Lesson 5: Identical Triangles

Lesson 5: Identical Triangles

When studying triangles, it is essential to be able to communicate about the parts of a triangle without any confusion. The following terms are used to identify particular angles or sides:

• between

• adjacent to

• opposite

• included (side/angle)

Between

• An angle can be between two sides

• A side can be between two angles

Multiple Choice

Use the figure △ 𝐴𝐵𝐶 to fill in the following blanks.

∠𝐴 is__________ sides 𝐴𝐵̅ and 𝐴𝐶̅̅.

• between

• adjacent to

• opposite

• included

Multiple Choice

Use the figure △ 𝐴𝐵𝐶 to fill in the following blanks.

Side 𝐴𝐵̅̅ is between ∠____ and ∠____ .

• $\angle A\ and\ \angle C$

• $\angle B\ \ and\ \angle C\$

• $\angle A\ \ and\ \ \angle B\$

Adjacent to

• The angle that is between two sides of a triangle are also adjacent to those two sides

Multiple Choice

________ is adjacent to sides AB and BC

•  $\angle A$  

•  $\angle B$  

•  $\angle C$  

Opposite to

• Each angle in a triangle has an opposite side

• If you draw a straight line through the center of an angle it will bisect the opposite side

Multiple Choice

 $\angle A\$  is opposite to 

•  $\angle B\$  

•  $\overline{BC}$  

•  $\overline{AB}$  

•  $\angle C$  

Multiple Choice

 $\overline{AB}$  is opposite 

•  $\overline{BC}$  

•  $\angle C$  

•  $\overline{AC}$  

•  $\angle B$  

Included

• An angle is included between the two connecting sides

• A side is included between two connected angles

Multiple Choice

Side_______ is the included side of ∠𝐵 and ∠𝐶.

• $\overline{BC\ }$

• $\overline{AB}$

• $\overline{BC}$

• $\angle B$

Multiple Choice

What is the included angle of sides 𝑨𝑩̅̅̅̅ and 𝑩𝑪̅̅̅̅?

•  $\angle A\$  

•  $\angle B\$  

•  $\angle C$  

•  $\overline{AB}$  

Corresponding angles and sides

Now that we know what to call the parts within a triangle, we consider how to discuss two triangles. We need to compare the parts of the triangles in a way that is easy to understand. To establish some alignment between the triangles, we pair up the vertices of the two triangles.

Corresponding Triangles

A correspondence between two triangles is a pairing of each vertex of one triangle with one (and only one) vertex of the other triangle. A correspondence provides a systematic way to compare parts of two triangles.

Multiple Select

Given the following triangle correspondences, use double arrows to show the correspondence between vertices/angles.  Triangle Correspondence $\Delta\ ABC\ \leftrightarrow\ \Delta\ STR$  (Choose all that apply)

•  $\angle A\ \leftrightarrow\ \angle S\$  

•  $\angle C\ \leftrightarrow\ \angle R\$  

•  $\angle\ A\ \leftrightarrow\ \angle\ B\$  

•  $\angle\ T\ \leftrightarrow\ \angle\ B\$  

Multiple Select

Given the following triangle correspondences, use double arrows to show the correspondence between sides.  Triangle Correspondence $\Delta\ ABC\ \leftrightarrow\ \Delta\ STR$  (Choose all that apply)

•  $\overline{AB}\ \leftrightarrow\ \overline{ST}$  

•  $\overline{ST\ }\ \leftrightarrow\ \overline{RT}$  
  

•  $\overline{BC}\ \ \leftrightarrow\ \overline{TR}$  
  

•  $\overline{AC}\ \leftrightarrow\ \overline{SR}$  
  

Angle and side congruence

In discussing identical triangles, it is useful to have a way to indicate those sides and angles that are equal. We mark sides with tick marks and angles with arcs if we want to draw attention to them. If two angles or two sides have the same number of marks, it means they are equal. 

Multiple Select

If two angles or two sides have the same number of marks, it means they are congruent.Two identical triangles are shown. Give a triangle correspondence that matches congruent sides when $\Delta ABC\ \leftrightarrow\ \Delta\ TSR\$  (choose all that apply)

 

•  $\overline{AB}\ \cong\overline{TS\ }$  

•  $\overline{BC}\ \cong\ \overline{SR\ }$  

•  $\overline{AC}\ \cong\ \overline{TR}$  

•  $\overline{AB}\ \cong\overline{TR}$  

Multiple Select

If two angles or two sides have the same number of marks, it means they are congruent.Two identical triangles are shown. Give a triangle correspondence that matches congruent angles when $\Delta ABC\ \leftrightarrow\ \Delta\ TSR\$  (choose all that apply)

 

•  $\angle A\ \cong\ \angle T\$  
  

•  $\angle A\ \cong\ \angle R\$  

•  $\angle C\ \cong\angle R\$  
  

•  $\angle B\ \cong\angle S$  
  

Lesson Summary

• Two triangles and their respective parts can be compared once a correspondence has been assigned to the two triangles. Once a correspondence is selected, corresponding sides and corresponding angles can also be determined

• Double arrows notate corresponding vertices. Triangle correspondences can also be notated with double arrows.

• Triangles are identical if there is a correspondence so that corresponding sides and angles are equal.

• An equal number of tick marks on two different sides indicates the sides are equal in measurement. An equal number of arcs on two different angles indicates the angles are equal in measurement.