Lesson 2

Congruent Parts, Part 2

  • Let’s name figures in ways that help us see the corresponding parts.

2.1: Math Talk: Which Are Congruent?

Each pair of figures is congruent. Decide whether each congruence statement is true or false. 

\(\triangle ABC \cong \triangle FED\)

Triangles A B C and D E F.

\(PZJM \cong LYXB\)

Concave quadrilaterals Z P M J and Y B L X.

Triangle \(ABC\) is congruent to triangle \(FED\).

Quadrilateral \(PZJM\) is congruent to quadrilateral \(LYXB\).

\(\triangle JKL \cong \triangle QRS\)

Triangles J K L and Q R S. Angles J L K and Q R S appear to be right angles.

\(ABCDE \cong PQRST\)

Pentagons A B C D E and P Q R S T.

Triangle \(JKL\) is congruent to triangle \(QRS\).

Pentagon \(ABCDE\) is congruent to pentagon \(PQRST\).

 

2.2: Which Triangles Are Congruent?

Here are three triangles.

 
  1. Triangle \(PQR\) is congruent to which triangle? Explain your reasoning.
  2. Show a sequence of rigid transformations that takes \(PQR\) to that triangle. Draw each step of the transformation.
  3. Explain why there can’t be a rigid transformation to the other triangle.

2.3: Are These Parts Congruent?

Triangles A B D and C D B. Point E located on side D B.
Two concave polygons, G F O N M and H I J K L. G, H, M, and L are collinear.

  1. Triangle \(ABD\) is a rotation of triangle \(CDB\) around point \(E\) by \(180^{\circ}\). Is angle \(ADB\) congruent to angle \(CDB\)? If so, explain your reasoning. If not, which angle is \(ADB\) congruent to?

  2. Polygon \(HIJKL\) is a reflection and translation of polygon \(GFONM\). Is segment \(KJ\) congruent to segment \(NM\)? If so, explain your reasoning. If not, which segment is \(NM\) congruent to?

  3. Quadrilateral \(PQRS\) is a rotation of polygon \(VZYW\). Is angle \(QRS\) congruent to angle \(ZYW\)? If so, explain your reasoning. If not, which angle is \(QRS\) congruent to?



Suppose quadrilateral \(PQRS\) was both a rotation of quadrilateral \(VZYW\) and also a reflection of quadrilateral \(YZVW\). What can we conclude about the shape of our quadrilaterals? Explain why.

Summary

Naming congruent figures so it’s clear from the name which parts correspond makes it easier to check whether 2 figures are congruent and to use corresponding parts. In this image, segment \(AB\) appears to be congruent to segment \(DE\). Also, segment \(EF\) appears to be congruent to segment \(BC\). So, it makes more sense to conjecture that triangle \(ABC\) is congruent to triangle \(DEF\) than to conjecture triangle \(ABC\) is congruent to triangle \(FDE\).

Triangles E F D and A B C.

If we are told quadrilateral \(MATH\) is congruent to quadrilateral \(LOVE\), without even looking at the figures we know:

  • Angle \(M\) is congruent to angle \(L\).
  • Angle \(A\) is congruent to angle \(O\).
  • Angle \(T\) is congruent to angle \(V\).
  • Angle \(H\) is congruent to angle \(E\).
  • Segments \(MA\) and \(LO\) are congruent.
  • Segments \(AT\) and \(OV\) are congruent.
  • Segments \(TH\) and \(VE\) are congruent.
  • Segments \(HM\) and \(EL\) are congruent.

Quadrilaterals \(MATH\) and \(LOVE\) can be named in many different ways so that they still correspond—such as \(ATHM \) is congruent to \(OVEL\) or \(THMA\) is congruent to \(VELO\). But \(ATMH\) is congruent to \(LOVE\) means there are different corresponding parts. Note that quadrilateral \(MATH\) refers to a different way of connecting the points than quadrilateral \(ATMH\).

3 quadrilaterals: M A T H, L O V E, and A T M H. In quadrilateral A T M H, sides T M and H A cross over each other.

Glossary Entries

  • corresponding

    For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.

    In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).