Congruence and Transformations
Recall that congruent means having the same size and shape. For instance:
Recall that congruent means having the same size and shape. For instance:
Two congruent segments have the same shape (segments) and the same size (length).
\(AB = CD\), so \(\overline{AB}\cong\overline{CD}\)
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Two congruent angles have the same shape (angles) and the same size (angle measure, such as degrees).
\(m\angle FEG = m\angle JHI\), so \(\angle FEG\cong\angle JHI\)
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In the Transformations Unit we learned that two figures are congruent if an isometry (congruence transformation) or composite of isometries can be used to map one figure onto the other. Which transformation maps one triangle onto the other?
For two figures to be congruent, all corresponding segments and all corresponding angles must be congruent. For this example there are many congruence statements: \(\overline{AB}\cong\overline{DF}\), \(\overline{AC}\cong\overline{DE}\), \(\overline{FE}\cong\overline{BC}\), \(\angle A\cong\angle D\), \(\angle C\cong\angle E\), \(\angle B\cong\angle F\).
When naming congruent polygons, ORDER MATTERS! Determining what type of isometry is needed to map one figure onto the other produces the correspondence and the order. In the example above \(\triangle ABC\cong\triangle DEF\) because of the translation. The congruence statement can be used to determine corresponding congruent angles or segments of two figures. Let's look at a couple of examples.
Example 1:
State all the congruent pairs of angles and sides given that \(MATHY\cong FBLUS\).
Solution:
No picture is needed if we follow the order of the vertices in the congruence statement.
\(\angle M\cong\angle F\) \(\overline{MA}\cong\overline{FB}\)
\(\angle A\cong\angle B\) \(\overline{AT}\cong\overline{BL}\)
\(\angle T\cong\angle L\) \(\overline{TH}\cong\overline{LU}\)
\(\angle H\cong\angle U\) \(\overline{HY}\cong\overline{US}\)
\(\angle Y\cong\angle S\) \(\overline{MY}\cong\overline{FS}\)
*It can be helpful to draw a picture. When drawing a picture from the name of a polygon, you must go around the polygon in order (clockwise or counterclockwise) and not skip vertices or go across the polygon. The example below is a translation. Marking the diagram with tick marks and arc marks can be helpful also.
When naming congruent polygons, ORDER MATTERS! Determining what type of isometry is needed to map one figure onto the other produces the correspondence and the order. In the example above \(\triangle ABC\cong\triangle DEF\) because of the translation. The congruence statement can be used to determine corresponding congruent angles or segments of two figures. Let's look at a couple of examples.
Example 1:
State all the congruent pairs of angles and sides given that \(MATHY\cong FBLUS\).
Solution:
No picture is needed if we follow the order of the vertices in the congruence statement.
\(\angle M\cong\angle F\) \(\overline{MA}\cong\overline{FB}\)
\(\angle A\cong\angle B\) \(\overline{AT}\cong\overline{BL}\)
\(\angle T\cong\angle L\) \(\overline{TH}\cong\overline{LU}\)
\(\angle H\cong\angle U\) \(\overline{HY}\cong\overline{US}\)
\(\angle Y\cong\angle S\) \(\overline{MY}\cong\overline{FS}\)
*It can be helpful to draw a picture. When drawing a picture from the name of a polygon, you must go around the polygon in order (clockwise or counterclockwise) and not skip vertices or go across the polygon. The example below is a translation. Marking the diagram with tick marks and arc marks can be helpful also.
Example 2:
Identify the isometry and prove the triangles are congruent.
Identify the isometry and prove the triangles are congruent.
Solution:
A \(180^{\circ}\) rotation is used with a center at \(O\) to map one triangle onto the other. From the diagram we are given:
\(\angle B\cong\angle T\) \(\overline{BO}\cong\overline{TO}\)
\(\angle I\cong\angle W\) \(\overline{IO}\cong\overline{WO}\)
\(\overline{IB}\cong\overline{WT}\)
\(\angle BOI\cong\angle TOW\) because vertical angles are congruent
Therefore \(\triangle BOI\cong\triangle TOW\) because all corresponding segments and angles are congruent.
Example 3:
Identify the isometry and prove the triangles are congruent.
A \(180^{\circ}\) rotation is used with a center at \(O\) to map one triangle onto the other. From the diagram we are given:
\(\angle B\cong\angle T\) \(\overline{BO}\cong\overline{TO}\)
\(\angle I\cong\angle W\) \(\overline{IO}\cong\overline{WO}\)
\(\overline{IB}\cong\overline{WT}\)
\(\angle BOI\cong\angle TOW\) because vertical angles are congruent
Therefore \(\triangle BOI\cong\triangle TOW\) because all corresponding segments and angles are congruent.
Example 3:
Identify the isometry and prove the triangles are congruent.
Solution:
A reflection is used with a reflection line \(\overleftrightarrow{EL}\) to map one triangle onto the other. From the diagram we are given:
\(\angle R\cong\angle F\) \(\overline{RE}\cong\overline{EF}\)
\(\overline{RL}\cong\overline{LF}\)
\(\overline{EL}\cong\overline{EL}\) by the reflexive property. \(\angle ELR\cong\angle ELF\) because right angles are congruent. \(\angle REL\cong\angle FEL\) because of the third angles theorem. Therefore \(\triangle REL\cong\triangle FEL\) because all corresponding segments and angles are congruent.
Notice that we can use theorems and properties that we have previously learned to justify congruent segments or congruent angles. Also, identifying the isometry shows the correspondence of the two figures.
A reflection is used with a reflection line \(\overleftrightarrow{EL}\) to map one triangle onto the other. From the diagram we are given:
\(\angle R\cong\angle F\) \(\overline{RE}\cong\overline{EF}\)
\(\overline{RL}\cong\overline{LF}\)
\(\overline{EL}\cong\overline{EL}\) by the reflexive property. \(\angle ELR\cong\angle ELF\) because right angles are congruent. \(\angle REL\cong\angle FEL\) because of the third angles theorem. Therefore \(\triangle REL\cong\triangle FEL\) because all corresponding segments and angles are congruent.
Notice that we can use theorems and properties that we have previously learned to justify congruent segments or congruent angles. Also, identifying the isometry shows the correspondence of the two figures.