Question

6-1 reteach to build understanding
the polygon angle-sum theorems

1. complete the following.

the diagram shows the _____ angles of a convex polygon, one at each vertex. the measures of those angles always add up to the same number, ____.
m∠1 + m∠2 + m∠3 + m∠4 = ______

the sum of the measures of the ____ angles of a convex polygon depends on the number of sides. the angles add up to 180°·(n − ____), where n is the number of sides of the polygon.
180°·(5 − 2) = ______

Explanation:

Left Part (Quadrilateral Exterior Angles):

Step1: Identify the angle type

The diagram shows the exterior angles of a convex polygon (quadrilateral here), one at each vertex.

Step2: Recall exterior angle sum theorem

For any convex polygon, the sum of the exterior angles is always \( 360^\circ \). So for a quadrilateral, \( m\angle1 + m\angle2 + m\angle3 + m\angle4 = 360^\circ \).

Right Part (Pentagon Interior Angles):

Step1: Identify the angle type

The sum of the measures of the interior angles of a convex polygon depends on the number of sides.

Step2: Recall interior angle sum formula

The formula for the sum of interior angles of a polygon is \( 180^\circ \cdot (n - 2) \), where \( n \) is the number of sides. For a pentagon, \( n = 5 \), so we use \( 180^\circ \cdot (5 - 2) \).

Step3: Calculate the sum

\( 180^\circ \cdot (5 - 2)=180^\circ\times3 = 540^\circ \).

Filling the Blanks:

• Left: The diagram shows the \(\boldsymbol{\text{exterior}}\) angles... The measures... add up to the same number, \(\boldsymbol{360^\circ}\). \( m\angle1 + m\angle2 + m\angle3 + m\angle4 = \boldsymbol{360^\circ} \)
• Right: The sum of the measures of the \(\boldsymbol{\text{interior}}\) angles... \( 180^\circ \cdot (n - \boldsymbol{2}) \)... \( 180^\circ \cdot (5 - 2) = \boldsymbol{540^\circ} \)

Answer:

s (Filled Blanks):

• Left: exterior, \( 360^\circ \), \( 360^\circ \)
• Right: interior, \( 2 \), \( 540^\circ \)