Question

in the figure below, suppose ( mangle1 = 67^circ ) and ( mangle3 = 56^circ ).
complete the statements below.
the sum of the interior angle measures of a triangle must be ( square^circ ).
so, ( mangle1 + mangle3 + mangle4 = square^circ ).
we are given that ( mangle1 = 67^circ ).
so, ( mangle3 + mangle4 = square^circ ).
from the figure, we can see that ( mangle1 + mangle2 = square^circ ).
since ( mangle1 = 67^circ ), it must be that ( mangle2 = square^circ ).
therefore, ( mangle2 ) select ( mangle3 + mangle4 ).
this result is an example of the exterior angle property of triangles.
for any triangle, the measure of an exterior angle
select

Explanation:

Step 1: Sum of interior angles of a triangle

The sum of the interior angle measures of a triangle is a well - known geometric fact. By the triangle angle - sum theorem, the sum of the interior angles of a triangle is \(180^{\circ}\). So, for \(\triangle\) with angles \(\angle1\), \(\angle3\) and \(\angle4\), we have \(m\angle1 + m\angle3+m\angle4 = 180^{\circ}\).

Step 2: Find \(m\angle3 + m\angle4\)

We know that \(m\angle1=67^{\circ}\) and \(m\angle1 + m\angle3 + m\angle4=180^{\circ}\). Substitute \(m\angle1 = 67^{\circ}\) into the equation \(m\angle1 + m\angle3 + m\angle4=180^{\circ}\), we get \(67^{\circ}+m\angle3 + m\angle4 = 180^{\circ}\). Then, by subtracting \(67^{\circ}\) from both sides of the equation, \(m\angle3 + m\angle4=180^{\circ}- 67^{\circ}=113^{\circ}\). We are also given that \(m\angle3 = 56^{\circ}\), but we can also calculate \(m\angle4\) from \(m\angle3 + m\angle4 = 113^{\circ}\) and \(m\angle3 = 56^{\circ}\), so \(m\angle4=113^{\circ}-56^{\circ} = 57^{\circ}\) (though this is not directly needed for the first few blanks).

Step 3: Find \(m\angle1 + m\angle2\)

From the figure, \(\angle1\) and \(\angle2\) form a linear pair. A linear pair of angles is supplementary, which means their sum is \(180^{\circ}\). So, \(m\angle1 + m\angle2 = 180^{\circ}\).

Step 4: Find \(m\angle2\)

Since \(m\angle1 = 67^{\circ}\) and \(m\angle1 + m\angle2=180^{\circ}\), we can find \(m\angle2\) by subtracting \(m\angle1\) from \(180^{\circ}\). So, \(m\angle2=180^{\circ}-67^{\circ}=113^{\circ}\).

Step 5: Compare \(m\angle2\) and \(m\angle3 + m\angle4\)

We found that \(m\angle2 = 113^{\circ}\) and \(m\angle3 + m\angle4=113^{\circ}\), so \(m\angle2=m\angle3 + m\angle4\).

Step 6: Exterior Angle Property

The exterior angle property of a triangle states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In this case, \(\angle2\) is an exterior angle of the triangle, and the two non - adjacent interior angles are \(\angle3\) and \(\angle4\), so \(m\angle2=m\angle3 + m\angle4\), which is in line with the exterior angle property.

Filling in the blanks:

• The sum of the interior angle measures of a triangle must be \(\boldsymbol{180}\)°. So, \(m\angle1 + m\angle3 + m\angle4=\boldsymbol{180}\)°.
• We are given that \(m\angle1 = 67^{\circ}\). So, \(m\angle3 + m\angle4=\boldsymbol{113}\)°.
• From the figure, we can see that \(m\angle1 + m\angle2=\boldsymbol{180}\)°. Since \(m\angle1 = 67^{\circ}\), it must be that \(m\angle2=\boldsymbol{113}\)°.
• Therefore, \(m\angle2\) \(\boldsymbol{=}\) \(m\angle3 + m\angle4\).
• This result is an example of the Exterior Angle Property of Triangles. For any triangle, the measure of an exterior angle is equal to the sum of the measures of the two non - adjacent interior angles.

Answer:

Step 1: Sum of interior angles of a triangle

The sum of the interior angle measures of a triangle is a well - known geometric fact. By the triangle angle - sum theorem, the sum of the interior angles of a triangle is \(180^{\circ}\). So, for \(\triangle\) with angles \(\angle1\), \(\angle3\) and \(\angle4\), we have \(m\angle1 + m\angle3+m\angle4 = 180^{\circ}\).

Step 2: Find \(m\angle3 + m\angle4\)

We know that \(m\angle1=67^{\circ}\) and \(m\angle1 + m\angle3 + m\angle4=180^{\circ}\). Substitute \(m\angle1 = 67^{\circ}\) into the equation \(m\angle1 + m\angle3 + m\angle4=180^{\circ}\), we get \(67^{\circ}+m\angle3 + m\angle4 = 180^{\circ}\). Then, by subtracting \(67^{\circ}\) from both sides of the equation, \(m\angle3 + m\angle4=180^{\circ}- 67^{\circ}=113^{\circ}\). We are also given that \(m\angle3 = 56^{\circ}\), but we can also calculate \(m\angle4\) from \(m\angle3 + m\angle4 = 113^{\circ}\) and \(m\angle3 = 56^{\circ}\), so \(m\angle4=113^{\circ}-56^{\circ} = 57^{\circ}\) (though this is not directly needed for the first few blanks).

Step 3: Find \(m\angle1 + m\angle2\)

From the figure, \(\angle1\) and \(\angle2\) form a linear pair. A linear pair of angles is supplementary, which means their sum is \(180^{\circ}\). So, \(m\angle1 + m\angle2 = 180^{\circ}\).

Step 4: Find \(m\angle2\)

Since \(m\angle1 = 67^{\circ}\) and \(m\angle1 + m\angle2=180^{\circ}\), we can find \(m\angle2\) by subtracting \(m\angle1\) from \(180^{\circ}\). So, \(m\angle2=180^{\circ}-67^{\circ}=113^{\circ}\).

Step 5: Compare \(m\angle2\) and \(m\angle3 + m\angle4\)

We found that \(m\angle2 = 113^{\circ}\) and \(m\angle3 + m\angle4=113^{\circ}\), so \(m\angle2=m\angle3 + m\angle4\).

Step 6: Exterior Angle Property

The exterior angle property of a triangle states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In this case, \(\angle2\) is an exterior angle of the triangle, and the two non - adjacent interior angles are \(\angle3\) and \(\angle4\), so \(m\angle2=m\angle3 + m\angle4\), which is in line with the exterior angle property.

Filling in the blanks:

• The sum of the interior angle measures of a triangle must be \(\boldsymbol{180}\)°. So, \(m\angle1 + m\angle3 + m\angle4=\boldsymbol{180}\)°.
• We are given that \(m\angle1 = 67^{\circ}\). So, \(m\angle3 + m\angle4=\boldsymbol{113}\)°.
• From the figure, we can see that \(m\angle1 + m\angle2=\boldsymbol{180}\)°. Since \(m\angle1 = 67^{\circ}\), it must be that \(m\angle2=\boldsymbol{113}\)°.
• Therefore, \(m\angle2\) \(\boldsymbol{=}\) \(m\angle3 + m\angle4\).
• This result is an example of the Exterior Angle Property of Triangles. For any triangle, the measure of an exterior angle is equal to the sum of the measures of the two non - adjacent interior angles.