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Question

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

38°
38 degrees

52°
52 degrees

128°
128 degrees

142°
142 degrees
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Answer 1

To find the sum of the measures of the interior angles at vertices I and J of triangle HIJ, we can use the property that the sum of the interior angles of any triangle is always 180 degrees.

Given that the exterior angle at vertex H is 38 degrees, we can determine the corresponding interior angle at vertex H. The relationship between an exterior angle and its adjacent interior angle is:

\[ \text{Exterior Angle} = \text{Interior Angle} + \text{Adjacent Interior Angle} \]

Thus, the interior angle at vertex H is:

\[ \text{Interior Angle at H} = 180° - 38° = 142° \]

Now, we can determine the sum of the interior angles at vertices I and J. Since the sum of all interior angles in triangle HIJ is 180°, we can express it as:

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

Plugging in the angle we've calculated for H:

\[ 142° + \text{Angle at I} + \text{Angle at J} = 180° \]

Now, we can simplify this to find the sum of the angles at I and J:

\[ \text{Angle at I} + \text{Angle at J} = 180° - 142° = 38° \]

So, the sum of the measures of the interior angles at vertex I and vertex J is 38 degrees.