Question

In triangle VWX, VW = 4.5 inches, WX = 5.9 inches, Measure of angle W = 28 degrees, and Measure of angle X = 47 degrees. If Triangle P Q R is congruent to triangle W V X, which statement is true?
QR = 4.5 cm
QR = 5.9 cm
Measure of angle R = 28 degrees
Measure of angle R = 47 degrees

Answer 1

Given that triangle \( PQR \) is congruent to triangle \( VWX \), we can use the properties of congruent triangles to determine the corresponding sides and angles.

In triangle \( VWX \):
- \( VW = 4.5 \) inches
- \( WX = 5.9 \) inches
- Measure of angle \( W = 28^\circ \)
- Measure of angle \( X = 47^\circ \)

We can find the measure of angle \( V \) in triangle \( VWX \) using the angle sum property of triangles:

\[
\text{Measure of angle } V = 180^\circ - \text{Measure of angle } W - \text{Measure of angle } X
\]

Substituting the known values:

\[
\text{Measure of angle } V = 180^\circ - 28^\circ - 47^\circ = 105^\circ
\]

Now, since \( \triangle PQR \) is congruent to \( \triangle VWX \), the corresponding angles and sides are equal, which gives us:

- \( P \) corresponds to \( V \)
- \( Q \) corresponds to \( W \)
- \( R \) corresponds to \( X \)

Therefore:

1. \( QR \) corresponds to \( VW \) and \( QR = VW = 4.5 \) inches.
2. \( PR \) corresponds to \( WX \) and \( PR = WX = 5.9 \) inches.
3. The measure of angle \( R \) corresponds to angle \( X \) and \( \text{Measure of angle } R = \text{Measure of angle } X = 47^\circ \).
4. The measure of angle \( Q \) corresponds to angle \( W \) and \( \text{Measure of angle } Q = \text{Measure of angle } W = 28^\circ \).

Given the choices:

- \( QR = 4.5 \) cm (True, since \( QR = VW = 4.5 \) inches)
- \( QR = 5.9 \) cm (False)
- Measure of angle \( R = 28 \) degrees (False, \( R = 47^\circ \))
- Measure of angle \( R = 47 \) degrees (True)

Thus, the true statement concerning triangle \( PQR \) is:
**QR = 4.5 cm**.