**) always intersect at a single point, called the **centroid** of the triangle. So, \( PT \), \( QU \), and \( RS \) (medians of triangle \( \Delta PQR \)) intersect at the centroid.

### 3. QU ≅ RS
- **False**: In general, there is no rule that the medians of a triangle are congruent (equal in length). The lengths of the medians depend on the side lengths of the triangle, so \( QU \) and \( RS \) are not necessarily equal.

### 4. The length of QU is half the length of RP.
- **False**: The line segment \( QU \) is the median from vertex \( Q \), connecting \( Q \) to the midpoint of side \( RP \). The length of a median is not necessarily half the length of the side it connects to. There is no general relationship between the length of a median and the length of the side it bisects.

### 5. The sum of the lengths of QU and RS is equal to the length of PT.
- **False**: There's no specific rule that relates the sum of two medians to the length of the third median in this way. The medians of a triangle do not follow such a relationship.

### 6. \( m\angle Q = m\angle R \)
- **False**: This would only be true if the triangle were isosceles, with \( Q \) and \( R \) being the base angles. However, for a general triangle \( \Delta PQR \), there is no reason to assume that \( m\angle Q \) and \( m\angle R \) are equal unless explicitly given.

### Summary of True/False for Each Statement:
1. **True**
2. **True**
3. **False**
4. **False**
5. **False**
6. **False**

Would you like further details or have any questions about these statements?  

Here are some related questions for practice:
1. What is the centroid of a triangle and how is it located?
2. How can you calculate the length of a median in a triangle?
3. Can the medians of an equilateral triangle be equal? Why or why not?
4. How do the angles of a triangle relate to each other in a right triangle?
5. What is the difference between medians, altitudes, and angle bisectors in a triangle?

**Tip**: The centroid of a triangle divides each median in a 2:1 ratio, with the longer segment being closer to the vertex.

Consider ∆PQR. Label the midpoint of PQ as point S, the midpoint of QR as point T, and the midpoint of RP as point U. Next, draw PT, QU, and RS. Which statements are true?

m∠P + m∠Q + m∠R = 180°
PT, QU, and RS intersect at the same point.
QU ≅ RS
The length of QU is half the length of RP.
The sum of the lengths of QU and RS is equal to the length of PT.
m∠Q = m∠R

Explore the properties of triangle ∆PQR, focusing on medians and the centroid. Learn why the medians intersect at a single point, the sum of angles in a triangle, and other key concepts. Suitable for Grades 8-10.

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