Angles of Triangles Quick Check

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An illustration shows triangle uppercase H uppercase I uppercase J. Uppercase H is the left vertex, uppercase I is the top vertex, and uppercase J is the right vertex. A line extending outward from uppercase H forms an exterior angle of 38 degrees.

What is the sum of the measures of the interior angles at vertex I and vertex J ?

(1 point)
Responses

128°
128 degrees

38°
38 degrees

142°
142 degrees

52°
52 degrees
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Answer 1

In a triangle, the sum of the interior angles is always 180 degrees. The exterior angle at vertex H measures 38 degrees, and it is equal to the sum of the interior angles at the opposite vertices I and J.

Using the exterior angle theorem:

\[ \text{Exterior angle} = \text{Interior angle at I} + \text{Interior angle at J} \]

Given that the exterior angle is 38 degrees:

\[ 38° = \text{Interior angle at I} + \text{Interior angle at J} \]

Thus, the sum of the measures of the interior angles at vertex I and vertex J is:

\[ \text{Interior angle at I} + \text{Interior angle at J} = 38° \]

Since the total sum of the angles in triangle HIJ is 180°, we can find the sum of the interior angles at vertices I and J:

\[ \text{Interior angle at H} + \text{Interior angle at I} + \text{Interior angle at J} = 180° \]

Now let's denote the sum of the angles at I and J as x:

\[ \text{Interior angle at H} + x = 180° \]

From the exterior angle theorem:

\[ x + 38° = 180° \]

So:

\[ x = 180° - 38° = 142° \]

Therefore, the sum of the measures of the interior angles at vertex I and vertex J is 142°.