Question

• angle — is included between sides ab and ac.
• angle — is included between sides ab and bc.
• angle — is included between sides ac and bc.

1. practice: using visual cues and summarizing

list all the possible ways to name the triangle at right.

1. practice: activating prior knowledge and organizing information

write the correct number in each blank to complete the review list.
review: triangle facts

• they are closed — dimensional figures.
• they are formed by — segments that connect — noncollinear points.
• they have exactly — sides, — angles, and — vertices.
• each side is included between — angles and is opposite — angle.
• each angle is included between — sides and is opposite — side.

1. practice: using visual cues

give one reason why each figure below is not a triangle.

Explanation:

Step1: Recall Triangle Definition

A triangle is a closed two - dimensional figure. So the first blank (for "They are closed ___ - dimensional figures") is filled with 2.

Step2: Segments and Points in Triangle

A triangle is formed by 3 segments that connect 3 non - collinear points. So the blanks "They are formed by _ segments that connect _ noncollinear points" are filled with 3 and 3 respectively.

Step3: Sides, Angles, Vertices of Triangle

A triangle has exactly 3 sides, 3 angles, and 3 vertices. So the blanks "They have exactly _ sides, _ angles, and ___ vertices" are filled with 3, 3, and 3 respectively.

Step4: Sides, Angles Relationship (Side - Angle)

Each side is included between 2 angles and is opposite 1 angle. So the blanks "Each side is included between _ angles and is opposite _ angle" are filled with 2 and 1 respectively.

Step5: Angles, Sides Relationship (Angle - Side)

Each angle is included between 2 sides and is opposite 1 side. So the blanks "Each angle is included between _ sides and is opposite _ side" are filled with 2 and 1 respectively.

Step6: Naming the Triangle (Question 5)

The triangle has vertices E, F, D. So the possible names are $\triangle EFD$, $\triangle EDF$, $\triangle FED$, $\triangle FDE$, $\triangle DEF$, $\triangle DFE$.

Step7: Why Figures are not Triangles (Question 7)

A triangle is a closed figure with 3 line segments connecting 3 non - collinear points. For a figure not to be a triangle, possible reasons could be: it is not closed, it has more or less than 3 sides, the sides are not line segments (e.g., curved), or the points are collinear. (Since the figures are not shown, we state the general reasons. If we had the figures, we could be more specific. For example, if a figure has 4 sides, it's a quadrilateral, not a triangle; if a figure has a curved side, it's not a polygon, hence not a triangle; if the three points are collinear, we can't form a triangle as the three segments would be colinear and form a line segment instead of a triangle.)

Answer:

For Question 5 (Naming the Triangle):

Possible names: $\triangle EFD$, $\triangle EDF$, $\triangle FED$, $\triangle FDE$, $\triangle DEF$, $\triangle DFE$

For Question 6 (Triangle Facts):

• They are closed $\boldsymbol{2}$ - dimensional figures.
• They are formed by $\boldsymbol{3}$ segments that connect $\boldsymbol{3}$ noncollinear points.
• They have exactly $\boldsymbol{3}$ sides, $\boldsymbol{3}$ angles, and $\boldsymbol{3}$ vertices.
• Each side is included between $\boldsymbol{2}$ angles and is opposite $\boldsymbol{1}$ angle.
• Each angle is included between $\boldsymbol{2}$ sides and is opposite $\boldsymbol{1}$ side.

For Question 7 (Why not a Triangle - General Reasons):

A figure is not a triangle if: it is not closed, has more/less than 3 sides, has non - linear (curved) sides, or its three points are collinear. (Specific reasons depend on the actual figures.)

For the Angle Inclusion Questions (First three bullet points):

• Angle $\angle BAC$ (or $\angle A$) is included between sides $AB$ and $AC$.
• Angle $\angle ABC$ (or $\angle B$) is included between sides $AB$ and $BC$.
• Angle $\angle ACB$ (or $\angle C$) is included between sides $AC$ and $BC$. (Assuming the triangle is $\triangle ABC$ as per the side labels $AB$, $BC$, $AC$)