Question

△XYZ∼△RST . m∠X=50° , m∠S=79° . What is the measure of angle T ?

Answer 1

Given that triangles \( \triangle XYZ \) and \( \triangle RST \) are similar, the corresponding angles of the triangles are equal.

We know the following angles:

- \( m\angle X = 50^\circ \)
- \( m\angle S = 79^\circ \)

To find the measure of angle \( T \), we first need to calculate the measure of the remaining angle in triangle \( XYZ \):

The sum of angles in a triangle is \( 180^\circ \). Thus, we can find \( m\angle Y \) in triangle \( XYZ \):

\[
m\angle Y = 180^\circ - m\angle X - m\angle Z
\]

Since the triangles are similar, we can identify the corresponding angles. Let \( m\angle Y \) correspond with \( m\angle R \) and \( m\angle T \) with \( m\angle Z \).

First, we calculate \( m\angle Z \):

1. Calculate \( m\angle Y \):
\[
m\angle Y = 180^\circ - 50^\circ - m\angle Z
\]

Next, in triangle \( RST \):
\[
m\angle R + m\angle S + m\angle T = 180^\circ
\]

Substituting \( m\angle S = 79^\circ \):
\[
m\angle R + 79^\circ + m\angle T = 180^\circ
\]

Now, we can also express \( m\angle R \):
\[
m\angle R = 180^\circ - 79^\circ - m\angle T
\]

Thus, we have:
\[
m\angle Y + m\angle R + m\angle S + m\angle T = 180^\circ
\]
This means we need:

To get both angles for similar triangles, use the calculated \( m\angle Y \):

Given that \( m\angle Y \) and \( m\angle R \) are corresponding angles, we need:

Continuing with \( m\angle X + m\angle S + m\angle T = 180^\circ \).

Let:
\[
m\angle T = 180^\circ - m\angle X - m\angle S
\]
Now we can plug in values:
\[
m\angle T = 180^\circ - 50^\circ - 79^\circ
\]

Calculating,
\[
m\angle T = 180^\circ - 129^\circ = 51^\circ
\]

Thus, the measure of angle \( T \) is:
\[
\boxed{51^\circ}
\]